# Mathematics Colloquium

Organizer(s)
Usual Time
Sunday, 12:00
Place
Room 201 in the Math Department

Upcoming Lectures
סמינרים | המחלקה למתמטיקה
Previous Lectures
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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.

- Eyal Kaplan, BIU
Eyal Kaplan, BIU
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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.

- Michael Megrelishvili, BIU
Michael Megrelishvili, BIU
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To every Banach space V one may associate a continuous dual action of the topological group Iso(V)
of all linear isometries on the weak-star compact unit ball B* of the dual space V*.
Which actions G x X --> X are "subactions" of Iso(V)  x B* --> B* for nice Banach spaces V ?
We study Banach representability for actions of topological groups on groups by automorphisms;
in particular, an action on itself by conjugations.
The natural question is to examine when we can find representations on low complexity Banach spaces.
In contrast to the standard left action of a locally compact second countable group G on itself,
the conjugation action need not be reflexively representable even for SL_2(R).
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.
On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l_1).
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.
- Eran Nevo, HUJI
Eran Nevo, HUJI
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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.

- Nathan Keller, BIU
Nathan Keller, BIU
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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

- Boris Solomyak, BIU
Boris Solomyak, BIU
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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.

- Natalia Tsilevich, St.Petersburg Dept. of Steklov Institute of Mathematics
Natalia Tsilevich, St.Petersburg Dept. of Steklov Institute of Mathematics
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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.

- Boris Kunyavskii  (BIU)
Boris Kunyavskii  (BIU)
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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.

- Avraham Aizenbud, Weizmann Institute
Avraham Aizenbud, Weizmann Institute
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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.

- Zlil Sela, HUJI
Zlil Sela, HUJI
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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.

- Boris Solomyak (BIU)
Boris Solomyak (BIU)
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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.

- Assaf Rinot, BIU
Assaf Rinot, BIU
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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.

- Hillel Furstenberg, HUJI
Hillel Furstenberg, HUJI
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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.

- Noga Alon
Noga Alon
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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.

- Lior Alon, IAS
Lior Alon, IAS
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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.

https://www.lioralon.net

- Yariv Aizenbud, Yale
Yariv Aizenbud, Yale
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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.

- Yaniv Ganor, Technion
Yaniv Ganor, Technion
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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.

- Tomer Goldfriend, Weizmann Institute of Science
Tomer Goldfriend, Weizmann Institute of Science
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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.

- Yotam Smilansky, Rutgers
Yotam Smilansky, Rutgers
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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.

- Alexei Kanel-Belov , Bar-Ilan University
Alexei Kanel-Belov , Bar-Ilan University
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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

