Mathematics Colloquium
Vector majorization is a preorder relation on the space of
n-dimensional real vectors. It has been generalized to matrices in a
number of different ways. The origins of Majorization Theory can be
traced back to Muirhead's results on symmetric polynomials, Schur's
work on Hadamard's inequality, and the notion of Lorenz curve.
The wide applicability of majorization to quantum mechanics arises as
a result of two fundamental theorems connecting majorization to
unitary matrices: Horn’s lemma and Uhlmann’s theorem, which
characterize vector and matrix majorization in terms of unitary
transformations. Notably, as means for measuring disorder,
majorization is more powerful than the classical notions of entropy.
We will discuss linear operators preserving majorization for
stochastic matrices and applications of this and similar problems to
quantum information theory.
Given a Banach function space X and an operator T on X, the phase retrieval problem concerns the question of whether every f in X is determined (up to a multiplicative factor) by |Tf|, i.e., the modulus of Tf. Problems of this nature were first studied by Pauli, who asked whether a square-integrable function is determined by its modulus and the modulus of its Fourier transform. Since then, the problem has been investigated from multiple directions. I will give an overview of important results in this research field and outline recent developments. The focus of the talk will be on approaches to the phase retrieval problem that draw from techniques in Complex, Harmonic, and Functional Analysis.
The talk is based on the joint works with Alexander Alexandrov, Petr Dunin-Barkowski, Maxim Kazarian and Sergey Shadrin.
The study of periodic Jacobi matrices on the line is a classical subject in spectral theory, motivated by condensed matter physics and having connections to orthogonal polynomial theory, potential theory and other areas of mathematics. In this talk we will discuss ongoing work that attempts to generalize the theory to more general trees, with an emphasis on Floquet theory which describes the structure of generalized eigenfunctions. We will describe some results obtained in joint works with Jess Banks, Jorge Garza Vargas, Eyal Seelig and Barry Simon.
Suppose u_j is an orthonormal basis of eigenfunctions of the Laplacian on a compact Riemannian manifold, and consider the probability measures |u_j|^2 dvol. The quantum ergodicity theorem of Shnirelman, Zelditch and Colin de Verdiere, states that if the geodesic flow on the manifold is ergodic then there is a density one subsequence of these measures that equidistributes. What about the remaining eigenfunctions? The quantum unique ergodicity conjecture of Rudnick and Sarnak states that for hyperbolic manifolds (or more generally for negatively curved manifolds), any sequence of measures as above equidistributes. In 2006 Lindenstrauss proved that this is true for Hecke-Maass forms on congruence surfaces. These are eigenfunctions of both the Laplacian and the Hecke operators, which are discrete averaging operators coming from the arithmetic structure of the manifold.
The Poisson summation formula states that the Fourier transform of a lattice’s counting measure is a discrete measure supported on the dual lattice. This fact characterizes periodicity via the Fourier transform and lies at the heart of number theory, harmonic analysis, and mathematical physics. Extending this idea, Fourier Quasicrystals (FQs) are almost periodic sets whose Fourier transform is supported on a discrete set, subject to growth conditions. For two decades, FQs were thought to be periodic or finite unions of periodic sets, a belief supported by Lagarias’s conjecture, proved by Lev and Olevskii in 2016. This changed with Kurasov and Sarnak (2020), who constructed the first genuinely non-periodic FQ. They provided a general construction of one-dimensional FQs by intersecting irrational lines in the torus with zero sets of Lee-Yang polynomials.
In this talk, I will show that all one-dimensional FQs can be constructed via the Kurasov-Sarnak method and present a new framework to generalize it to higher dimensions using a novel class of high-codimension algebraic varieties, termed Lee-Yang varieties.
Based on joint work with Alex Cohen, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov, this presentation explores the interplay between Fourier analysis, algebraic geometry, and is designed for a broad audience without requiring prior expertise.
zoom link: https://biu-ac-il.zoom.us/j/83293533633?pwd=QuYptGtg3dhheD4DWERu8EaOjzS…
The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.
In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.
This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.
No special background in probability theory or complexity theory will be assumed.
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Since the very beginning, the role of bases in simple modules over finite and Lie groups was critical: at the very least they were indispensable in computing dimensions and characters. This role was played well by Young tableaux bases in Specht modules for symmetric groups S_n and Gelfand-Tsetlin bases for simple GL_n(C)-modules, to name a few.
After the groundbreaking Kazhdan-Lusztig paper of 1979, it became clear that in order to obtain an interesting basis in each simple module of a group algebra of a finite group, one has to first deform it and then construct a (canonical) basis in the so deformed algebra. This canonical basis projects to each simple module and becomes a basis there.
A similar idea on the Lie-theoretic side was coined by Gelfand and Zelevinsky in 1983: to find bases in all simple modules V_lambda over GL_n(C) one can naturally realize V_lambda as a subspace of the coordinate algebra C[N] of the group N of unipotent nxn matrices and find a good basis there which will naturally descend to each V_lambda under the embedding into C[N], and will eventually solve the tensor product multiplicity problem (for reductive or semisimple Lie groups G with the maximal unipotent subgroup N they expected same outcome).
With the discovery of quantum groups in 1986, this idea was implemented in a very surprising (and in a sense dual way) way by George Lusztig in 1989 who noticed that the quantized enveloping algebra U_q(nn), where nn is the Lie algebra of N, admits a canonical basis B which, on the one hand, was constructed in Kazhdan-Lusztig fashion (by means of the celebrated "Lusztig's Lemma"), and on the other hand, descends to each simple module V_lambda over the full quantum group U_q(gg), where gg is the Lie algebra of G (shortly after Lusztig's amazing discovery, Masaki Kashiwara in 1990 recovered Lusztig's constructions via the theory of crystal bases which emerge when q=0).
In our work with Andrei Zelevinsky in 1993 we adapted a dual approach by taking advantage of the fact that U_q(nn) is not only a q-deformation of the universal enveloping algebra U(nn), but also is a q-deformation of its Hopf dual C[N], thus claiming that the dual basis B^dual of B fits Gelfand-Zelevinsky framework (our approach had an unexpected byproduct: the modern theory of cluster algebras emerged from the study of B^dual for various semisimple or even even Kac-Moody Lie groups).
In the second part of the talk, I will reveal that the aforementioned dual canonical bases of U_q(nn) and of its remarkable subalgeras U_q(w), known as quantum Schubert cells, are also obtained by the application of the (very innocently looking!) Lusztig Lemma and if time permit, I will demonstrate how to obtain a canonical basis in quantum Heisenberg algebra H_q(gg) and, ultimately, in U_q(gg) along these lines. This is joint work with Jacob Greenstein, 2014-2017.
Central simple algebras are the building blocks of the associative structure theory. One of their key properties is that by extending the scalars, they become matrix algebras. Although nothing in algebra remains the same if we drop the associativity assumption, there is strong motivation (which I will discuss briefly) to try to recover this theory in the nonassociative realm. For this idea to have some hope, we must assume that the algebra contains a maximal subfield in the nucleus (the part which does associate with everything). Indeed, by extending the scalars, these algebras, termed "semiassociative'', become "skew matrix algebras''. I will show how this new structure leads to a nonassociative generalization of the Brauer group, in a way that should be understandable even if you momentarily forgot what the classical Brauer group is.
Based on joint work with Eliyahu Matzri, our students Guy Blachar and Edan Rein, and Darrell Haile.
The Sierpinski carpet and Menger sponge are well-studied connected generalizations of the Cantor set. They are also members of a two-parameter family of connected higher-dimensional fractals that can be constructed from the $n$-cube as iterated function systems.
In this talk we focus on taxicab paths—piece-wise linear paths that always travel parallel to a coordinate axis with possible limiting behavior at the endpoints—between any two points $x$ and $y$ in members of this fractal family. In particular, given points $x$ and $y$ in such a fractal, we show how to explicitly construct taxicab geodesics between them. As an application, we compare the taxicab metric to the standard Euclidean metric in the fractals we consider, providing a sharp bound on their ratio. We will also briefly address non-taxicab geodesics.
This is joint work with Elene Karangozishvili and Derek Smith.
Given an algebraic variety X, and two subvarieties Y, Y', can Y be deformed to Y' within X?
More generally, we would like to understand the different "deformation classes" of subvarieties of X of a given dimension. For topological deformations, the answer is given by singular cohomology and is rather well understood. For algebraic deformations, the answer is quite mysterious and depends heavily on the ground field. In codimension 1, this amounts to studying rational points on abelian varieties (such as elliptic curves). In higher codimension, much less is understood.
I'll survey this topic and conclude by giving some new results in codimension two, concerning the so-called Ceresa cycles.
In the search for mathematical explanations for physical phenomena, algebraic geometry and combinatorics have made surprising appearances.
In this talk, I will overview some exciting results connecting combinatorics and algebraic geometry to water waves (coming from the Kadomtsev-Petviashvili equation) and to particle physics (coming from scattering amplitudes).
Extremal combinatorics poses a fundamental question: How large can a system be while avoiding certain configurations? A classic instance of this inquiry arises in extremal graph theory: Given a fixed graph H, what is the maximum number ex(n, H) of edges a graph G on n vertices can have if it excludes H as a subgraph? This problem remains widely open for H being a complete bipartite graph and is known as Zarankiewicz’s problem.
Even when considering algebraic constraints on the hosting graph G, such as being the incidence graph of points and bi-variate polynomials of fixed degree, Zarankiewicz’s problem remains notoriously challenging. This geometric interpretation of Zarankiewicz’s problem has led to the emergence of Incidence Geometry.
In this talk, I will provide an overview of notable results in this domain and will introduce a novel approach to Zarankiewicz’s problem.
Based on joint work with Chaya Keller.
In the talk, I plan to discuss the notion of measurable entire functions, introduced by Benjy Weiss around 25 years ago, as well as several related results and questions.
Statistical physics models for disordered materials provide precise predictions about the typical complexity of certain combinatorial optimization problems. The underlying common structure is that of many discrete variables, whose interaction is represented by a random sparse graph.
I will discuss recent progress in proving some of these predictions. In particular, on the emerging theory of nonlinear large deviations, yielding mean field approximation for certain Gibbs measures and representing such measures as mixtures of not too many product measures.
The notion of matrix integration was introduced in 2007 by Bhat and Mukherjee [1], as a natural counterpart to the classical notion of matrix differentiation [2]. It has several motivations in various areas, including the geometry of polynomials -- an extensive topic with many beautiful open problems, for example, Sendov's conjecture.
Not every matrix has an integral, and the problem of existence of integrable and non-integrable matrices with a given Jordan structure has remained open since [1]. We present a complete solution to this problem. Moreover, we have established an easy-to-check criterion for matrix integrability. Our solution is based on special types of polynomials, namely Shabat polynomials and conservative polynomials. Shabat polynomials are complex polynomials with at most 2 finite critical values, and conservative polynomials are polynomials leaving their critical points fixed [3]. To construct the necessary polynomials of both types we rely on their relation with bicolored plane trees, see [4, 5].
Additionally, we explore the properties of matrix integrals and their applications in the geometry of polynomials.
The talk is based on several joint works with Alexander Guterman, Elena Kreines, Patrick Ng, and Fedor Pakovich.
References:
[1] B. V. R. Bhat and M. Mukherjee, Integrators of matrices, Linear Algebra Appl. 426 (2007), 71--82.
[2] C. Davis, Eigenvalues of compressions, Bull. Math. Soc. Sci. Math. Phys. RPR 51 (1959), 3--5.
[3] A. Kostrikin, Conservative polynomials, in ``Stud. Algebra Tbilisi'' (1984). 115--129.
[4] F. Pakovich, Conservative polynomials and yet another action of Gal(Q/Q) on plane trees, J. Théor. Nr. Bordx. 20 (2008), 205--218.
[5] G. B. Shabat and V. A. Voevodsky, Drawing curves over number fields, The Grothendieck Festschrift 3 (1990), 199--227.
The origins of majorization theory can be found in a variety of different contexts. The most notable examples are Muirhead's work on symmetric polynomials, Schur's work on Hadamard's inequality, and works by Lorenz and Dalton on income distributions. Hardy, Littlewood, and Polya laid the foundations of majorization as an independent theory.
Vector majorization is a preorder notion on the real vector space. A vector x is majorized by a vector y if, for any k, the sum of the k largest coordinates of x is not less than the sum for y, and the total sums coincide. Later this notion was generalized to matrices and other structures. Matrix and vector majorizations have numerous applications in many areas of mathematics.
We will discuss the notion of majorization for matrix classes. Using a geometric approach, we solve the problem of finding minimal covering classes. We characterize majorizations for (0, 1) and (0, \pm 1)-matrices and discuss connections between these results and linear programming and graph theory. Finally, we will consider linear preserver problems connected to majorization. With the help of combinatorial matrix theory, we are able to prove the classical results of Ando and Li, Poon under weaker assumptions. In particular, we characterize linear operators preserving strong majorization for (0, 1)-matrices.
The talk is based on results of joint work with Geir Dahl and Alexander Guterman.
Survival analysis (SA) prediction involves the prediction of the time until an event of interest occurs (TTE), based on input attributes. The main challenge of SA is instances where the event is not observed (censored), typically through an alternative (censoring) event. Most SA prediction methods suffer from drawbacks limiting the usage of advanced machine learning methods: Ignoring the input of the censored samples, no separation between model and loss, typical small datasets and high input dimensions.
We show that current approaches misinterpret the event likelihood, and propose a loss function, denoted suRvival Analysis lefT barrIer lOss (RATIO), that explicitly incorporates the censored samples input in the prediction. RATIO accounts for the difference between censored and uncensored samples, by only considering censoring events occurring after the predicted, and through a linear term on the uncensored data event time. RATIO can be used with any prediction model. We further propose FIESTA, a data augmentation method, combining the TTE of uncensored samples with the input of censored samples.
We show that RATIO drastically improves the precision and reduces the bias of SA prediction in both models and real-life SA problems, and FIESTA allows for the inclusion of high-dimension data in SA methods even with a small number of uncensored samples.
Based on joint work with Oshrit Shtossel and Omry Koren.
Valuations are finitely additive measures on convex sets. The space of translation-invariant valuations carries a number of structures ( i.e., product and convolution), satisfying non-trivial properties. One such recently discovered property is an analogue of the classical Hodge-Riemann relations from algebraic geometry. In special cases, they lead to new inequalities on convex bodies, to be discussed in the talk.
No preliminary knowledge of valuation theory or algebraic geometry is necessary to understand the talk.
Given a valued field (K,v) with valuation ring O and an algebraic group G over K, a model of G is a group scheme \mathcal{G} over O for which \mathcal{G}\times_O K=G. We exhibit the existence of such models for G an elliptic curve and K an algebraically closed field and on the way we classify the different group schemes one can get. The proof uses a combination of tools from model theory and algebraic geometry.
The aim of this talk is to state the theorem and to introduce some of the model theoretic notions required for the proof.
No prior knowledge in model theory is required, though a basic course in logic will help.