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) - Itay Londner, Weizmann Institute of Science Itay Londner, Weizmann Institute of Science - 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. - Tali Pinsky, Technion Tali Pinsky, Technion - The Lorenz equations are a classical example of a chaotic flow in R^3. As they are hard to analyze, a simpler geometric model that is chaotic in the same way has been proposed. Smale’s 14th problem was to prove that the original Lorenz flow is equivalent to the geometric model. 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. - Dani Szpruch (Open University) Dani Szpruch (Open University) - 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 - Boris Zilber (University of Oxford) Boris Zilber (University of Oxford) - 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 - Danylo Radchenko (ETH, Zurich) Danylo Radchenko (ETH, Zurich) - 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… - Alexandre Borovik, University of Manchester Alexandre Borovik, University of Manchester - 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) - Ofer Busani, University of Bristol Ofer Busani, University of Bristol - 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. - Nikolai Vavilov, St. Petersburg State University Nikolai Vavilov, St. Petersburg State University - 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. - Zemer Kosloff, Hebrew University of Jerusalem Zemer Kosloff, Hebrew University of Jerusalem - 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. - Eran Assaf, Dartmouth College Eran Assaf, Dartmouth College - 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. - Yael Karshon, University of Toronto Yael Karshon, University of Toronto - 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. - Nadav Dym (Duke University) Nadav Dym (Duke University) - 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 https://arxiv.org/abs/2006.11709 - Yatir Halevi (Ben Gurion University) Yatir Halevi (Ben Gurion University) - 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. - Chaim Even-Zohar (The Alan Turing Institute, London) Chaim Even-Zohar (The Alan Turing Institute, London) - 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. - Galit Ashkenazi-Golan Galit Ashkenazi-Golan - 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 - Barak Sober (Duke University) Barak Sober (Duke University) - 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 - Adi Glucksam (University of Toronto) Adi Glucksam (University of Toronto) - 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. - Tal Orenshtein (Technical University of Berlin and Weierstrass Institute) Tal Orenshtein (Technical University of Berlin and Weierstrass Institute) - 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. - Ariel Rapaport (University of Cambridge) Ariel Rapaport (University of Cambridge) - 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 - Adam Dor On (University of Copenhagen) Adam Dor On (University of Copenhagen) - 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. - Louis Rowen (Bar-Ilan University) Louis Rowen (Bar-Ilan University) - 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. - Ruth Lawrence-Naimark, Hebrew University Ruth Lawrence-Naimark, Hebrew University - 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. - Lev Shneerson, Hunter College, CUNY Lev Shneerson, Hunter College, CUNY - We will speak about growth and Gelfand-Kirillov dimensions of subsemigroups of free inverse semigroups (joint work with Stuart Margolis). - Asaf Katz, University of Chicago Asaf Katz, University of Chicago - 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. - Nir Lev, Bar-Ilan University Nir Lev, Bar-Ilan University - 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. - Cyril Tintarev (Uppsala, visiting Technion) Cyril Tintarev (Uppsala, visiting Technion) - 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. - Mikhail Katz, Bar-Ilan University Mikhail Katz, Bar-Ilan University - 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. - Mikhail Zaidenberg, Fourier Institute, Grenoble Mikhail Zaidenberg, Fourier Institute, Grenoble - 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. - Sara Tukachinsky, IAS, Princeton Sara Tukachinsky, IAS, Princeton - 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. - פרופ' ישראל אומן פרופ' ישראל אומן - - (Adam Dor-On (University of Copenhagen and UIU (Adam Dor-On (University of Copenhagen and UIU - 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. - Ofer Zeituni, The Weizmann Institute Ofer Zeituni, The Weizmann Institute - 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. - Benjamin Weiss, The Hebrew University Benjamin Weiss, The Hebrew University - 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. - Andre Reznikov, Bar-Ilan University Andre Reznikov, Bar-Ilan University - 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". - Dimitri Gourevitch, University of Valenciennes, France Dimitri Gourevitch, University of Valenciennes, France - 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. - Simcha Haber, Bar-Ilan University Simcha Haber, Bar-Ilan University - A graph property is first order expressible if it can be written as a formal sentence using the universal and existential quantifiers with variables ranging over the vertices of the graph, the usual connectives and the relations = and ~, where x~y stands for adjacency. 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. - Daniel Berend, Ben-Gurion University Daniel Berend, Ben-Gurion University - 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. - Karoly Simon, Technical University of Budapest Karoly Simon, Technical University of Budapest - 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. - Yair Hayut, Kurt Goedel Research Center, Vienna Yair Hayut, Kurt Goedel Research Center, Vienna - 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. - Assaf Rinot, Bar-Ilan University Assaf Rinot, Bar-Ilan University - 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. - Gady Kozma, Weizmann Institute Gady Kozma, Weizmann Institute - 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. - Alexei Kanel-Belov, Bar-Ilan University Alexei Kanel-Belov, Bar-Ilan University - 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. - Dmitry Novikov, Weizmann Institute Dmitry Novikov, Weizmann Institute - 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. - Amos Nevo, the Technion Amos Nevo, the Technion - 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). - Yotam Smilansky, The Hebrew University Yotam Smilansky, The Hebrew 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. - Gregory Soifer, Bar-Ilan Gregory Soifer, Bar-Ilan - 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. - Dr. Rita Gitik, Department of Mathematics, University of Michigan Dr. Rita Gitik, Department of Mathematics, University of Michigan - 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. - Prof. Yuval Peres, Microsoft Research Prof. Yuval Peres, Microsoft Research - 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. - Sara Tukachinsky, Princeton University Sara Tukachinsky, Princeton University - 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. - Alexei Kanel-Belov, Bar-Ilan University Alexei Kanel-Belov, Bar-Ilan University - 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. - Tali Pinsky, Technion Tali Pinsky, Technion - 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. - Fedor Pakovich, Ben-Gurion University Fedor Pakovich, Ben-Gurion University - 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. - Zeev Rudnick, Tel-Aviv University Zeev Rudnick, Tel-Aviv University - 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. - Shimon Brooks, Bar-Ilan University Shimon Brooks, Bar-Ilan University - 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. - Lev Buhovski, Tel-Aviv University Lev Buhovski, Tel-Aviv University - 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. - Nathan Keller, Bar-Ilan University Nathan Keller, Bar-Ilan University - 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. - Nishant Chandgotia, Tel-Aviv University Nishant Chandgotia, Tel-Aviv University - 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. - Prof. Ron Peled, Tel-Aviv University Prof. Ron Peled, Tel-Aviv University - 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. - Prof. Jake Solomon, Hebrew University Prof. Jake Solomon, Hebrew University - 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. - Prof. Victor Vinnikov (Ben-Gurion University) Prof. Victor Vinnikov (Ben-Gurion University) - 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). - Dr. Danny Neftin, Technion Dr. Danny Neftin, Technion - 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. - Prof. Gil Ariel, Bar-Ilan University Prof. Gil Ariel, Bar-Ilan University - 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). - Dr. Menachem Shlossberg, University of Udine Dr. Menachem Shlossberg, University of Udine - 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. - Prof. Eliyahu Rips, Hebrew University Prof. Eliyahu Rips, Hebrew University - 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 - Prof. Victor Vinnikov, Ben-Gurion University Prof. Victor Vinnikov, Ben-Gurion University - 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 - Dr. Shira Zerbib,University of Michigan and MSRI Dr. Shira Zerbib,University of Michigan and MSRI - 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. - Dr. Naomi Feldheim (Weizmann Institute) Dr. Naomi Feldheim (Weizmann Institute) - 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. - Prof. Mark Agranovsky Prof. Mark Agranovsky - 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 - Prof. Shmuel Weinberger (University of Chicago) Prof. Shmuel Weinberger (University of Chicago) - 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. - "Prof. Shmuel Weinberger, "University of Chicago "Prof. Shmuel Weinberger, "University of Chicago - 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. - (Prof. Shmuel Weinberger (University of Chicago (Prof. Shmuel Weinberger (University of Chicago - 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. - Prof. Tahl Nowik Prof. Tahl Nowik - Introductory lecture for Prof. Weinberger's lecture - Prof. Misha Bialy (Tel Aviv University) Prof. Misha Bialy (Tel Aviv University) - Birkhoff's conjecture states that the only integrable billiards in the plane are ellipses. I am going to give a survey of recent progress in this conjecture and to discuss geometric results and questions around it. No prior knowledge of the subject will be assumed. - (Prof. Pedro Silva (CMUP, University of Porto (Prof. Pedro Silva (CMUP, University of Porto - 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). - Efim Zelmanov, University of California at San Diego Efim Zelmanov, University of California at San Diego - Algebras with Polynomial Identities is a well developed theory with strong Israeli roots. We will discuss a group theoretic analog of this theory. - Eyal Kaplan, Bar-Ilan University Eyal Kaplan, Bar-Ilan University - . 