In many applications that involve large volumes of data (whether low- or high-dimensional), identifying and exploiting the underlying geometry is an essential ingredient; this is also the case in many of the research projects on which I have worked. In my talk, I will concentrate mainly on two projects.
In the first project, optimization techniques are used to determine an optimal low-dimensional smooth manifold (without imposing its dimension a priori) approximating a large family of noisy (possibly high-dimensional) data points. Theoretical analysis of the resulting algorithm shows that the proposed optimization solution converges to a quasi-uniform reconstruction of the manifold within a bounded time. This nonparametric approach can then be extended to address various approximation tasks in high dimensions, such as function approximation or recovering missing information in the data. We illustrate the results with toy data and real applications.
After years of extensive research, there were classes of functions for which a parametric approximation was unknown. In the second project, I will introduce a new reflecto-multiscale function, that is a generalization of the well-known refinable functions. I will rigorously show that the limit function of the reflecto-multiscale refinement process is Holder continuous and has Holder continuity of the highest-order well-defined derivatives.
Stochastic gradient descent (SGD) stands as a cornerstone of optimization and modern machine learning. However, a complete understanding of why SGD performs so well remains a major challenge.
In this talk, I will present a mathematical theory for SGD in high dimensions when both the number of samples and problem dimensions are large. We show that the dynamics of SGD, applied to generalized linear models and multi-index problems, with data possessing a general covariance matrix, become deterministic in the large sample and dimensional limit. In particular, the limiting dynamics are governed by a set of low-dimensional ordinary differential equations (ODEs).
Our setup encompasses a wide spectrum of optimization problems, including linear regression, logistic regression, and two-layer neural networks. In addition, it unveils the implicit bias inherent in SGD. For each of these problems, the deterministic equivalent of SGD enables us to derive a close approximation of the statistical risk (with explicit and vanishing error bounds). Furthermore, we leverage our theorem to establish explicit conditions on the step size, ensuring the convergence and stability of SGD within high-dimensional settings.
This is a joint work with E. Collins-Woodfin, C. Paquette, and E. Paquette, for more details see https://arxiv.org/abs/2308.08977.
The Gowers uniformity k-norm on a finite abelian group measures the averages of complex functions on such groups over k-dimensional arithmetic cubes. The inverse question about these norms asks if a large norm implies correlation with a function of an algebraic origin.
The analogue of the Gowers uniformity norms for measure-preserving abelian actions are the Host-Kra-Gowers seminorms, which are intimately connected to the Host-Kra-Ziegler factors of such systems. The corresponding inverse question, in the dynamical setting, asks for a description of such factors in terms of systems of an algebraic origin.
In this talk, we survey recent results about the inverse question in the dynamical and combinatorial settings, and in particular how an answer in the former setting can imply one in the latter.
This talk is based on joint works with Asgar Jamneshan and Terence Tao.
Following Kneser, we survey the notion of neighboring lattices and how the study of the neighboring relation on lattices gives rise to modular forms, with a focus on applications to the Arthur-Langlands conjectures, concerning automorphic forms for orthogonal and symplectic groups.
The rational Cherednik algebra H_c is an algebra of interest in modern representation theory, which is a degeneration of the double affine Hecke algebra, introduced by Cherednik to prove Macdonald's conjectures about properties of Macdonald polynomials. For values of c lying in chambers separated by walls, representations of H_c are labeled by partitions. Combinatorial wall-crossing is a bijection from the set of irreducible representations of H_c to the set of irreducible representations of H_c', where c and c' lie in adjacent chambers separated by a wall. Combinatorial wall-crossing across one wall was proven by Losev to be equal in large positive characteristic to an extension of the Mullineux involution from modular representation theory of the symmetric group.
We will exhibit interesting patterns in combinatorial wall-crossing, both proven and observed in computer experiments, and use them to prove and refine parts of a conjecture of Bezrukavnikov.
It is possible to measure oscillations of a function by means of the theory of persistence modules and barcodes. I will explain how Sobolev norms can control such measurements. Applications include generalizations of Courant's nodal domain theorem and Bezout's theorem. The talk is based on a joint work with Jordan Payette, Iosif Polterovich, Leonid Polterovich, Egor Shelukhin, and Vukašin Stojisavljević. No prior knowledge of spectral geometry and topological persistence will be assumed.
In this talk, we will introduce infinite-dimensional Lie algebras which are direct limits of finite-dimensional Lie algebras, our main example being sl(infty). As time permits, we will discuss some recent motivation in modern mathematics for studying these Lie algebras.
The classical isoperimetric inequality in Euclidean space R^n states that among all sets (“bubbles”) of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the n-sphere S^n and on n-dimensional Gaussian space G^n (i.e. R^n endowed with the standard Gaussian measure). Furthermore, one may consider the “multi-bubble” isoperimetric problem, in which one prescribes the volume of p \geq 2 bubbles (possibly disconnected) and minimizes their total surface area – as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to p=1; the case p=2 is called the double-bubble problem, and so on.
In 2000, Hutchings, Morgan, Ritoré and Ros resolved the double-bubble conjecture in Euclidean space R^3 (and this was subsequently resolved in R^n as well) – the boundary of a minimizing double-bubble is given by three spherical caps meeting at 120°-degree angles. A more general conjecture of J. Sullivan from the 1990's asserts that when p \leq n+1, the optimal multi-bubble in R^n (as well as in S^n) is obtained by taking the Voronoi cells of p+1 equidistant points in S^n and applying appropriate stereographic projections to R^n (and backwards).
In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for p \leq n bubbles in Gaussian space G^n– the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) p+1 equidistant points. In the talk, we describe our approach in that work, as well as recent progress on the multi-bubble problem on R^n and S^n. In particular, we show that minimizing bubbles in R^n and S^n are always spherical when p \leq n, and we resolve the latter conjectures when in addition p \leq 5 (e.g. the triple-bubble conjectures when n \geq 3 and the quadruple-bubble conjectures when n \geq 4).
Periods are complex numbers given as values of integrals of algebraic functions defined over domains, bounded by algebraic equations and inequalities with coefficients in Q. In this talk, we will deal with a class of periods, Frobenius constants, arising as matrix entries of the monodromy representations of certain geometric differential operators. More precisely, we will consider seven special Picard - Fuchs type second order linear differential operators corresponding to families of elliptic curves. Using periods of modular forms, we will witness some of these Frobenius constants in terms of zeta values. This is a joint work with Masha Vlasenko.
I will talk about elementary equivalence, elementary definability, regular bi-interpretability and universal equivalence of general linear groups and Chevalley groups over different classes of rings.
A group is said to have the infinite conjugacy class (ICC) property if every non-identity element has an infinite conjugacy class. In this talk I will survey some ideas in geometric group theory, random walks and harmonic functions on groups, and topological dynamics and show how the ICC property sheds light on these three seemingly distinct areas. In particular I will discuss when a group has only constant bounded harmonic functions, when every proximal dynamical system has a fixed point, and what this all has to do with the growth of a group. No prior knowledge of geometric group theory, random walks and harmonic functions on groups or topological dynamics will be assumed. This is based on joint works with Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowsi.
The length of a finite system of generators for a finite-dimensional (not necessarily associative) algebra over a field is the least positive integer k such that the products of length not exceeding k span this algebra as a vector space. The maximum length for the systems of generators of an algebra is called the length of this algebra. Length function is an important invariant widely used to study finite dimensional algebras since 1959. The length evaluation can be a difficult problem, for example, the length of the full matrix algebra is unknown and it was conjectured by Paz in 1984 to be a linear function on the size of matrices. We investigate different algebraic properties of the length function for associative and non-associative algebras and in particular present our recent results on the upper bounds for the length.
The talk is based on the series of joint works with Dmitry Kudryavtsev and Olga Markova.
A Gaussian stationary process is a random function f:R-->R whose distribution is invariant under real shifts, and whose evaluation at any finite number of points is a centered Gaussian random vector. The mathematical study of these random functions goes back at least 80 years, with pioneering works by Kac, Rice and Wiener. Nonetheless, many basic questions about them turned out to have complicated answers, or remained open for many years. One prominent example is estimating the probability that the process does not cross a certain level (one-sided barrier) or a certain pair of levels (two-sided barrier) during a long period of time. A more advanced question (most interesting to physicists) is: when such a rare event occurs, how does it occur?
In this talk we will provide a quick introduction to Gaussian stationary processes, and describe how a spectral perspective, combined with tools from probability and harmonic analysis, sheds new light on these long-lasting questions.
In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g is a primitive root mod p. By observing numerical deviations from Artin's originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor; leading to a modified conjecture that was eventually proved correct by Hooley (1967), under the assumption of the Generalized Riemann Hypothesis (GRH). In this talk we discuss several variants of Artin's conjecture: namely an "Artin Twin Primes Conjecture", as well as an appropriate analogue of Artin's primitive root conjecture for algebraic function fields.
Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results. No mathematical prerequisites will be assumed (beyond the concept of Lebesgue measure and vector spaces over finite fields).
In linear partial differential equations, hypoellipticity is the condition that if Df=g, with g smooth, then f is necessarily smooth too. The best-known hypoelliptic equations are the elliptic equations, which are characterized by an isotropy property that can be readily checked point-by-point. Various more general point-by-point sufficiency criteria for hypoellipticity have been studied, beginning with famous work of Lars Hormander in the 1960’s. Quite recently these criteria have been used to formulate and prove index theorems for hypoelliptic operators in the spirit of the famous Atiyah-Singer index theorem, and to apply index-theoretic techniques to new problems. I shall give a survey of some of these developments. Of special interest is Jean-Michel Bismut's hypoelliptic Laplacian, which is a remarkable family of operators interpolating between the Laplace operator on a Riemannian manifold and the geodesic flow on its tangent bundle.
Let f=(f_1,...,f_m) be a tuple of m polynomials with integer coefficients in n variables.
Then f can naturally be considered as a complex map f:C^n->C^m, and one may approach studying f from several different perspectives:
1) Arithmetic, by counting the number of solutions to the system of equations {f_i = c_i mod r}, for different choices of integers c_i and r.
2) Geometric, by studying the geometry and singularities of the fibers (level-sets) of the complex map f:C^n->C^m.
3) Analytic, by taking a smooth measure \mu with compact support in C^n, and studying analytic
(e.g. integrability) properties of the resulting pushforward measure f_*(\mu) on C^m.
In this talk I will explain how these different approaches relate to one another. I will then discuss a natural algebraic convolution operation
between maps, which allows one to study the behavior of certain algebraic families of random walks using these connections.
This is based on a series of works from recent years joint with I. Glazer.
The study of descents of permutations may be traced back to Euler, and is fundamental to contemporary algebraic combinatorics and its applications. A cyclic extension of this notion was introduced in the late 20th century.
The talk will focus on aspects of descents and cyclic descents for permutations and for standard Young tableaux. Following an axiomatization of the notion of a cyclic descent extension, we will characterize sets of combinatorial objects for which such an extension exists. The main results concern sets of standard Young tableaux of a fixed shape and sets of permutations of a fixed cycle type, i.e., conjugacy classes. The original proofs of both results were algebraic and not constructive, but in the case of tableaux a constructive proof was later given by Brice Huang.
Based on joint works with Pál Hegedüs, Vic Reiner and Yuval Roichman
No prior acquaintance is assumed.
Uniqueness properties are ubiquitous in many fields of study.
I will discuss uniqueness in the context of representation theory, and attempt
to motivate and explain the study of local factors, based on uniqueness.
The talk is motivated by a recent collaboration with Aizenbud and Gourevitch.
In contrast to the standard left action of a locally compact second countable group G on itself,
The conjugation action of SL_n(R) is not Asplund representable for every n > 3.
The linear action of GL_n(R) on R^n, for every n > 1, is not representable on Asplund Banach spaces.
The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). This is unclear for SL_2(R).
As a byproduct we obtain some counterexamples about Banach representations of homogeneous G-actions G/H.
Given a graph G and an embedding of its vertices in R^d, what continuous motions of the vertices preserve all edge lengths?
Clearly all motions induced by an isometry of R^d do, these are the trivial motions; are there any others?
If the answer is NO for all (equivalently, for one) generic embedding, G is called d-rigid. What are the d-rigid graphs?
This problem has been extensively studied since the 70s, and is still widely open for d>=3.
It is studied mainly from algebraic geometry, linear algebra, and combinatorics points of view.
Variants of it, especially in dimensions 2 and 3, are of importance also beyond mathematics, e.g. in structural engineering, computational biology and more.
I will discuss various directions and open problems in graph rigidity, including:
combinatorial characterization, numerical consequences, non-generic rigid embeddings, rigidity of random graphs,
and a quantitative version of rigidity via spectral analysis of the related stiffness matrix.
Discrete Fourier analysis studies functions on the discrete cube {-1,1}^n, using their discrete Fourier expansion and functional-analytic tools. Results in discrete Fourier analysis have applications in diverse fields, ranging from social choice and machine learning to mathematical physics. Cryptanalysis studies the practical security of the encryption schemes we use. The central object in cryptanalysis is "attack techniques" – which are algorithms that allow an adversary to intercept communications, forge digital signatures, etc.
In this talk we propose a new approach to understanding cryptanalytic attacks, using seemingly unrelated techniques from discrete Fourier analysis. We will show that Fourier-analytic techniques can be helpful in addressing core questions, such as: "Can we prove that certain cryptanalytic attacks are optimal?" and "Is there a need for post-quantum secret-key cryptosystems?". We will mostly concentrate on open questions, but will also show promising results following the new approach.
Based on joint works with Itai Dinur and Ohad Klein
A finite positive measure on the Euclidean space is called Rajchman if its Fourier transform tends to zero at infinity.
Absolutely continuous measures are Rajchman by the Riemann-Lebesgue Lemma, but it is a delicate question to decide which singular measures are such.
For many purposes simple convergence if the Fourier transform to zero is not sufficient and some quantitative decay is needed.
Recently there has been renewed interest and a lot of activity in studying such questions for ``fractaI'' measures.
I will survey results on Fourier decay for some classes of fractal measures, such as the ``natural'' uniform measure on a Cantor set of a uniform contraction ratio and their generalizations.
Some of these results are classical, going back to Erdos, Salem, and Kahane, and some are recent.
I will survey the basic ideas and results of asymptotic representation
theory (ART), mostly of symmetric groups, and then focus on a recent
novel approach to Thoma's theorem based on the combinatorial
Robinson-Schensted-Knuth correspondence, as well as other recent
contributions.
We give a survey of recent developments in the study of equations in groups and Lie algebras and related local-global invariants, focusing on parallels between the two algebraic structures.
The study of multiplicities is a central question in abstract harmonic analysis. I will present this question in various settings, and introduce a central conjecture about the boundedness of multiplicities. If time allows, I will present several recent results regarding this conjecture.
In the late 1970s W. Thurston proved that the countable set of volumes of
closed hyperbolic 3-manifolds is well-ordered, that only finitely many closed hyperbolic
3-manifolds can have the same volume, and that the ordinal of the set of these volumes is
${\omega_0}^{\omega_0}$.
We prove analogous results for the rates of growth of hyperbolic (and other) groups.
We study the countable set of rates of growth of a hyperbolic group with respect to all
its finite sets of generators, the countable set of rates of growth of all the finitely generated
subgroups of a hyperbolic group (with respect to all their finite generating sets), and the rates
of growth of all the finitely generated subsemigroups of a hyperbolic group.
Our results suggests a polynomial invariant for generating sets (and tuples) in (some) hyperbolic groups.
Joint work with Koji Fujiwara.
A finite positive measure on the Euclidean space is called Rajchman if its Fourier transform tends to zero at infinity.
Absolutely continuous measures are Rajchman by the Riemann-Lebesgue Lemma, but it is a delicate question to decide which singular measures are such.
For many purposes simple convergence if the Fourier transform to zero is not sufficient and some quantitative decay is needed.
Recently there has been renewed interest and a lot of activity in studying such questions for ``fractaI'' measures.
I will survey results on Fourier decay for some classes of fractal measures, such as the ``natural'' uniform measure on a Cantor set of a uniform contraction ratio and their generalizations.
Some of these results are classical, going back to Erdos, Salem, and Kahane, and some are recent.
In a paper published at the Annals of Mathematics in 1977,
Magidor proved that it is consistent that the first cardinal at which
the generalized continuum hypothesis (GCH) fails be the very first
singular cardinal Aleph_w. This demonstrates the failure of a
compactness property for Aleph_w. In a paper from 1982, Magidor proved
that Aleph_w can satisfy yet another compactness property. It remained
open ever since whether the two results can coexist in the same model.
In this talk, I will report on a joint work with A. Poveda and D.
Sinapova in which we obtain the desired model.
If Q is a convex set, a transformation T: Q ->Q is affine if it preserves the convex structure of Q. An affine representation of a group is a homomorphism of G to the group of invertible affine transformations of a compact convex Q. It is irreducible if no proper closed, convex subset of Q is left invariant. Abelian and compact groups have no non-trivial irreducible affine representations. From the classical theory of harmonic functions we show that any bounded harmonic function on the upper half plane leads to an irreducible affine representation of SL(2,R).
We discuss generalizations leading to the notion of the Poisson boundary of a group.
A basic result in Algebraic Geometry implies that a nonzero multivariate polynomial of low degree cannot vanish over a large box. This can be used to obtain results in extremal combinatorics, graph theory, additive number theory and combinatorial geometry.
I will describe recent and less recent applications of the technique, and will mention several intriguing open problems.
If you choose a random symmetric matrix, would it have two equal eigenvalues? Probably not, since symmetric matrices with degenerate eigenvalues capture a "very small portion" of the entire space of symmetric matrices. This is an example of a genericty result. A more complicated and famous result in spectral geometry is Uhlenbeck's genericity theorem ('72) which states that given any manifold, for a Baire-generic choice of Riemannian metric, all eigenvalues of the associated Laplalacian are simple, with eigenfunctions that are Morse and does not vanish at their critical point. A similar result for metric graphs was given by Friedlander ('05) and Berkolaiko and Liu ('17). Given any fixed graph structure, for a Baire-generic choice of edge lengths, all eigenvalues of the Laplacian (with standard vertex conditions) are simple, with eigenfunctions that are Morse and do not vanish on vertices.
However, apriori, the Baire-generic set of ``good'' edge lengths can have zero Lebesgue measure (and so random sampling would probably miss it).
In this talk, I will introduce a stronger notion of genericity, that implies full Lebesgue measure. I will give an independent proof of the previous results, showing that they are in fact strongly generic. To do so, I will define the ``Trace Space'', a moduli space of all eigenpairs associated with a graph structure. I will also show that any non-trivial homogenous relation on the traces must fail generically. This proof is algebraic in nature, uses ergodic theory, and is based on a conjecture of Colin de Verdiere which was recently proved by Kurasov and Sarnak.
I will not assume any prior knowledge in spectral geometry or metric graphs.
In many data-driven applications, the data follows some geometric structure, and the goal is to recover this structure. In many cases, the observed data is noisy and the recovery task is even more challenging. A common assumption is that the data lies on a low dimensional manifold. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there was no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise.
In various areas of mathematics there exist "big fiber theorems", these are theorems of the following type: "For any map in a certain class, there exists a 'big' fiber", where the class of maps and the notion of size changes from case to case. We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov's notion of ideal-valued measures. We adapt the latter notion to the realm of symplectic topology, using an enhancement of a certain cohomology theory on symplectic manifolds introduced by Varolgunes, allowing us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results. Necessary preliminaries will be explained.
The talk is based on a joint work with Adi Dickstein, Leonid Polterovich and Frol Zapolsky.
Hamiltonian integrable models are essentially the simplest volume preserving dynamical systems: their flow is quasi-periodic along invariant tori, defined by a large set of conserved functions. These systems are non-ergodic and non-chaotic. The interest in near-integrable systems--- where the set of conservation laws is slightly perturbed--- goes back to Laplace and Lagrange studies on the stability of the Solar System. While integrability breaking in low dimensions can be fully characterized by the Kolmogorov-Arnold-Moser theorem, the case of high, or infinite dimensional systems is not fully understood. I will introduce two complementary points of view to describe the dynamics of many-body, near-integrable systems: (I) by a fast ergodic dynamics tangent to the invariant tori together with a slow drift normal to it, and (II) as a weak stochastic perturbation to an integrable model. Both ideas will be illustrated via numerical simulations for the paradigmatic example of the Fermi-Pasta-Ulam-Tsingou (FPUT) chain. I will show how the former picture can be employed to devise a fast numerical integration scheme to study very long equilibration times of the system. As for point (II), I will present a simple stochastic model which explains the basic mechanism ruling the onset of chaos in large FPUT chains. Finally, I will discuss implications to other models such as the classical Solar System and quantum chains.
The study of aperiodic order and mathematical models of quasicrystals is concerned with ways in which disordered structures can nevertheless manifest aspects of order. In the talk I will describe examples such as the aperiodic Penrose and pinwheel tilings, together with several geometric, functional, dynamical and spectral properties that enable us to measure how far such constructions are from demonstrating lattice-like behavior. A particular focus will be given to new results on multiscale substitution tilings, a class of tilings that was recently introduced jointly with Yaar Solomon.
Let $M$ be a compact metric space, $U\subset M$ be its open subspace, $f:M\to M$ be a homeomorphism of the compact into itself, and $x\in M$ be an initial point. It determines sequence of points
$x,f(x),\ldots,f^{(n)}(x),\ldots$ With the sequence of iterations, one can associate an infinite binary word $w_n=a$ for $f^{(n)}(x_0)\in U$, $w_n=b$ for $f^{(n)}(x_0)\notin U$
which is called the {\it evolution} of point $x_0$. If $f$ is invertible then $n\in \mathbb{Z}$, otherwise $n\in \mathbb{N}$. Symbolic dynamics investigates the interrelation between the
properties of the dynamical system $(M,f)$ and the combinatorial properties of the word $W_n$. For words over alphabets which comprise more symbols, several characteristic sets should be
considered: $U_1,\ldots,U_n$.
The famous Sturmian sequences and some of their generalizations
present situation of ``combinatorial paradise''. It provides patterns for further investigation. Let us remind the classical result.
{\bf Equivalence theorem} {\it Let $W$ be an infinite recurrent word over the binary alphabet
$A=\{a,b\}$. The following conditions are equivalent: 1) The word $W$ is a {\em Sturmian word}, i.e., for any $n\geq 1$,
the number of different subwords of length $n$ that occur in $W$ is equal to $T_n(W)=n+1$. 2) The word is not periodic and is {\em balanced}, i.e., any
two subwords $u,v\subset W$ of the same length satisfy the inequality $||v|_a-|u|_a|\leq 1$, where $|w|_a$ denotes the number
of occurrences of symbol $a$ in the word $w$. 3) The word $W=(w_n)$ is a {\em mechanical} word with irrational
$\alpha$, i.e. there exist an irrational $\alpha$, $x_0 \in [0,1]$, and interval $U\subset \mathbb{S}^1$,
$|U|=\alpha$, such that $w_n=a$ for $T_{\alpha}^n(x_0)\in U$, $w_n=b$ for $T_{\alpha}^n(x_0)\notin U$
4) Word $W$ can be obtained as a limit of the sequence of finite words $\{w_i\}_{i=1}^\infty$, such that $w_{i+1}$ can be
obtained from $w_{i}$ via substitution of the following type $a^{k_i}b\to b, a^{k_i+1}b\to a$ or $b^{k_i}a\to a, b^{k_i+1}a\to
b$. Iff sequence of these substitutions is periodic, then $\alpha$ is
a quadratic irrational.}
There are several different ways of generalizing Sturmian words. One of the most important is interval exchange transformation.
It can be done in terms of {\em Rauzy graphs}. The {\it Rauzy graph} of order $k$ (the {\it $k$-graph}) of the word $W$ is the directed graph whose
vertices biuniquely correspond to the factors of length $k$ of the word $W$ and the vertex $A$ is connected to vertex $B$ by directed
arc iff $W$ has a factor of length $k+1$ such that its first $k$ letters make the subword that corresponds to $A$ and the last $k$
symbols make the subword that corresponds to $B$. By the {\it follower} of the directed $k$-graph $G$ we call the directed graph
$\Fol(G)$ constructed as follows: the vertices of graph $\Fol(G)$ are in one-to-one correspondence with the arcs of graph $G$ and
there exists an arc from vertex $A$ to vertex $B$ if and only if the head of the arc $A$ in the graph $G$ is at the notch end of
$B$. The $(k+1)$-graph is a subgraph of the follower of the $k$-graph; it results from the latter by removing some arcs.
Vertices which are tails of (or heads of) at least two arcs correspond to {\it special factors}; vertices which are heads and
tails of more than one arc correspond to {\it bispecial factors}. The sequence of the Rauzy $k$-graphs constitutes the {\it
evolution} of the Rauzy graphs of the word $W$.
We discourse factor-dynamics properties, universal sequences (for example language consisting all words which appear in $k$-interval exchange for some $k$, deformations of universal language
and the first digit sequence $2^{n^2}$.
Let $P(n)$ be a polynomial, having an irrational coefficient of the highest degree. A word $W=(w_n)$, (n\in \mathbb{N})$ consists of a sequence of first binary numbers of $\{P(n)\}$ i.e. $w_n=[2\{P(n)\}]$. Denote by $T(k)$ the number of different subwords of $W$ of length $k$.
{\bf Theorem}. There exists a polynomial $Q(k)$, depending only on the power of the polynomial $P$, such that $T(k)=Q(k)$ for all sufficiently great $k$.
similar is true for the first digit sequence $2^{n^2}$.
(Joint work with , I.Reshetnikov)
It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M.
In joint work with Izabella Laba (UBC), we proved that this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce some ingredients from the proof.
In this talk I intend to describe the geometric model and a recent extension that preserves more of the global topology of the flow. I’ll explain how this new model is strongly related to the geodesic flow on the modular surface, and how it yields a way to approach analytically Smale’s 14th problem.
This is joint work with Christian Bonatti.
Langlands-Shahidi method is one of the main ways to study automorphic L-functions for quasi-split algebraic groups. This method is centered around Shahidi local coefficients arising from a certain uniqueness result. In this talk we shall recall the definition of these coefficients and discuss a local application. Then we shall describe an analog of these coefficients for covering groups defined in a setting where the uniqueness mentioned above fails. Finally, we shall discuss the behavior of this analog under restriction to the derived group. This is a joint project with Fan Gao and Freydoon Shahidi.
ZOOM:
https://us02web.zoom.us/j/89074854473?pwd=R1BWVEZ4NG5yMkhNYVB2RGRLVnNMdz09
I am going to discuss a progress in a project which aims to formalise the notion of an analytic covering space of a complex algebraic variety
in such a way that the formal pseudo-analytic cover is unique up to abstract isomorphisms. At this stage our interest focuses on the upper half-plane as the cover of modular curves.
On the model theory side it is based on Shelah's theory of abstract elementary classes. On the geometric side it uses the rich theory of complex multiplication and
canonical models of Shimura varieties as developed by Shimura, Langlands, Deligne, Milne, Borovoi and others.
ZOOM:
https://us02web.zoom.us/j/89074854473?pwd=R1BWVEZ4NG5yMkhNYVB2RGRLVnNMdz09
In this talk I will explain the Cohn-Kumar conjecture
about energy minimization in Euclidean spaces, its connection to
the sphere packing problem and its recent solution in 8 and 24 dimensions
based on some novel interpolation formulas for radial Fourier eigenfunctions.
The talk is based on a joint work with H. Cohn, A. Kumar, S.D. Miller, and M. Viazovska.
ZOOM:
https://us02web.zoom.us/j/89074854473?pwd=R1BWVEZ4NG5yMkhNYVB2RGRLVnNMd…
We offer a systematic approach to a class of attacks on communication
channels protected by homomorphic encryption based on black box
algebraic analysis. Our conclusion is that wide classes of algebraic
structures should not be used as ambient structures for homomorphic
encryption. We give some examples for groups and rings, but our general
methodology is much wider applicable.
Black box algebra deals with a category where objects are finite
algebraic structures (fields, rings, group,s projective planes etc.)
with elements implemented as 0-1 strings of length L (perhaps different
for different objects) and operations are performed by external devices
or algorithms which work in time bounded by a polynomial in L).
Similarly, morphisms are homomorphisms computable in polynomial time.
We will show that this is a fascinating theory with many unusual
features and a huge range of open problems.
(Joint work with Sukru Yalcinkaya)
ZOOM: https://us02web.zoom.us/j/89074854473?pwd=R1BWVEZ4NG5yMkhNYVB2RGRLVnNMdz09
To each vertex x\in Z^2 assign a positive weight \omega_x. A geodesic between two ordered points on the lattice is an up-right path maximizing the cumulative weight along itself. A bi-infinite geodesic is an infinite path taking up-right steps on the lattice and such that for every two points on the path, its restriction to between the points is a geodesic. Assume the weights across the lattice are i.i.d., does there exist a bi-infinite geodesic with some positive probability?
In the case the weights are Exponentially distributed, we answer this question in the negative. We show an analogous result for the positive-temperature variant of this model.
Joint work with Marton Balazs and Timo Seppalainen.
In the last decades there was much ado about computer proofs, computer aided proofs,
computer verified proofs, and the like. It is obvious that the advent and proliferation of
computers have drastically changed applications of mathematics.
What one discusses much less, however, and what I find much more interesting, is how
computers have changed mathematics itself, and mathematicians’ stance in regard of
mathematical reality, both as far as the possibilities to immediately observe it, and the
apprehension of what we can hope to prove.
Computers have already changed mathematics in what concerns very basic ideas much
more fundamental than any individual theories: balance of ideas and computations,
large and small (and, most importantly, intermediate size!), finite and infinite, feasible
and unfeasible, deterministic and random, etc.