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. - Tobias Hatnick, The Technion Tobias Hatnick, The Technion - 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. - Tobias Hartnick, The Technion Tobias Hartnick, The Technion - 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. - Gregory Soifer, Bar-Ilan University and Weizmann Institute Gregory Soifer, Bar-Ilan University and Weizmann Institute - 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. - Gregory Soifer, Bar-Ilan University and Weizmann Institute Gregory Soifer, Bar-Ilan University and Weizmann Institute - 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. - Chloé Perin, Hebrew University of Jerusalem Chloé Perin, Hebrew University of Jerusalem - 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. - Karoly Simon, Budapest Technical University Karoly Simon, Budapest Technical University - 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? - Uri Bader, The Weizmann Institute of Science Uri Bader, The Weizmann Institute of Science - In recent decades it became clear that applications of Ergodic Theory are useful for the study of linear groups, remarkable examples being Mostow and Margulis Rigidity Theorems. What began as a collection of ad-hoc methods is getting now the shape of an organized theory. In my talk I survey some old and new ideas and results in this direction. No prior knowledge of Ergodic Theory will be assumed. - Semyon Alesker, Tel-Aviv University Semyon Alesker, Tel-Aviv University - 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. - Elon Lindenstrauss Elon Lindenstrauss - Arithmetic quotients of algebraic groups such as the space of unit volume lattices in R^n can be studied fruitfully from many directions and contain deep and subtle arithmetic information. Homogeneous dynamics studies these spaces by considering the action of a subgroup of the algebraic group on such a quotient. Of particular interest is the action of multiparameter diagonal groups: these display remarkable rigidity properties that are absent in the context of one parameter diagonalizable group actions. One aspect of this rigidity is joining rigidity: under suitable conditions, this rigidity implies that knowing that an orbit of a multiparameter diagonalizable group in a product of two arithmetic quotients is equidistributed in each one of these quotients individually implies joint equidistribution. I would explain this phenomena as well as some arithmetic consequences. - Gal Binyamini Gal Binyamini - The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate for a very wide class of transcendental sets. This result plays a key role in Pila-Zannier's proof of the Manin-Mumford conjecture, Pila's proof of the Andre-Oort conjecture for modular curves, Masser-Zannier's work on torsion anomalous points in elliptic families, and many more recent developments. I will briefly sketch the Pila-Wilkie theorem and the way it enters into the arithmetic applications. I will then discuss recent work on an effective form of the Pila-Wilkie theorem (for certain sets) which leads to effective versions of many of the applications. I will also discuss a joint work with Dmitry Novikov on sharpening the asymptotic from sub-polynomial to poly-logarithmic for certain structures, leading to a proof of the restricted Wilkie conjecture. The structure of the systems of differential equations satisfied by various transcendental functions plays a key role for both of these directions. - Prof. Omri Sarig, Weizmann Institute of Science Prof. Omri Sarig, Weizmann Institute of Science - "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. - Prof. Ron Adin Prof. Ron Adin - 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 - Iosif Polterovich, University of Montreal Iosif Polterovich, University of Montreal - 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. - Ilan Barnea Ilan Barnea - 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. - Alexander Fish, School of Mathematics and Statistics, University of Sydney, Australia Alexander Fish, School of Mathematics and Statistics, University of Sydney, Australia - 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. - Boris Khesin, University of Toronto Boris Khesin, University of Toronto - 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. - Adam Sheffer Adam Sheffer - 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. - Edva Roditty-Gershon, Bristol University Edva Roditty-Gershon, Bristol University - 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. - Assaf Rinot Assaf Rinot - In the early 1970's, Hindman proved a beautiful theorem in additive Ramsey theory asserting that for any partition of the set of natural numbers into finitely many cells, there exists some infinite set such that all of its finite sums belong to a single cell. In this talk, we shall study generalizations of this statement. Among other things, we shall present a negative partition relation for the real line which simultaneously generalizes a recent theorem of Hindman, Leader and Strauss, and a classic theorem of Galvin and Shelah. This is joint work with D.J. Fernandez Breton from the University of Michigan. - Prof. Arkady Berenstein, University of Oregon Prof. Arkady Berenstein, University of Oregon - 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. - Piotr Szewczak Piotr Szewczak - 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. - Nir Lev, Dept. of Mathematics, Bar-Ilan University Nir Lev, Dept. of Mathematics, Bar-Ilan University - 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. - Ari Meir Brodsky Ari Meir Brodsky - 