Much of the recent progress in mathematics would had been impossible without
computers. In particular, they allowed to reconnect to the history of mathematics
and solve problems in the absolute sense, as they were posed by the XVII--XVIII
centuries classics, rather than merely in asymptotic forms.
I recount my personal experience of using computers as a mathematical tool, and
the experience of such similar use in the works of my colleagues that I could observe
at close range, especially in group theory, and number theory.
This experience has radically changed my perception of many aspects of mathematics,
what it is, how it functions, and especially, how it should be taught.
The Bernoulli shift model which is the action on a sequence of i.i.d. random variables by time shifts in one of the central examples of classical ergodic theory. To this date much is known on the ergodic theoretic properties of this model. A notable example is Sinai factor theorem which says that if a given system has positive entropy (is chaotic) then it has Bernoulli shift models as factors (subsystems) which leads to one of the most common definitions of chaos. In addition, Ornstein theorem says that two Bernoulli systems are isomorphic if and only if they have the same measure theoretic entropy while Dye's theorem implies that all Bernoulli shifts are orbit equivalent.
Relaxing the i.i.d. hypothesis to independence one arrives to a very natural model of Bernoulli shifts which are not at equilibrium. Up to recent years very little was known on the ergodic theoretic properties of these important systems and in the recent decade a beautiful theory emerged with connections and applications to probability theory (exchangeability), smooth dynamics (new classes of Anosov diffeomorphisms) and geometric group theory ($l^2$ Betti numbers, Stallings ends theorem). I will survey (some of) these results and some of the future challenges. Based on several works, among them is a joint work with Michael Bjorklund and Stefaan Vaes and two joint works with Terry Soo.
ZOOM: https://us02web.zoom.us/j/89074854473?pwd=R1BWVEZ4NG5yMkhNYVB2RGRLVnNMdz09
Modular curves are fundamental objects in modern number theory, which allow us to relate geometric objects and arithmetic ones. Recent advances towards long standing conjectures depend crucially on explicit computation of these objects. The talk will describe novel algorithms that achieve this goal.
I will report on some old and new classification results in
equivariant symplectic geometry, including my classification, joint with
Sue Tolman, of Hamiltonian torus actions with two dimensional quotients.
Phase retrieval is the inverse problem of reconstructing a signal from linear measurements, when the phase of the measurements is lost and only the magnitude is known. This problem occurs in many applications including crystallography, optics, and acoustics.
In the talk I will discuss results on invertibility and stability of phase retrieval. I will focus on a recent paradigm which characterizes phase retrieval stability via appropriate notions of graph connectivity, and in particular our recent results relating real phase retrieval to the Cheeger constant, and complex phase retrieval to the spectral gap of the graph Laplacian. As corollaries we obtain sharp estimates for the dependence of the stability constant on the dimension of the ambient space, and examples of (in)stable signals in infinite dimensions.
The talk is based on the paper
Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements
Classification theory is a program initiated by Shelah in the early 70s with the aim of classifying complete first order theories into "tame" and "wild" theories by some relatively simple combinatorial invariants. Examples of such theories are stable, dependent and simple theories.
From the beginning, the applications stemming from this work moved in two intertwining directions.
The first, a detailed study of the geometry arising in such theories and applying these notions to algebra and geometry (e.g. Hrushovski's proof of Mordell-Lang).
The second, studying and classifying algebraic objects whose theories enjoy these properties (e.g. Shelah and Cherlin proved that every superstable field is algebraically closed).
In this talk I will introduce the main model theoretic notions and mainly follow the second direction of applications. I will review some old results and some newer results by myself and others especially in the dependent setting, where an open conjecture of Shelah's leads the way.
No prior knowledge in model theory or logic is required.
Patterns in a large combinatorial structure, such as a graph, a word, or a permutation, are the induced substructures on small subsets of vertices or entries. If a permutation describes the relation between the y-ranks and x-ranks of n points in the xy-plane, then every subset of k points induces one of k! possible patterns. Similarly, in an n-letter word over a finite alphabet, every k positions induce a k-letter subword. Some important questions are: How frequently does each pattern occur in typical settings? How efficiently can we count these occurrences? What do they tell us about the global properties of the structure?
Consider the k!-dimensional vector of pattern statistics in a uniformly random permutation. I will present a decomposition of its distribution into k asymptotically uncorrelated components of different orders in n. This extends work by Janson, Nakamura, and Zeilberger (2015) who characterized the first and last components. I will also discuss new algorithms for counting permutation patterns (SODA 2021). Our methods lead to near linear time computation of consistent variants of Hoeffding’s independence test, improving upon the so far most efficient quadratic time algorithms. Finally, I will discuss the full spectral decomposition of the distribution of subword statistics under different random models, with relations to several questions in statistics and combinatorics and potential applications to machine learning.
Joint work with Calvin Leng, and with Ran Tessler and Tsviqa Lakrec.
We prove the existence of equilibrium in repeated Blackwell games with Borel tail-measurable winning sets, and with Borel tail-measurable payoffs. The proof uses a regularity result of the minmax value
A common observation in data-driven applications is that high dimensional data has a low intrinsic dimension, at least locally. Thus, when one wishes to work with data that is not governed by a clear set of equations, but still wishes to perform statistical or other scientific analysis, an optional model is the assumption of an underlying manifold from which the data was sampled. This manifold, however, is not given explicitly but we can obtain samples of it (i.e., the individual data points). In this talk, we will consider the mathematical problem of estimating manifolds from a finite set of samples, possibly recorded with noise. Using a Manifold Moving Least-Squares approach we provide an approximant manifold with high approximation order in both the Hausdorff sense as well as in the Riemannian metric (i.e., a nearly isometry). In the case of bounded noise, we present an algorithm that is guaranteed to converge in probability when the number of samples tends to infinity. The motivation for this work is based on the analysis of the evolution of shapes through time (e.g., handwriting or primates' teeth) and we will show how this framework can be utilized to answer scientific questions in paleontology and archaeology
Let T be the action of the complex plane on the space of entire functions defined by translations, i.e T_w takes the entire function f(z) to the entire function f(z+w). B.Weiss showed in `97 that there exists a probability measure defined on the space of entire functions, which is invariant under this action. In this talk I will present optimal bounds on the minimal possible growth of functions in the support of such measures, and discuss other growth related problems inspired by this work. In particular, I will focus on the question of minimal possible growth of frequently oscillating subharmonic functions
The talk is partly based on a joint work with L. Buhovsky, A. Logunov, and M. Sodin.
Random walks in random environment (RWRE) have been extensively studied in the last half-century. Two prototypical cases are the reversible and the ballistic classes and even though they are fundamentally different, functional central limit theorems (FCLT) are known to hold in both. This is done using rather different techniques; Kipnis-Varadhan's theory for additive functional of Markov processes is applicable in the reversible case while for the ballistic class the main feature is a regenerative structure which guarantees that the process contains a random walk as a subsequence.
Rough path theory is a deterministic theory of integration which allows path integration with respect to singular signals in a continuous manner. It typically provides a framework to solve stochastic differential equations (ordinary and partial) driven by a singular noise.
We shall discuss some recent contributions, in which we lift additive functionals of Markov processes and regenerative processes to the rough path space and obtain an enhanced FCLT.
The first level of the limiting rough path is naturally the Brownian motion but, somewhat surprisingly a deterministic linear perturbation appears in the second level. We characterize the latter in various ways. Except for the immediate application to SDE, this provides some new information on the structure of the limiting path. The aforementioned classes of RWRE are covered as special cases.
Based on collaborations with Johaness Bäumler, Noam Berger, Jean-Dominique Deuschel, Olga Lopusanschi, Martin Slowik and Nicolas Perkowski.
A well known conjecture in fractal geometry says that the dimension of a self-similar measure is strictly smaller than its natural upper bound only in the presence of exact overlaps. That is, only if the maps in the generating iterated function system do not generate a free semigroup. I will present recent developments regarding this conjecture, focusing on my joint work with P. Varjú regarding homogeneous systems of three maps
The Toeplitz algebra of a directed graph is the norm-closed $*$-algebra generated by edge and vertex concatenation operators on the inner-product space of square summable sequences indexed by finite paths of the graph. A canonical quotient of it is the celebrated Cuntz-Krieger algebra, which is deeply connected to the associated subshift of finite type of the directed graph. Understanding representations of Cuntz-Krieger algebras has become useful for producing wavelet bases on self-similar sets, for encoding properties of slowed continued-fractions expansions and for the development of non-commutative function theory.
In this talk I will present a complete characterization of those finite directed graphs that admit a Cuntz-Krieger representation in which the weak operator topology closed algebra generated by edge and vertex operators (without their adjoints !) is automatically $*$-closed (and hence a von-Neumann algebra). The first example of this counter-intuitive phenomenon was produced by Charles Read in the case where the graph has a single vertex and two loops. I will explain how the Road Coloring theorem of Trahtman (and a periodic version of the theorem due to B\'eal and Perrin) is used to reduce the problem to Read's example.
*Based on joint work with Christopher Linden.
Our goal is to present an axiomatic algebraic theory which unifies
and “explains” aspects of tropical algebra, hyperfields, and fuzzy rings in terms
of classical algebraic concepts, especially negation, which may not exist a priori.
It was motivated by an attempt to understand whether or not it is coincidental
that basic algebraic theorems are mirrored in supertropical algebra, and was
spurred by the realization that some of the same results are obtained in parallel
research on hyperfields, fuzzy rings, and matroids.
These and many other algebraic theories involve the study of a set T with
incomplete structure that can be understood better by embedding T in a larger
set A endowed with more algebraic structure.
Often the algebra A is a semi-algebra, such as the max-plus algebra, which
does not necessarily have negatives, so much of the research has become a
project of developing a general theory of semi-algebras and their modules,
with suitable axioms that permit one to prove theorems in valuation theory
(tropicalization), linear algebra (joint with Akian and Gaubert), matrix theory
and quadratic forms (joint with Chapman and Niv), geometry, Grassmann
algebras (joint with Gatto), and homology (joint with Jun and Mincheva).
We include a 5-minute introduction to tropical mathematics, to keep the
talk self-contained.
As is well-known, discretisations often behave rather differently from the continuous models which they are meant to approximate, at the very least they may contain a much more delicate structure. We will discuss two manifestations of this. While trying to construct discrete analogues of the continuum picture of the differential graded algebra of differential forms on a manifold, we find that there is not just one, but rather there are three pictures, depending upon which one of the three properties of commutativity, associativity and Leibniz is dropped. Each is a world on its own in which different tools can be applied.
We will also discuss connections with Sullivan's proposed model of lattice hydrodynamics which is `naturally' derived from physical principles in a discrete setting, rather than as an artificial discretisation of a continuous equation.
This is joint work with Dennis Sullivan and Nissim Ranade.
We will speak about growth and Gelfand-Kirillov dimensions of subsemigroups of free inverse semigroups (joint work with Stuart Margolis).
In the talk we will introduce those flows and their dynamical behavior.
Moreover, we show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure rigidity for the moduli space of translation surfaces, can be adapted to smooth ergodic theory and in particular towards the study of Anosov flows.
Using this adaption, we show that for a quantitatively non-integrable Anosov flow, every generalized u-Gibbs measure is absolutely continuous with respect to the whole unstable manifold.
No prior knowledge is assumed, all the concepts will be explained in the talk.
Which domains in Euclidean space admit an orthogonal basis of exponential functions? For example, the cube is such a domain, but the ball is not. In 1974, Fuglede made a fascinating conjecture that these domains could be characterized geometrically as the domains which can tile the space by translations. I will survey the subject and report about the recent developments on Fuglede's conjecture.
We present a structural theorem for bounded sequences in Banach spaces based on ideas of concentration compactness that is, on the role of non-compact invariance for isolating lack of convergence in a sum of localized "bubbles" . Surprisingly the argument requires not weak convergence but its less known cousin, Delta-convergence.
In the 1950s, Carl Loewner proved an inequality relating the
shortest closed geodesics on a 2-torus to its area. Many
generalisations have been developed since, by Gromov and others. We
show that the shortest closed geodesic on an area-minimizing surface S
for a generic metric on CP^2 is controlled by the total volume, even
though the area of S is not. We exploit the Croke--Rotman inequality,
Gromov's systolic inequalities, the Kronheimer--Mrowka proof of the
Thom conjecture, and White's regularity results for area minimizers.
An infinite group G is called highly transitive if it acts on some infiniteset m-transitively for any natural number m. We give a brief survey on some recent results on abstract highly transitive groups.
Then we pass to examples of affine algebraic varieties with the automorphism group acting highly transitively; specifically, of toric affine varieties. We show that a highly transitive group can be generated by a finite number of one-parameter subgroups; for the affine spaces, three such subgroups suffice. We formulate some open problems related to group growth.
Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from Riemann surfaces with boundary to a symplectic manifold, with boundary conditions and various constraints on boundary and interior marked points. The presence of boundary leads to bubbling phenomena that pose a fundamental obstacle to invariance. In a joint work with J. Solomon, we developed a general approach to defining genus zero OGW invariants.
The construction uses the heavy machinery of Fukaya A_\infty algebras. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW invariants has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called the open WDVV equations. For projective spaces, this system of PDE generates recursion relations that allow the computation of all invariants.
We further reinterpret the open WDVV equations as the associativity relation for a relative quantum product.
No prior knowledge of any of the above notions will be assumed.
The theory of Markov chains has applications in diverse areas of research such as group theory, dynamical systems, electrical networks and information theory. These days, connections with operator algebras seem to manifest mostly in quantum information theory, where Markov chains are generalized to quantum channels.
In this talk we will show how questions about operator algebras constructed from stochastic matrices, studied by Markiewicz and myself, still motivate new problems in classical Markov chain theory. More precisely, we characterize coincidence of conditional probabilities in terms of generalized Doob transforms, which then leads to stronger classification results for the associated operator algebras. This turns out to be intimately related to determining positive harmonic functions for the stochastic matrix. Time permitting, I will explain how non-commutative peak points of the associated operator algebra can be completely characterized in terms of the stochastic matrix.
*This is based on joint work with Xinxin Chen, Langwen Hui, Christopher Linden and Yifan Zhang, conducted as part of an undergraduate research project in Illinois Geometry Lab at UIUC.
We discuss the spectrum of high dimensional non-Hermitian matrices under small noisy perturbations. That spectrum can be extremely unstable, as the maximal nilpotent matrix J_N with J_N(i,j)=1 iff j=i+1 demonstrates. Numerical analysts studied worst case perturbations, using the notion of pseudo-spectrum. Our focus is on finding the locus of most eigenvalues (limits of density of states), as well as studying stray eigenvalues ("outliers"). I will describe the background, show some fun and intriguing simulations, and present some theorems. No background will be assumed. The talk is based on joint work with Anirban Basak and Elliot Paquette.
A subset of the integers P is called predictive if for any zero entropy finite-valued stationary process {X_i} , X_0 is measurable with respect to the sigma-algebra generated by the {X_i ; i \in P}. Zero entropy processes are exactly those for which N itself is a predictive set. I will discuss come necessary and some sufficient conditions for a set to be predictive. It turns out that this notion is related to the classical Riesz sets in harmonic analysis that were defined many years ago by Y. Meyer. All of the relevant notions will be defined ab initio.
I will discuss new (analytic) developments in the old paradigm in automorphic functions connecting L-functions to representation theory via the fundamental notion of a period. I revisit the orbit method of Kirillov with the aim of making it "effective".
Quantum determinant is an analog of the usual determinant
which lives in a Quantum Matrix Algebra (QMA). I plan
to introduce the notion of QMAs and quantum analogs
of some symmetric polynomials in these algebras.
In particular, such "quantum symmetric elements"
in Generalized Yangians will be exhibited. Also, I plan
to discuss the role of these objects in Integrable
Systems theory.
First order expressible properties have been studied using random finite models. That is, by looking on the possible behavior of first order properties given a probability space of graphs, e.g., G(n,p). A number of very attractive and surprising results have been obtained along the years. In the talk I'll mention some of the classic results, demonstrate proof techniques and present two new results and a few open problems. No knowledge of logic is assumed.
Suppose that a company distributes a commercial product and that each package contains a coupon. There are $n$ types of coupons, and a customer wants to collect at least one of each. How many packages need to be bought on the average until getting all coupons? This is referred to as the coupon collector problem, and goes back at least as far as de Moivre.
Clearly, one expects that, in the beginning of the process, most coupons obtained will be new ones. As we continue, it takes more and more time to obtain a new coupon. We will start from the question what is the maximum time between two consecutive new coupons throughout the whole process. Then we will discuss some related problems.
This talk is an overview on the dimension theory of some dynamically defined function graphs, like Takagi and Weierstrass function. In particular, we study the dimension of Markovian fractal interpolation functions (which play important role in various applications) and generalized Takagi functions generated by non-Markovian dynamics.
One of the basic results in model theory is Lowenheim-Skolem. It states that every infinite model has infinite sub-models of any size. Elementary substructures basically catch all intrinsic properties of the large structure (first order properties). Obtaining substructure with more similarity to the original structure (second order properties) is more subtle, and often independent of the standard axioms of set theory.
In this talk I will discuss a special case of a second-order version of the Lowenheim-Skolem theorem - Chang's Conjecture. This principle is deeply connected to large cardinal axioms, in a ways that are not fully understood yet. I will present some of the definitions in this area and discuss some cases in which the Chang's Conjecture holds.
It is well-known that the product of two compact topological
spaces is again compact, but that the product of two Lindelof spaces
need not be Lindelof. In this talk, we shall address questions of these
flavor, focusing on a joint project of the speaker together with Chris
Lambie-Hanson.
In 1870 Cantor proved that a trigonometric series which converges to zero everywhere must be trivial. In the 50s it was asked: under what conditions is this still true if the convergence is only along a subsequence? We will show a number of results on this topic, hopefully some hints of the proofs will also be showed. Joint work with A. Olevskii.
The famous Jacobian Conjecture states that locally invertible polynomial mapping over ${\mathbb C}$ is globally invertible. Dixmier conjecture says that any endomorphism of the ring of differential operators (Weil algebra $A_n$), is an automorphism. In the paper A. Ya. Kanel-Belov and M. L. Kontsevich, "The Jacobian conjecture is stably equivalent to the Dixmier conjecture", Mosc. Math. J., 7:2 (2007), 209-218, arXiv: math0512171. we constructed a homomorphism between the automorphism semigroup of $A_n$ and polynomial symplectomorphisms of ${\mathbb C}^{2n}$. Kontsevich Conjecture states that this homomorphism is an isomorphism of invertible mappings.
In fact, Kontsevich's Conjecture states that deformation quantization of affine space preserves the group of symplectic polynomial automorphisms, i.e. the group of polynomial symplectomorphisms in dimension $2n$ is canonically isomorphic to the group of automorphisms of the corresponding $n$-th Weyl algebra. The conjecture is confirmed for $n=1$ and open for $n>1$. We play with Plank constants and use singularity trick to confirm the general case of the conjecture, see Alexei Kanel-Belov, Andrey Elishev, Jie-Tai Yu, Augmented Polynomial Symplectomorphisms and Quantization, 2018 , 21 pp., arXiv: 1812.02859.
Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of cells homeomorphic to cubes. The cell decomposition can be built effectively, and is one of the most powerful tools in studying properties of real semialgebraic sets.
Another most useful tool, the Gromov-Yomdin Lemma, builds a uniform in parameters cover of real algebraic sets by images of $C^r$-smooth mappings of cubes.
There is a non-trivial obstruction to complexification of this result, related to inner hyperbolic metric properties of complex holomorphic sets.
We proved a new simple lemma about functions in one complex variables. This allowed us to construct a proper holomorphic version of the above results,
for complex (sub)analytic and semialgebraic sets, combining best properties of both. As a corollary, we prove an old Yomdin's conjecture on $\epsilon$-tail entropy for analytic maps.
This is a joint work with Gal Binyamini.
In recent years, the classical theory of entropy for a dynamical system has been revolutionized by the ground-breaking work of several researchers. Two definitions were proposed and developed for actions of general groups : sofic entropy (initiated by L. Bowen) and Rokhlin entropy (initiated by B. Seward). We will start with a very brief account of the latter, and then describe our own recently developed approach to entropy theory for free probability-measure-preserving actions of all countable groups. We will then formulate our main result, namely that Rokhlin entropy satisfies a Shannon-McMillan-Breiman pointwise convergence theorem. We will demonstrate the geometric significance of this convergence theorem in the case of actions of free non-Abelian groups
Based on joint work with F. Pogorzelski (Leipzig University).
Substitution schemes provide a classical method for
constructing tilings of Euclidean space. Allowing multiple scales to
appear in the substitution rule, multiscale substitution schemes are
introduced. In the talk we will consider some interesting new
geometric objects which are generated by such multiscale schemes.
We will focus on Kakutani sequences of partitions, in which every
element is defined by the substitution of all tiles of maximal measure
in the previous partition, and include the sequences of partitions of
the unit interval considered by Kakutani as a special case. Applying
new path counting results for directed weighted graphs, we will show
that such sequences of partitions are uniformly distributed, thus
extending Kakutani's result. Furthermore, we will describe certain
limiting frequencies associated with sequences of partitions, which
relate to the distribution of tiles of a given type and the volume
they occupy.
We prove the discreteness of small deformations of a
discrete co-compact subgroup of the group of isometries of a locally
compact metric space under some natural assumptions.
This is a joint work with G. Margulis.
The talk will be self-contained and should be accessible to master's students.
We describe several examples of tame subgroups of finitely presented groups and prove that the fundamental groups of certain finite graphs of groups are locally tame.
Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf or the PNAS article http://www.pnas.org/content/early/2018/09/06/1720804115). Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984). The talk will conclude with open problems on the behavior of greedy matching schemes. Joint work with Nina Holden and Alex Zhai.
In 2003, Welschinger defined invariants of real symplectic
manifolds of complex dimensions 2 and 3, which are related to counts
of pseudo-holomorphic disks with boundary and interior point
constraints (Solomon, 2006). The problem of extending the definition
to higher dimensions remained open until recently (Georgieva, 2013,
and Solomon-Tukachinsky, 2016-17).
In the talk I will give some background on the problem, and describe
a generalization of Welschinger's invariants to higher dimensions,
with boundary and interior constraints, a.k.a. open Gromov-Witten
invariants. This generalization is constructed in the language of
$A_\infty$-algebras and bounding chains, where bounding chains play
the role of boundary point constraints. If time permits, we will
describe equations, a version of the open WDVV equations, which the
resulting invariants satisfy. These equations give rise to recursive
formulae that allow the computation of all invariants of
$\mathbb{C}P^n$ for odd $n$.
This is joint work with Jake Solomon.
No previous knowledge of any of the objects mentioned above will be assumed.
Consider a set of convex figures in R^2. It can be proven
that one of these figures can be moved out of the set by translation
without disturbing the others. Therefore, any set of planar figures
can be disassembled by moving all figures one by one. However,
attempts to generalize it to R^3 have been unsuccessful and finally,
quite unexpectedly, interlocking structures of convex bodies were
found. These structures can be used in engineering. In a small grain
there is no room for cracks, and crack propagation should be arrested
on the boundary of the grain. On the other hand, grains "keep" each
other. So it is possible to get "materials without crack propagation"
and get new use of sparse materials, say ceramics. Surprisingly, such
structures can be assembled with any type of platonic polyhedra, and
they have a geometric beauty.
I will describe some nice connections between closed
geodesics on surfaces, knot theory, continued fractions and hyperbolic
three-manifolds. Using a certain gadget called a "template" for the
modular surface, found by Ghys, it is possible to obtain an upper
bound for the volume of a geodesic (or its complement in the unit
tangent bundle) in terms of its length. This is joint work with Maxime
Bergeron and Lior Silberman.
Let $A$ and $B$ be rational functions on the Riemann sphere. The function $B$ is said to be semi-conjugate to the function $A$ if
there exists a non-constant rational function $X$ such that
$$
A\circ X=X\circ B. (*)
$$
The semi-conjugacy condition generalises both the classical conjugacy relation and the commutativity condition. In the talk we present a description of solutions of functional equation (*) in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere, and discuss numerous relations of this equation with complex dynamics and number theory.
Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Do these angles exhibit order or randomness? I will discuss the statistics of these angles and present a conjecture, motivated by a random matrix model and by function field considerations.
Do “chaotic” waves spread out randomly, or can they
concentrate near a point? In the 70s, Berard gave non-trivial bounds
for the sup-norm of a Laplace eigenfunction on a manifold of negative
sectional curvature; though far from the conjectured bounds for
surfaces of negative curvature, and those predicted by the random-wave
model, the bound has not been improved on since. Recently, Hassel and
Tacy extended Berard’s result to L^p norms, for all p>6.
In this talk we will focus on the analogous problem for large regular
graphs, and show how to get estimates analogous to Berard and
Hassel-Tacy, for all p>2. We will also discuss how the methods can be
applied to get Hassel-Tacy bounds for joint eigenfunctions on the
sphere. This is joint work with E. Le Masson.
I will talk about the Eliashberg-Gromov theorem on C^0 rigidity of symplectic diffeomorphisms, and its extensions obtained recently in the framework of C^0 symplectic geometry.
Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an 'enlarged' copy H^+ of a fixed hypergraph H. These include well-known problems such as the Erdos-Sos 'forbidding one intersection' problem and the Frankl-Furedi 'special simplex' problem.
In this talk we present a general approach to such problems, using a 'junta approximation method' that originates from analysis of Boolean functions. We prove that any (H^+)-free hypergraph is essentially contained in a 'junta' -- a hypergraph determined by a small number of vertices -- that is also (H^+)-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all C<k<n/C, a complete solution of the extremal problem for a large class of H's, which includes the aforementioned problems, and solves them for a large new set of parameters. Joint work with Noam Lifshitz.
Suppose that we are given a stationary stochastic process
{X_n}_{n\in Z}. Can we model it by another stationary stochastic
process {Y_n}_{n\in Z} where Y_n can take only two values? In 1971,
Krieger answered with an affirmative under certain natural
assumptions. It is now well-known that the analogous result holds true
for modelling stationary random fields {X_n}_{n\in Z^d} as well. What
if we now constrain the stationary stochastic process {Y_n}_{n\in Z^d}
to take only three values such that adjacent values are distinct?
Along with Tom Meyerovitch, we find that this is true thereby
answering a question of Şahin and Robinson. No background in
stochastic processes or ergodic theory will be assumed.
We consider the task of coloring the vertices of a large discrete box in the integer lattice Z^d with q colors so that no two adjacent vertices are colored the same. In how many ways can this be done? How does a typical coloring look like? What is the proportion of proper colorings in which two opposite corners of the box receive the same color? Is it about one in q?
We discuss these questions and the way their answers depend on the dimension d and the number of colors q, presenting recent results with Yinon Spinka.
Motivations are provided from statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice.
A classical result of Lojasiewicz says that a bounded gradient flow trajectory of a real analytic function converges to a unique limit. I will discuss an analogous result for maps from a Riemann surface into a symplectic manifold that satisfy the non-linear Cauchy-Riemann equation with real analytic Lagrangian boundary conditions. The proof relies on an isoperimetric inequality that controls the singularities of real analytic Lagrangian intersections.
The Floer cohomology of a pair of Lagrangian submanifolds is defined using solutions of the non-linear Cauchy-Riemann equation, and depends in general on the global geometry of the ambient symplectic manifold. However, as a consequence of our result and Gromov's compactness theorem, we see that in certain situations, the Floer cohomology of a pair of Lagrangian submanifolds is a local invariant. This fits nicely with conjectures relating Floer cohomology and algebraic invariants of singular Lagrangian intersections arising from deformation quantization and perverse sheaves.
No background in symplectic geometry will be assumed. This talk is based on joint work with M. Verbitsky.
Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Their study originated in
the groundbreaking work of J.L. Taylor on noncommutative spectral theory in the 1970s, but it is mostly in the last decade that the theory established itself as a new and active research area. I will survey some aspects of these developments, including (if time permits) recent work on interpolation and extension problems. The talk will be aimed at a general mathematical audience and should be accessible for graduate students (or even advanced undergraduates).
Many properties of rational functions f arising from problems in number theory, dynamics, and complex analysis, can be studied by writing f as a composition f_1 o ... o f_r, where the f_i's are indecomposable rational functions, i.e. cannot be decomposed nontrivially further. However basic questions such as determining the relationship between two such decompositions of f remain unknown. We shall describe progress towards a description and its applications to various problems.
Bacterial swarming is a collective mode of motion in which cells migrate rapidly over surfaces. Swarming is typically characterized by densely packed groups moving in irregular, yet coherent patterns of whirls and flows.
Analysis of individual cell trajectories within dense swarms reviles that the interplay between the single cell motion and the collective flow results in chaotic dynamics. Moreover, trajectories are consistent with Lévy walks – random processes in which the Gaussian central limit theorem fails. A model suggests a new route in which Lévy walking can result from chaotic dynamics.
The talk will explain these observations – no prior knowledge is required. More generally, I will try to convey how the phenomenon of collective bacterial movement draws from and can contribute new ideas to a range of mathematical subjects such as stochastic processes, hydrodynamics and dynamical systems.
Joint work with Avraham Be'er (BGU) and Andy Reynolds (Rothamsted Research, UK).
A Hausdorff topological group $(G, \tau)$ is called minimal if there exists no Hausdorff group topology on $G$ which is strictly coarser than $\tau$.
We say that a topological group $G$ is hereditarily minimal, if every subgroup of $G$ is minimal.
By Prodanov's Theorem an infinite compact abelian group $K$ is isomorphic to $\Z_p$ (p-adic integers) for some prime $p$ if and only if $K$ is hereditarily minimal.
We study hereditarily minimal groups. The following theorem is one of our main results.
Theorem
Let $G$ be an infinite hereditarily minimal locally compact group that is either compact or locally solvable. Then $G$ is either center-free or isomorphic to $\Z_p$, for some prime $p$.