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. - Dr. Victoria Sevostyanova (Haifa U.) Dr. Victoria Sevostyanova (Haifa U.) - 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. - Nikolai Gordeev, Russia State Pedagogical University, St. Petersburg Nikolai Gordeev, Russia State Pedagogical University, St. Petersburg - Abstract is attached. - Erick Herbin, Ecole Centrale Paris Erick Herbin, Ecole Centrale Paris - 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. Assaf Rinot Assaf Rinot 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. - Yoav Segev (BGU) Yoav Segev (BGU) - 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. - Boris Zilber (Oxford) Boris Zilber (Oxford) - In 1988 Hrushovski found counterexamples to the speaker's conjectures that categorical theories are in a certain sense reducible to algebraic geometry. Actually, the counterexamples are the outcomes of a very special abstract construction which is based on a combinatorial inequality in terms of dimensions of algebraic origin. The counterexamples were originally perceived as unwelcome mathematical pathologies. We will explain how Hrushovski's construction can be linked to the theory of classical transcendental functions and how it leads to certain conjectures which eventually can be recognised as a form of Grothendieck - Andre period conjecture. - Prof. Jerzy Kąkol Prof. Jerzy Kąkol - 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. - Dr. M. Schein, (Bar Ilan) Dr. M. Schein, (Bar Ilan) - 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 computation of these Dirichlet series turns out to be equivalent to the computation of some p-adic integrals over algebraic groups; integrals of this type have been studied extensively since the 1960's. Each of these two ways of approaching what turns out to be the same problem sheds light on the other. The talk will discuss the connection between pro-isomorphic subgroups and p-adic integrals. It will also discuss recent joint work with Mark Berman on the behavior of pro-isomorphic zeta functions under base extension. No knowledge of the subject will be assumed. - Prof. Eli Shamir, Hebrew University, Jerusalem Prof. Eli Shamir, Hebrew University, Jerusalem - The key concept of our discussion is that of a perfect matching (PM) in a bipartite graph. 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. - Leonid Makar -Limanov Leonid Makar -Limanov - Yitang Zhang who became famous recently thanks to brake-through in the twins conjecture wrote his PhD thesis on the plane Jacobian conjecture where he gave an estimate of the geometric degree of the corresponding mapping of a plane via algebraic degrees of the images of the coordinate functions. In my talk I'll explain how to gets a better estimate via the Newton polytope approach. - Mira Shamis Mira Shamis - 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. - Eugenii Shustin Eugenii Shustin - 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.) - William I. Newman (UCLA) William I. Newman (UCLA) - 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. - Ido Efrat (BGU) Ido Efrat (BGU) - 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. - Yaar Solomon (Stony Brook University) Yaar Solomon (Stony Brook University) - 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]. - Eli Matzri Eli Matzri - The Merkurjev-Suslin Theorem tell us that the n-torsion part of the Brauer group of a field containing a primitive n-th root of 1 is generated by symbol algebras. A natural question is:What can be said on the minimal number of symbols needed. In this talk I will survey some of the known results and give the idea for the proof for a bound in a geometric situation (by which I mean when the base field contains an algebraically closed field). - Boris Kunyavskii (Bar-Ilan University) Boris Kunyavskii (Bar-Ilan University) - 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. - Mikhail G. Tkachenko Mikhail G. Tkachenko - - Prof. Boris Levit, Queen's University, Kingston, Canada Prof. Boris Levit, Queen's University, Kingston, Canada - The classical Abel-Jacobi elliptic functions have been extensively used in the Approximation Theory and, in particular, in the Optimal Recovery problems. Such functions posses a variety of very attractive properties, being much more exible and versatile in comparison to the circular functions. I will consider several examples of linear interpolating spaces generated by such functions. A notion of an optimal interpolating space will be discussed, drawing on some applications in statistical problems of random data approximation. Conditions will be presented under which the interpolating spaces generated by the Abel-Jacobi elliptic functions contain constants and are optimal, in the case of equidistant interpolating design. I will also mention some open problems related to a possible extension of these results to the more general class of automorphic functions. - Prof. Boris Levit, Queen's University, Kingston, Canada Prof. Boris Levit, Queen's University, Kingston, Canada - The classical Abel-Jacobi elliptic functions have been extensively used in the Approximation Theory and, in particular, in the Optimal Recovery problems. Such functions posses a variety of very attractive properties, being much more exible and versatile in comparison to the circular functions. I will consider several examples of linear interpolating spaces generated by such functions. A notion of an optimal interpolating space will be discussed, drawing on some applications in statistical problems of random data approximation. Conditions