In particular,
Corollary
If $G$ is an infinite hereditarily minimal locally compact nilpotent group, then $G$ is isomorphic to $\Z_p$ for some prime $p$.
This is a joint work with D. Dikranjan, D. Toller and W. Xi.
Small cancellation groups and their generalizations are used for constructing groups with various exotic properties. The theory of small cancellation groups can be developed both geometrically (via van Kampen diagrams) and combinatorially. Van Kampen diagrams for small cancellation groups display negative curvature features. For the combinatorial approach we are able to develop its ring theoretic analog
he topological KKMS Theorem is a powerful extension of Brouwer's Fixed-Point
Theorem, which was proved by Shapley in 1973 in the context of game theory.
We prove a colorful and polytopal generalization of the KKMS Theorem, and show
that our theorem implies some seemingly unrelated results in discrete geometry
and combinatorics involving colorful settings.
For example, we apply our theorem to provide a new proof of the celebrated
Colorful Caratheodory Theorem due to Barany, which asserts that if 0 is in the
convex hull of n+1 sets of points in R^n, then there exists a colorful
selection of points, one from each set, containing 0 in its convex hull. We
further apply our theorem to obtain an upper bound on the piercing numbers in
colorful collections of d-interval families (namely, d+1 families of sets in
R, every set being a union of d intervals); this generalizes results of
Tardos, Kaiser and Alon for the non-colored case. Finally, we apply our
theorem to questions regarding envy-free fair division of goods (e.g., cakes)
among a set of players.
Joint with Florian Frick.
Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Their study originated in
the groundbreaking work of J.L. Taylor on noncommutative spectral theory in the 1970s, but it is mostly in the last decade that the theory established itself as a new and active research area. I will survey some aspects of these developments, including (if time permits) recent work on interpolation and extension problems. The talk will be aimed at a general mathematical audience and should be accessible for graduate students or even advanced undergraduates
A Gaussian stationary process is a random function f:R-->R or f:C-->C,
whose distribution is invariant under real shifts, and whose evaluation at
any finite number of points is a centered Gaussian random vector.
The mathematical study of these random functions goes back at least 75 years,
with pioneering works by Kac, Rice and Wiener.
Nonetheless, many basic questions about them, such as the fluctuations of their number of zeroes,
or the probability of having no zeroes in a large region, remained unanswered for many years.
In this talk, we will provide an introduction to Gaussian stationary process and
describe how a new spectral perspective, combined with tools from harmonic, real and
complex analysis, yields new results about such long-lasting questions.
In 1687, Sir Isaac Newton established that the area cut off from an oval in $\mathbb R^2$
by a straight line never depends algebraically on the line (the question was motivated by
Kepler's law in celestial mechanics). In 1987, V. I. Arnold proposed to generalize Newton's
observation to higher dimensions and conjectured that all smooth bodies, with the exception
of ellipsoids in odd-dimensional spaces, have an analogous property. The talk is devoted to
the current status of the conjecture
Special Lecture Series on “Variations on a Theme of Borel” - 3rd Lecture
Over 60 years ago, Borel, on the basis of theorems of Mostow, conjectured a topological rigidity statement that has become central to topology. During these lectures, I will use his heuristic to suggest other statements, some true, some false, and some conjectural. Overall, this area is devoted to a profound influence that the fundamental group has on topology and geometry.
Over 60 years ago, Borel, on the basis of theorems of Mostow, conjectured a topological rigidity statement that has become central to topology. During these lectures, I will use his heuristic to suggest other statements, some true, some false, and some conjectural. Overall, this area is devoted to a profound influence that the fundamental group has on topology and geometry.
Over 60 years ago, Borel, on the basis of theorems of Mostow, conjectured
a topological rigidity statement that has become central to topology. During these lectures, I will use his heuristic to suggest other statements, some true, some false, and some conjectural. Overall, this area is devoted to a profound influence that the fundamental group has on topology and geometry.
Introductory lecture for Prof. Weinberger's lecture
Taking as departure point an article by Cameron, Gadouleau, Mitchell and Peresse on maximal
lengths of subsemigroup chains, we introduce the subsemigroup complex H(S) of a
nite semigroup
S as a (boolean representable) simplicial complex de
fined through chains in the lattice of subsemi-
groups of S. The rank of H(S) is the above maximal length minus one and H(S) provides other
useful invariants concerning the lattice of subsemigroups of S. We present a research program for
such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The
results include alternative characterizations of independence and bases, asymptotical estimates on
the number of bases, or establishing when the complex is pure or a matroid.
This is joint work with Stuart Margolis (Bar-Ilan University, Ramat Gan, Israel) and John
Rhodes (University of California, Berkeley, USA).
Algebras with Polynomial Identities is a well
developed theory with strong Israeli roots. We will discuss a group theoretic analog of this theory.
. One of the primary goals of number theory is to understand the absolute Galois group of the rational numbers. An important goal of the Langlands Program, is to understand the finite dimensional representations of this group, using automorphic representations. In the other direction, passing from the Galois side to the automorphic setting, this can be regarded as an arithmetic parametrization of local and automorphic representations. This parametrization, which is conjectural in part, predicts a transfer or lifting of local and automorphic representations between two reductive algebraic groups.
The global lift has been established in the celebrated work of Cogdell, Kim, Piatetski-Shapiro and Shahidi for automorphic representations admitting a certain Fourier functional, using the Converse Theorem; and in general by Arthur and by Mok, using the trace formula.
In a joint work with Cai and Friedberg, we present a new proof of functoriality, using integral representations, which generalizes the work of Cogdell et al. to arbitrary automorphic representations. This proof is based on our recent collaboration with Ginzburg, where we generalized the classical doubling method. It is expected to have further applications to the problems of descent and to covering groups.
In this talk I will first explain the basic ideology behind geometric group theory: How and to what extend can we understand (finitely-generated) groups as geometric objects? I will discuss the classical Schwarz-Milnor lemma which provides a translation mechanism between groups and geometry. In particular I will discuss a certain class of isometric actions called geometric actions. I will then explain that the Schwarz-Milnor machinery not only applies to isometric actions, but also to quasi-isometric quasi-actions of groups, and try to convince you that this is actually the more natural context of modern geometric group theory.
In the final part of my talk, I will discuss some very recent developments which show that one can not only “quasify" the notion of an isometric action but also the notion of a group itself. This allows us to not only interpret groups, but also more general algebraic structures called approximate groups as geometric objects. Time permitting I will comment on various algebraic, geometric and analytic aspects of approximate groups. This final part is based on joint work with Michael Björklund and Matthew Cordes.
No prior knowledge of geometric group theory is required and large parts of the talk should be understandable to master and PhD students.
In this talk I will first explain the basic ideology behind geometric group theory: How and to what extend can we understand (finitely-generated) groups as geometric objects? I will discuss the classical Schwarz-Milnor lemma which provides a translation mechanism between groups and geometry. In particular I will discuss a certain class of isometric actions called geometric actions. I will then explain that the Schwarz-Milnor machinery not only applies to isometric actions, but also to quasi-isometric quasi-actions of groups, and try to convince you that this is actually the more natural context of modern geometric group theory.
In the final part of my talk, I will discuss some very recent developments which show that one can not only “quasify" the notion of an isometric action but also the notion of a group itself. This allows us to not only interpret groups, but also more general algebraic structures called approximate groups as geometric objects. Time permitting I will comment on various algebraic, geometric and analytic aspects of approximate groups. This final part is based on joint work with Michael Björklund and Matthew Cordes.
No prior knowledge of geometric group theory is required and large parts of the talk should be understandable to master and PhD students.
The study of affine crystallographic groups has a long history which goes back to Hilbert's 18th problem. More precisely Hilbert (essentially) asked if there is only a finite number, up to conjugacy in Aff(R^n) of crystallographic groups G acting isometrically on R^n. In a series of papers Bieberbach showed that this was so. The key result is the following famous theorem of Bieberbach. A crystallographic group G acting isometrically on the n-dimensional Euclidean space R^n contains a subgroup of finite index consisting of translations. In particular, such a group G is virtually abelian, i.e. G contains an abelian subgroup of finite index. In 1964 Auslander proposed the following conjecture
The Auslander Conjecture: Every crystallographic subgroup G of Aff(R^n) is virtually solvable, i.e. contains a solvable subgroup of finite index.
In 1977 J. Milnor stated the following question:
Question: Does there exist a complete affinely flat manifold M such that the fundamental group of M contains a free group?
We will explain ideas and methods, recent and old results related to the above problems.
The study of affine crystallographic groups has a long history which goes back
to Hilbert's 18th problem. More precisely Hilbert (essentially) asked if there is only a finite number, up to conjugacy in \text{Aff}$(\mathbb{R}^n)$, of crystallographic groups $\G $ acting isometrically on $\mathbb{R}^n$. In a series of papers Bieberbach showed that this was so. The key result is the following famous theorem of Bieberbach. A crystallographic group $\G$ acting isometrically on the $n$--dimensional Euclidean space $\mathbb R^n$ contains a subgroup of
finite index consisting of translations. In particular, such a group $\Gamma$ is virtually abelian, i.e. $\Gamma$ contains an
abelian subgroup of finite index.
In 1964 Auslander proposed the following conjecture \bigskip \\
\pro {\it The Auslander Conjecture.} Every crystallographic subgroup $\Gamma$ of \text{Aff}$(\mathbb{R}^n)$
is virtually solvable, i.e. contains a solvable subgroup of finite
index. \endpro\\
In 1977 J. Milnor stated the following question:\\
\pro {\it Question.} Does there exist a complete affinely flat manifold $M$ such that $\pi_1(M) $ contains a free group ? \endpro \\
We will explain ideas and methods, recent and old results related to the above problems.
We will give an overview of questions one might ask about the first-order theory of free groups and related groups: how much information can first-order formulas convey about these groups or their elements, what algebraic interpretation can be given for model theoretic notions. It turns out that techniques from geometric group theory are very useful to tackle such problems. We will assume no special knowledge of model theory.
Recently considerable attention has been paid to the study of arithmetic sums of two planar sets A+G:={a+g: a in A, g in G}. We focus on the case when G is a piecewise C^2 curve, in particular when G is the unit circle. In this case there is a natural guess what the size (Hausdorff dimension, Lebesgue measure) of A+G should be. We verify it under some simple natural assumptions. We also address the more difficult question: under which condition does the set A+G have non-empty interior?
Translation invariant valuations which are continuous in the Hausdorff metric play a special role in the theory and its applications to integral geometry. Theory of such valuations is an active topic in convexity. In recent years it was realized that the space of such valuations admits rich structures, in particular the multiplicative structure. The latter turned out to be useful in integral geometry. First I will explain some of the classical background and examples. Then I will discuss more recent results mentioned above.
"Symbolic dynamics" is a powerful technique for describing the combinatorial structure of large collections of orbits of dynamical systems with "chaotic" behaviour. I will describe this technique, and will report on recent advances on the question what sort of "chaos" is needed to this method to succeed. The talk is meant for a general audience, including people with little or no background in dynamical systems.
Descents of permutations have been studied for more than a century. This concept was vastly generalized, in particular to standard Young tableaux (SYT). More recently, cyclic descents of permutations were introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding concept for SYT, Rhoades found a very elegant solution for rectangular shapes. In an attempt to extend the concept of cyclic descents, explicit combinatorial definitions for two-row and certain other shapes have been found, implying the Schur-positivity of various quasi-symmetric functions. In all cases, the cyclic descent set admits a cyclic group action and restricts to the usual descent set when the letter n is ignored. Consequently,
the existence of a cyclic descent set with these properties was conjectured for all shapes, even the skew ones. This talk will report on the surprising resolution of this conjecture: Cyclic descent sets exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies non-negativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants. We shall also comment on issues of uniqueness. Based on joint works with Sergi Elizalde, Vic Reiner, Yuval Roichman.
Dr. Menachem Shlossberg invites you to a “haramat kosit” in celebration of obtaining a postdoctorate
at the University of Udine, Italy, at 1:30 PM next to the Colloquium Room
ד"ר מנחם שלוסברג מזמין אתכם להרמת כוסית לרגל קבלת משרת פוסט דוקטורט באונ' אודין באיטליה
בשעה 13:30 בחדר ע"י חדר המחלקה
The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.
Model categories, introduced by Quillen, provide a very general context in which it is possible to set up the basic machinery of homotopy theory. In particalar they enable to define derived functors, homotopy limits and colimits, cohomology theories and spectral sequences to catculate them. However, the structure of a model category is usually hard to verify, and in some interesating cases even impossible to define. In this lecture I will define a much simpler notion then a model category, called a weak fibration category. By a theorem due to T. Schlank and myself, a weak fibration category gives rise in a natural way to a model category structure on its pro category, provided some technical assumptions are satisfied. This result can be used to construct new model structures in different mathematical fields, and thus to import the methods of homotopy theory to these situations. Examples will be given from the categories of simplicial presheaves, C*-algebras and complexes in Abelian categories. Applications will be discussed with each example.
The above encompasses joint work with Tomer M. Schlank, Yonatan Harpaz, Geoffroy Horel, Michael Joachim Snigdhayan Mahanta and Matan Prezma.
We present a new approach (joint with M. Bjorklund (Chalmers)) for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.
By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.
By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.
A plane curve is called nondegenerate if it has no inflection points.
How many classes of closed nondegenerate curves exist on a sphere?
We are going to see how this geometric problem, solved in 1970, reappeared along with its generalizations in the context of the Korteweg-de Vries and Boussinesq equations. Its discrete version is related to the 2D pentagram map defined by R. Schwartz in 1992.
We will also describe its generalizations, pentagram maps on polygons in any dimension and discuss their integrability properties.
While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis.
A simple example of an incidences problem: Given a set of n points and set of n lines, both in R^2, what is the maximum number of point-line pairs such that the point is on the line. Studying incidence problems often involves the uncovering of hidden structure and symmetries.
In this talk we introduce and survey the topic of geometric incidences, focusing on the recent polynomial techniques and results (some by the speaker). We will see how various algebraic and analysis tools can be used to solve such combinatorial problems.
In the talk I will discuss classical problems concerning the distribution of square-full numbers and their analogues over function fields. The results described are in the context of the ring Fq[T ] of polynomials over a finite field Fq of q elements, in the limit q → ∞.
I will also present some recent generalization of these kind of classical problems.
Hecke algebras H_q(W) of Coxeter groups W first emerged in the study of Chevalley groups in mid sixties and since then became central objects in Representation Theory of Coxteter groups and semisimple Lie groups over finite fields. In particular, as a one-parameter deformation of the group algebra kW of W, the Hecke algebra H_q(W) helps to classify representations of W and to equip each simple kW-module with the canonical Kazhdan-Lusztig basis.
Unfortunately, unlike the group algebra kW, the Hecke algebra H_q(W) lacks a Hopf algebra structure, that is, it is not clear how to tensor multiply H_q(W)-modules. Moreover, there is a general consensus that a naive Hopf structure on H_q(W), if exists, would essentially coincide with that on kW, so we would not gain any new information.
In my talk (based on joint work with D. Kazhdan) I suggest a roundabout: instead of forcing a naive Hopf structure on H_q(W), we find a ``reasonably small" Hopf algebra H(W) (we call it Hecke-Hopf algebra of W) that "naturally" contains H_q(W) as a coideal subalgebra.
The immediate benefit of this enlargement of H_q(W) is that each representation of H(W) and each representation of H_q(W) can be tensor multiplied into a new representation of H_q(W), thus allowing to create infinitely many new H_q(W)-modules out of a single one.
Hecke-Hopf algebras have some other applications, most spectacular of which is the construction of new infinite families of solutions to the quantum Yang-Baxter equation.
The theory of selection principles deals with the possibility of obtaining mathematically significant objects by selecting elements from sequences of sets. The studied properties mainly include covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially functions spaces. Often, the characterization of a mathematical property using selectionprinciple is a nontrivial task leading to new insights on the characterized property.
I will give an overview of this theory and, if time permits, present some resent results obtained jointly with Boaz Tsaban and Lyubomyr Zdomskyy.
We know by classical Fourier analysis that the unit cube in R^d has an orthogonal basis consisting of exponential functions. Which other domains admit such a basis? Fuglede conjectured (1974) that these so-called "spectral domains" could be characterized geometrically by their possibility to tile the space by translations. I will survey the subject and then discuss some recent results, joint with Rachel Greenfeld, where we focus on the conjecture for convex polytopes.
We explore results of Ramsey theory (also known as partition calculus) and show how they apply to cardinals, ordinals, trees, and arbitrary partial orders, leading up to the main result which is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem.
A full exposition of the results is contained in my PhD thesis, available at http://hdl.handle.net/1807/68124.
Hilbert’s fourteenth problem asks whether the algebra of invariants for an action of a linear algebraic group is finitely generated.
This is true for reductive groups and the problem is open for unipotent groups. We discuss the case of the adjoint action of a maximal unipotent subgroup U in GL_n(K) on the nilradical m of any parabolic subalgebra, where K is an algebraically closed field of zero characteristic. This action is extended to a representation in the algebra K[m]. I will show that the algebra of invariants K[m]^U is finitely generated. Besides, a set of algebraically independent invariants generating the field K(m)^U will be presented.
Abstract is attached.
We consider the issue of generalized stochastic processes, indexed by an abstract set of indices. What should the minimal required conditions on the indexing collection be, to study some of the usual properties of these processes, such as in- crement stationarity, martingale and Markov properties or integration question? The already known examples of processes indexed by functions or metric spaces can be addressed by this way.
We show how the set-indexed framework of Ivanoff-Merzbach allows to study these generalized processes.
Some set-indexed processes can be considered as random measures on some δ- ring. Some generalized processes can be defined as an integral with respect to some measure on the indexing collection. The example of set-indexed Lévy processes is considered. The links with function-indexed processes could be discussed.
If time permits, we could also discuss regularity issue : continuity or Hölder regularity.
This talk is based on works in collaboration with Ely Merzbach and Alexandre Richard.
Recall that the real line is that unique separable, dense linear ordering with no endpoints in which every bounded set has a least upper bound.
Around the year of 1920, Souslin asked whether the term *separable* in the above characterization may be weakened to *ccc*. (A linear order is said to be separable if it has a countable dense subset. It is ccc if every pairwise-disjoint family of open intervals is countable.)
Amazingly enough, the resolution of this single problem led to many key discoveries in set theory. Also, consistent counterexamples to this problem play a prominent role in infinite combinatorics.
In this talk, we shall tell the story of the Souslin problem, and report on an advance recently obtained after 40 years of waiting.
Jordan algebras J of charateristic not 2 sometimes contain
a set of idempotents (e^2=e) that generate J such that their adjoint
map ad_e: u \mapsto ue (u\in J) has the minimal polynomial
x(x-1)(x-1/2), and with additional restrictions on products
of elements in the eigenspaces of ad_e (for each e).
Generalizing these properties (not only of such Jordan
algebras) Hall, Rehren, Shpectorov (HRS) introduced ``Axial algebras
of Jordan type''. In my talk I will present structural results
on Axial algebras of Jordan type 1/2 (a case which was not
dealt with in HRS), I will discuss their idempotents e, the corresponding
``Miyamoto involutions'' \tau(e) and the group that these involutions
generate.
This is joint work with J. Hall, S. Shpectorov.
Corson (1961) started a systematic study of certaintopological properties of the weak topology w of Banach spaces E. This
line of research provided more general classes such as reflexive
Banach spaces, Weakly Compactly Generated Banach spaces and the class
of weakly K-analytic and weakly K-countably determined Banach spaces.
On the other hand, various topological properties generalizing
metrizability have been studied intensively by topologists and
analysts. Let us mention, for example, the first countability,
Frechet-Urysohn property, sequentiality, k-space property, and
countable tightness. Each property (apart the countable tightness)
forces a Banach space E to be finite-dimensional, whenever E with the
weak topology w is assumed to be a space of the above type. This is a
simple consequence of a theorem of Schluchtermann and Wheeler that an
infinite-dimensional Banach space is never a k-space in the weak
topology. These results show also that the question when a Banach
space endowed with the weak topology is homeomorphic to a certain
fixed model space from the infinite-dimensional topology is very
restrictive and motivated specialists to detect the above properties
only for some natural classes of subsets of E, e.g., balls or bounded
subsets of E. We collect some classical and recent results of this
type, and characterize those Banach spaces E whose unit ball B_w is
k_R-space or even has the Ascoli property. Some basic concepts from
probability theory and measure theoretic properties of the space l_1
will be used.
A finitely generated group $G$ has only a finite number, say $a_n(G)$, of subgroups of any given index $n$. The study of subgroup growth, i.e. of the behavior of this sequence, has been an active area of research for several decades. A variant problem investigates the sequence $a_n^\wedge (G)$ counting subgroups of index $n$ whose profinite completion is isomorphic to that of the original group $G$, and in particular the analytic properties of the Dirichlet series derived from this sequence.
The expansion condition in Hall's marriage theorem can be extended to an unbiased 2-sided one.
This enables an alternative (and simpler) proof of Evans' (proven) Conjecture:
A partial nxn Latin square with n-1 dictated entries admits a completion to a full Latin square.
PMs are used to successively fill the square by rows, columns or diagonals. Latin square tables correspond to quasi-groups; the ones corresponding to groups are only a tiny fraction of them, as n grows. However, for Sudoku tables of order mnxmn, the completion (say by diagonals) usually fails, even if there are no dictated entries, unless they are conjugates of a twisted product of two groups, of orders n and m.
We shall discuss the Chirikov standard map, an area-preserving map of the torus to itself in which quasi-periodic and chaotic dynamics are believed to coexist. We shall describe how the problem can be related to the spectral properties of a one-dimensional discrete Schroedinger operator, and present a recent result.
Based on joint work with T. Spencer.
We define refined tropical enumerative invariants counting plane tropical curves of a given degree and a given positive genus and having marked points on edges and at vertices. This extends Block-Goettsche and Goettsche-Schroeter refined tropical invariants. As a consequence we obtain tropical (complex) descendant invariants and (real) broccoli invariants of positive genus.
(Joint work with F. Schroeter.)
We consider both time series as well as spatial distributions (in 1-4 dimensions). In the first, we observe that time series for individual and independently deviating random variables can manifest pattern through the emergence of peak-to-peak sequences that are visible to the eye yet fail all Fourier analysis schemes and reveal a seeming periodicity of 3-events per cycle. We note that this can explain observations of apparent cycles in mammalian animal populations. We consider models, as well, based on the Langevin equation of kinetic theory and the Smolouchowski relation that present circumstances where the apparent period can vary from 3-4 and, for a special subclass of problems, to periods between 2 and 3. We explore how cataloged observational data from global earthquake catalogues, magnetospheric AL index observations, Old Faithful Geyser eruption data, and the performance of the Standard & Poor's 500 index (percent daily variation) manifest different degrees of statistical agreement with the theory we derived. We present a simple model for many mammalian population cycles whose underlying phenomenological basis has strong biological implications. We then employ directed graphs to explore nearest-neighbor relationships and isolate the character of spatial clustering in 1-4 dimension. We observe that the one-dimensional problem is formally equivalent to that presented by peak-to-peak sequences in time series and also demonstrates a mean number of points per cluster of 3 in one dimension. We then take the first moment of each of the clusters formed, and observed that they too form clusters. We observe the emergence of a hierarchy of clusters and the emergence of universal cluster numbers, analogous to branching ratios and, possibly, Feigenbaum numbers. These, in turn, are related to fractals as well as succularity and lacunarity, although the exact nature of this connection has not been identified. Finally, we show how hierarchical clustering emerging from random distributions may help provide an explanation for observations of hierarchical clustering in cosmology via the virial theorem and simulation results relating to the gravitational stabilization in a self-similar way of very large self-gravitating ensembles.
These symmetry patterns are described by the absolute Galois group of the field, whose
structure is in general still a mystery.
We will describe what is known about this symmetry group: classical facts, consequences
of the epochal work by Veovodsky and Rost, and very recent structural results and conjectures
related to higher cohomology operations and intersection theorems.
Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers' question. The second proof is direct and it has nice applications in combinatorics. The talk will be accessible to a general audience. [This is a joint work with Omri Solan and Barak Weiss].
In 1846, Arthur Cayley defined a correspondence between orthogonal
matrices of determinant one and skew-symmetric matrices. This
observation was a starting point of a long (and yet unfinished)
story. In the talk we will overview its highlights, with a focus on
the achievements obtained during the past decade and some open
problems.
Let mu_{m,n} be the canonical invariant measure on the Grassmann manifold
of m-dimensional subspaces in C^{m+n}; the flat coordinates on the Grassmann
manifold allow us to consider mu_{m,n} as a measure on the space Mat(m x n) of
complex matrices. By definition, the family of measures mu_{m,n} has
the property of consistency under natural projections
Mat((m + 1) n) ---> Mat(m n) ; Mat(m x (n + 1)) ---> Mat(m x n)
and consequently defines a probability measure on the space Mat of infinite
complex matrices. The measure mu is by definition unitarily-invariant and admits
a natural one-parameter family of unitarily-invariant deformations mu^(s), called
the Pickrell measures. The Pickrell measures are finite for s > -1 and infinite
for s < 0.
The first main result of the talk is the solution to the problem, posed by
Borodin and Olshanski in 2000, of the explicit description of the ergodic decomposition of infinite Pickrell measures. The decomposing measures are naturally identified with sigma-finite processes on the half-line R+ and can be viewed as sigma-finite analogues of determinantal point processes. For different values of the parameter s, these measures are mutually singular.
In the second part of the talk we will discuss absolute continuity and singularity of determinantal point processes. The main result here is that determinantal point processes on Z induced by integrable kernels are indeed quasi-invariant under the action of the in nite symmetric group. The Radon-Nikodym derivative is found explicitly. A key example is the discrete sine-process of Borodin, Okounkov and Olshanski. This result has a continuous counterpart: namely, that
determinantal point processes with integrable kernels on R, a class that includes processes arising in random matrix theory such as the sine-process, the process with the Bessel kernel or the Airy kernel, are quasi-invariant under the action of the group of di eomorphisms with compact support.
The first part of the talk is based on the preprint http://arxiv.org/abs/1312.3161;
the second part, on the preprint http://arxiv.org/abs/1409.2068.
In 1990 Helmut Hofer introduced a bi-invariant metric on symplectomorphism
groups which nowadays plays an important role in symplectic topology and Hamiltonian dynamics.
I will review some old, new and yet unproved results in this direction.
Although higher structures have been around for quite some time, they recently have come back into focus through renewed interest in higher categories. There are several reasons for this.
In geometry one is trying to interpret extended cobordism theories, where the higher structures are meant to mimic higher codimensions. An analogue in algebra is known to the 2-categorical level, the prime example being the 2-category of rings, bi-modules and bi-module morphisms. Beyond this there are many open questions of fundamental nature. The central problem is what type of coherence to require.
In physics higher structures naturally appear in two related fashions. The first is through the extended field theories and the second through field theories with defects. This is mathematically mimicked by cobordisms and defect lines and points abstractly interpreted as inclusions into higher dimensional objects.
The "truncated" versions of higher structures can be assembled into infinity up to a homotopy everything version. This is the setting of the influential program of Lurie which provides firm foundations to derived algebraic geometry, and, hopefully, to higher differential geometry which is not yet that well established.
Geometric and physical points of view combine in the constructions of string topology and in the proofs of the cobordism hypothesis. One approach to this, which will also be an integral part of this program, is the operadic/monadic point of view as many liigher categorical structures can be interpreted as actions of certain liigher dimensional operads/monads. The classical homotopy theory teaches us that this is the correct way to encode higher homotopies and homotopical algebra in general.
The complexity of higher dimensional structures and necessity to work with them efficiently has required reconsideration of the foundations of mathematics. A new theory called univalent foundations or homotopy type theory emerges in recent years which has a potential to become a common language for mathematicians working with higher categorical structures. We wish to include this theory as a supplement to our main topics, but also as a possible future direction of research.
The Cerny conjecture, concerned with the minimal length of a reset word in a finite automata, is considered one of the most longstanding open problem in the theory of finite automata. In this talk, we discuss the background of the conjecture, attempts at a proof, and partial results obtained so far by various researchers. In the second part, we present our recent results, which shade a light on the question of why the conjecture is so hard to prove.
Suppose a light source is placed in a polygonal hall of mirrors (so light can bounce off the walls). Does every point in the room get illuminated? This elementary geometrical question was open from the 1950s until Tokarsky (1995) found an example of a polygonal room in which there are two points which do not illuminate each other. Resolving a conjecture of Hubert-Schmoll-Troubetzkoy, in joint work with Lelievre and Monteil we prove that if the angles between walls is rational, every point illuminates all but at most finitely many other points. The proof is based on recent work by Eskin, Mirzakhani and Mohammadi in the ergodic theory of the SL(2,R) action on the moduli space of translation surfaces. The talk will serve as a gentle introduction to the amazing results of Eskin, Mirzakhani and Mohammadi.
Non-Archimedean analytic geometry is an analog of complex analytic geometry over non-Archimedean (e.g., p-adic) fields. In the talk, I'll explain what non-Archimedean analytic spaces are, list basic facts about them, and tell about their applications
Hyperbolic groups can be defined through the geometry of Cayley graphs, viewed as geodesic metric spaces. One important feature of hyperbolic groups is the concept of boundary, which can be defined through the topological completion for an appropriate metric (such as the visual metrics), and has the advantages of compactness. An endomorphism of a hyperbolic group admits a continuous extension to the boundary if and only if it is uniformly continuous with respect to a visual metric, and a Hölder condition is a particularly nice way of achieving uniform continuity. In joint work with Vítor Araújo (Universidade Federal da Bahia), we have proved that an endomorphism of a hyperbolic group satisfies a Hölder condition with respect to a visual metric if and only if it is virtually injective and its image is a quasi-convex subgroup. Moreover, if the group is virtually free or torsion-free co-hopfian, then the endomorphism is uniformly continuous if and only if it satisfies a Hölder condition if and only if it is virtually injective. However, this stronger claim does not necessarily hold for arbitrary hyperbolic groups.
Cooperative interactions, their stability and evolution, provide an interesting context in which to study the interface between cellular and population levels of organization. Such interactions also open the way for the discovery of new population dynamics mechanisms.
We have studied a version of the public goods model relevant to microorganism populations actively extracting a growth resource from their environment. Cells can display one of two phenotypes – a productive phenotype that extracts the resources at a cost, and a non-productive phenotype that only consumes the same resource. We analyze the continuous differential equation model as well as simulate stochastically the full dynamics. It is found that the two sub-populations, which cannot coexist in a well-mixed environment, develop spatio-temporal patterns that enable long-term coexistence in the shared environment. These patterns are solely fluctuation-driven, since the continuous system does not display Turing instability. The average stability of the coexistence patterns derives from a dynamic mechanism in which one sub-population holds the environmental resource close to an extinction transition of the other, causing it to constantly hover around its critical transition point, forming a mechanism reminiscent of selforganized criticality. Accordingly, power-law distributions and long-range correlations are found.
When a time scale separation occurs between two dynamic parameters is defined, a structurally unstable point emerges and any small perturbation of the dynamics with additive noise leads to an equilibrium distribution in which both species coexist in context of additive but not multiplicative noise.
For three quarters of a century Linear Programming (LP) was the main tool for solving resource allocation problems (RAP)- one of the main problem in economics.
In 1975 L. V. Kantorovich and T. C. Koopmans shared the Nobel Prize in Economics Nonlinear Equilibrium vs. Linear Programming for resource allocation problems.“for their contributions to the theory of optimum allocation of limited resources."
When LP is used for RAP the prices for goods and the resource availability are given a priori and independent on the production output and prices for the resources. It often leads to solutions, which are not practical, because they contradict to the basic market law of supply and demand.
We consider an alternative to LP approach to RAP, which is based on Nonlinear Equilibrium (NE). The NE is a generalisation of Walras-Wald equilibrium, which is equivalent to J Nash equilibrium in n-person concave game.
NE eliminates the basic drawbacks of LP. Finding NE is equivalent to solving a variation inequality (VI) on the Cartesian product of the primal and dual non negative octants, projection on which is a very simple operation. For solving the VI we consider two methods: projected pseudo-gradient (PPG) and extra pseudo-gradient (EPG), for which projection is the main operation at each step.
We established convergence, proved global Q-linear rate and estimated complexity of both methods under various assumptions on the input data.
Both PPG and EPG can be viewed as pricing mechanisms for establishing economic equilibrium.
Let f be a power series (in several variables) or a C^\infty-smooth function. In many cases just a finite part of Taylor expansion is enough to determine f up to the change of coordinates. Alternatively, the deformations of f by terms of high enough orders are trivial. This phenomenon is called the finite determinacy.
An immediate application is the algebraization: f has a polynomial representative.
More generally, for maps of smooth spaces the finite determinacy (under various group-actions) has been intensively studied for about 50 years (by Mather, Tougeron, Arnol'd, Wall and many others).
The chip ring game Bjorner, Lovasz and Shor (BLS) introduced the following game in 1991: N chips are placed on the vertices on a n-vertex graph and at every turn, the solitaire player chooses a vertex i of degree di which has at least di chips on it and "fires" i by shifting a chip from i to each of i's neighbours.
The game duration problem BLS have proved the remarkable result that whenever this game terminates, it always does so in the same number of moves, irrespective of gameplay! (I will explain the background for this). They also gave an elegant upper bound on the number of moves. However, computer simulation reveals that the game actually ends in far fewer moves than the BLS bound in all examined cases.
The new results I will show a new approach to obtaining upper bounds on the game duration, based on a re nement of the classic BLS analysis together with a simple but potent new observation.
The new bounds are always at least as good as the BLS bound and in some cases the improvement is dramatic. For example, for the strongly regular graphs BLS reduces to O(nN) while the new bound reduces to O(n+N). For dense regular graphs BLS reduces to O(N) while the new bound reduces to O(n) (for such it holds that n = O(N)).
The proof technique involves a careful analysis of the pseudo-inverse of the graph's discrete Laplacian.
The wider context Time permitting, I will also discuss the appearance of chip ring (and its very close relative, the sandpile model) in diverse mathematical and scientific contexts.
The Inverse Galois Problem, asking which groups can be realizable as
Galois groups of fields, is a major problem in Galois theory.
For example the fact that there is no general formula for the roots of
a polynomial of degree five follows from the fact that
the symmetric group S_5, which is not solvable, is realizable as a
Galois group of a field.
Minac and Tan conjectured that if G is the Galois group of a field,
then G has vanishing triple Massey products (to be defined in the lecture).
In the talk I will give some general background on this new property
and its relation to the inverse Galois problem via a work of Dwyer, and try to give a
flavor of my proof of the Minac-Tan conjecture.
Ergodic theory studies actions of a group G by measure preserving transformations on a probability space. Usually the focus is on "essentially free" actions, namely actions for which almost all stabilizers are tirival. Classically the methods are analitic and combinatorial.
Recently it becomes more and more clear that in the study of non essentially free actions - sophisticated group theoretic tools also come into the picture. I will try to demonstrate this by an array of recent results due to Bader-Lacreux-Duchnese, Tucker-Drob, as well as some joint papers with Abert and Virag and myself.
בשעה 11:15 יערך טכס חלוקת הפרסים לזוכים בתחרות בר-אילן במתמטיקה לסטודנטים
Light refreshments will be served at 11:45 AM next to the Colloquium Room
כיבוד קל יוגש לפני ההרצאה בשעה 11:45 בחדר ע"י חדר המחלקה
The subject of this talk is the analysis of pure point distributions that have a pure point spectrum.
It will be discussed in the framework of “quasi-crystals" inspired by the experimental discovery in the middle of the '80s s of non-periodic atomic structures with diffraction patterns consisting of spots.
Based on joint work with Alexander Olevskii
~~ It is well known that the generating function of a generic finitely presented algebra is rational.
The purpose of this talk is to present an answer on a similar question in the case of algebraic operads.
Namely, I will show that the generating series of a generic nonsymmetric operad is an algebraic function
and the generating series of a generic symmetric operad is differentially algebraic.
Despite the motivation coming from the operad theory, a substantial part of the talk will only deal with the avoidance problems for (labeled) rooted trees hence will be accessible to nonspecialists.
based on joint work with D.Piontkovsky (arXiv:1202.5170).
~~One of the major achievements of statistical mechanics is the development of theoretical tools to bridge between the microscopic description of a system and its observed macroscopic behavior, tracking the emergence of large-scale phenomena from the mechanistic description of the system’s interacting components. A key factor in determining this emergent behavior is associated with the underlying geometry of the system’s interactions - a natural notion when treating structured systems, yet difficult to generalize when approaching complex systems. Indeed, social, biological and technological systems feature highly random and non-localized interaction patterns, which challenge the classical connection between structure, dimensionality and dynamics, and hence confront us with a potentially new class of dynamical behaviors. To observe these behaviors we developed a perturbative formalism that enables us to predict an array of pertinent macroscopic functions directly form the microscopic model describing the system’s dynamics. We find that while microscopically complex systems follow diverse rules of interaction, their macroscopic behavior condenses into a discrete set of dynamical universality classes.
Relevant papers:
Universality in network dynamics. Nature Physics 9, 673–681 (2013) doi:10.1038/nphys2741
Network link prediction by global silencing of indirect correlations. Nature Biotechnology 31, 720–725 (2013) doi:10.1038/nbt.2601
~~We compare several growth models on the two dimensional lattice. In some models, like internal DLA and rotor-router aggregation, the scaling limits are universal; in particular, starting from a point source yields a disk. In the abelian sandpile, particles are added at the origin and whenever a site has four particles or more, the top four particles topple, with one going to each neighbor. Despite similarities to other models, for the sandpile, the intriguing pattern that arises is not circular and depends on the particular lattice. A scaling limit exists for the sandpile, as was recently shown by Pegden and Smart, but it is not universal and still mysterious. This research has been greatly influenced by pictures of the relevant sets, which I will show in the talk. They suggest a connection to conformal mapping which has not been established yet.
Talk based on joint works with Lionel Levine
~~
In this talk we will present a model based on random walks for the density of subsets in finitely generated groups.
The main focus will be on the group large sieve method which is a tool for estimating the density by investigating the finite quotients of the group. We will describe applications of this method for linear groups as well as for mapping class groups.
~~
The concept of group duality is fundamental in the analysis of locally compact abelian groups.
The theory of (analytic) quantum groups was developed in order to provide a framework for duality of general locally compact groups.
The simple set of axioms describing "locally compact quantum groups" (LCQGs) introduced in '00 by Kustermans and Vaes is built on preceding, deep works of Kac and Vainerman, Enock and Schwartz, Woronowicz, Baaj and Skandalis, Masuda and Nakagami and many others.
LCQGs have an intriguing structure theory, and numerous results on locally compact groups have already been generalized to LCQGs.
In this talk we will motivate and introduce the definition of LCQGs, explain and exemplify how they are constructed and mention some of their applications.
Afterwards, we shall describe a generalization of recent work on aspects of ergodic theory of semigroup actions on von Neumann algebras to the context of quantum semigroups.
These results give a Jacobs-de Leeuw-Glicksberg splitting at the von Neumann algebra level.
TBA
: I will introduce a variational problem that consists of the classic isoperimetric problem, i.e. minimization of perimeter subject to a volume constraint, perturbed by a nonlocal term modeling long-range interactions. This geometry problem is the focus of much activity these days and I will survey some results of my own and of others aimed at better understanding the rich energy landscape that emerges from the interplay between these two competing terms in the problem.
We start with a simple fact: the fundamental solutions of the Laplacian in Rn can be continued as multi-valued
analytic functions in Cn up to the complex bicharacteristic conoid. This extension ramies around the complex
isotropic cone: z2
1 + z2n
= 0 and has "moderate growth".
For an elliptic linear partial dierential operator of the second order with analytic coe-cients and simple complex characteristics in an open set Rn, this may be generalized: every fundamental solution can be continued at least locally as a multi-valued analytic function in Cn up to the complex bicharacteristic conoid.
This holomorphic extension is ramied around the bicharacteristic conoid and belongs to the so-called Nilsson
class ("moderate growth").
In fact, those results remain true for such operators with degree bigger than 4 , but the proofs are different due to the lack of natural geodesic distance associated to the operators
Those results may be connected with D-module theory, and more precisely with regular holonomic D-Modules.
We'll explain this link and state a general conjecture
Paul Cohen showed that the Continuum Hypothesis is independent of the usual axioms of set theory. His solution involved a new apparatus for constructing models of set theory - the method of *forcing*. As Cohen predicted, the method of forcing became very successful in establishing the independence of various statements from the usual axioms of set theory. What Cohen never imagined, is that forcing would be found useful in proving theorems.
In this talk, we shall present a few results in combinatorics whose proof uses the method of forcing, including our recent resolution of the infinite weak Hedetniemi conjecture.
The talk will be targeted to a general audience.
I will discuss recent progress in the problem of counting nodal domains of eigenfunctions of Laplacian (i.e., counting connected components of the complement to the zero set of real valued eigenfunctions). This in an old question taken up by Courant and his school. One of many intractable questions is under what conditions one have the number of nodal domains to be unbounded as the eigenvalue goes to infinity. The main difficulty in this problem is that it is known not to be a local property.
The example I will consider concerns with eigenfunctions of the Laplace-Beltrami operator on compact hyperbolic surfaces. The distinctive property of such a setup is its Quantum (Unique) Ergodicity (to be explained). I will discuss how this could be used in order to deduce strong bounds on eigenfunctions and how this forces the number of nodal domains to grow with the eigenvalue.
(Joint work with J. Bernstein, A. Gosh, P. Sarnak)
Wolf Prize Day 2012
A central question in the theory of random walks on groups is how symmetries
of the underlying space gives rise to structure and rigidity of the random
walks. For example, for nilpotent groups, it is known that random walks have
diffusive behavior, namely that the rate of escape, defined
as the expected distance of the walk from the identity satisfies
E|Xn|~=n^{1/2}. On nonamenable groups, on the other hand we have E|Xn|
~= n. (~= meaning upto (multiplicative) constants )-
In this work, for every 3/4 <= \beta< 1 we construct a finitely generated
group so that the expected distance of the simple random walk from its
starting point after n steps is n^\beta (up to constants). This answers a
question of Vershik, Naor and Peres. In other examples, the speed exponent can
fluctuate between any two values in this interval.
Previous examples were only of exponents of the form 1-1/2^k or 1 , and were
based on lamplighter (wreath product) constructions.
(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups)
In this lecture we will describe how a variation of the lamplighter
construction, namely the permutational wreath product, can be used to get
precise bounds on the rate of escape in terms of return probabilities of the
random walk on some Schreier graphs. We will then show how groups of
automorphisms of rooted trees, related to automata groups , can then be
constructed and analyzed to get the desired rate of escape. This is joint
work with Balint Virag of the University of Toronto.
No previous knowledge of randopm walks, automaton groups or wreath products is
assumed.
Joint work with Sebastian Neumayer, Gil Zussman and Eytan Modiano
Communication networks are vulnerable to natural
disasters, such as earthquakes or floods, as well as to physical
attacks, such as an Electromagnetic Pulse (EMP) attack. Such
real-world events happen in specific geographical locations and
disrupt specific parts of the network. Therefore, the geographical
layout of the network determines the impact of such events on
the network's connectivity. Thus, it is desirable to assess
the vulnerability of geographical networks to such disasters.
I will discuss several algorithms, based on mixed
linear planning and computational geometry, to locate such
vulnerabilities, and present some case studies on real networks.
The Jacobian conjecture is a famous open problem in affine algebraic geometry which says
that a polynomial mapping in n complex variables with constant non zero determinant is injective
and surjective witha polynomial inverse mapping.
In this talk we will outline a proof of the surjectivity for the case of n=2 An abstract is attached
A group $\Gamma$ if of type $F_k$ if it admits an
Eilenberg MacLane complex with finite k-skeleton.
For such groups one can define the (k-1)-dimensional Dehn function,
which measures the difficulty to fill (k-1)-cycles by k-chains.
I will describe the optimal higher-dimensional Dehn functions for uniform
S-arithmetic subgroups of reductive groups over global fields.
I will also discuss a conjectural picture for non-uniform S-arithmetic
groups.
The Riemann Zeta-Function is simple to define but utterly
impossible to find all its zeros. Euler looked at it a century before
Riemann to study the prime numbers and the value of the zeta function at the
integers. By 1860, many far-reaching mysteries were uncovered.
We shall describe them, as well as today's Conjectures of Langlands
and Iwasawa which are built upon them.