will be presented under which the interpolating spaces generated by the Abel-Jacobi elliptic functions contain constants and are optimal, in the case of equidistant interpolating design. I will also mention some open problems related to a possible extension of these results to the more general class of automorphic functions. - Alexander I. Bufetov (CNRS, Steklov, IITP, NRU-HSE) Alexander I. Bufetov (CNRS, Steklov, IITP, NRU-HSE) - 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. - Leonid Polterovich (Tel-Aviv University) Leonid Polterovich (Tel-Aviv University) - 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. - Martin Markl (Academy of Sciences of the Czech Republic) Martin Markl (Academy of Sciences of the Czech Republic) - 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. - Andrzej Kisielewicz (Wroclaw) Andrzej Kisielewicz (Wroclaw) - 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, w​e ​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. - Barak Weiss (TAU) Barak Weiss (TAU) - 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. - Valdimir Berkovich Valdimir Berkovich - 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 - Pedro V. Silva (University of Porto) Pedro V. Silva (University of Porto) - 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. - Remi Coulon Remi Coulon - The free Burnside group of exponent n, B(r,n), is the quotient of the free group of rank r by the subgroup generated by all n-th powers. This group was introduced in 1902 by W. Burnside who asked whether it is finite or not. This problem motivated many developments in group theory. In 1968 P.S. Novikov and S.I. Adian made a breakthrough by proving that if n is sufficiently large then B(r,n) is infinite. In this talk we will focus on the symmetries of B(r,n). More precisely we will consider on the outer automomorphism group of B(r,n). Among other things, we will see that it inherits some properties coming from the outer automorphism group of free groups. - Yoram Luzon Yoram Luzon - 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. - Roman Polyak Roman Polyak - 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. - Shifi Reif Shifi Reif - Lie superalgebras and their representations were introduced to mathematics to study supersymmetry in theoretical physics and have since been applicable in algebra, combinatroics, number theory and various other fields. We shall discuss the recently proved Kac-Wakimoto character formula for representations of Lie superalgebras and its specializations to formulas in combinatorics and number theory such as the Jacobi formula for counting the number of presentations of an integer as a sum of k squares. - Zalman Balanov (UT at Dallas) Zalman Balanov (UT at Dallas) - The van der Pol oscillator (VDPO) consists of an LCR-contour and a negative feedback loop which can be implemented on the basis of a triode. The corresponding second order (autonomous) van der Pol equation is the simplest nonlinear mathematical model widely used in electrical engineering. In practice, one is usually dealing with networks of VDPOs coupled symmetrically. Studying the impact of symmetries of a system to the actual dynamics (in particular, symmetric properties and minimal number of periodic regimes) constitutes a problem of great importance and complexity which can be traced back to the classical 16-th Hilbert problem. Periodic solutions to symmetric autonomous systems very often are studied via the so-called equivariant Hopf bifurcation in a parameterized system (i.e. a phenomenon occurring when the parameter crosses some "critical" value causing a change of stability of the "trivial solution", which results in appearance of small amplitude non-constant periodic solutions near the trivial one). The commonly used method (M. Golubitsky et al.) is based on the equivariant singularity theory (EST) combined with a Lyapunov-Schmidt reduction or Central Manifold Theorem. However, this method cannot be applied if the system in question is not smooth enough and/or the phase space does not admit a local linear structure. On the other hand, if an inductance element in the van der Pol oscillator contains a ferromagnetic core, the ferromagnetic material can introduce a hysteresis relation between the magnetic induction and magnetic field. In many cases (for example, in the presence of the ferroresonance phenomenon), the hysteresis effect cannot be neglected. As a matter of fact, systems with hysteresis almost always are non-smooth and the corresponding phase spaces do not admit a local linear structure. Therefore, the EST based method cannot be applied to them. In my talk, I will show how an alternative method based on the usage of the new invariant ‚Äì twisted equivariant degree (introduced by Z. Balanov and W. Krawcewicz) -- can be effectively applied to symmetric networks of VDPO with hysteresis. In particular, a direct link between physics, topology, algebra and analysis underlying the VDPOs will be established. This talk is based on a joint work with W. Krawcewicz, D. Rachinskii and A. Zhezherun. - Dmitry Kerner (BGU) Dmitry Kerner (BGU) - 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). - Felix Goldberg Felix Goldberg - The chip ring game Bjorner, 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. - Eli Matzri Eli Matzri - 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. - Yair Glasner (BGU) Yair Glasner (BGU) - 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. - Joel Hass (UC Davis) Joel Hass (UC Davis) - I will talk about recent work showing that the problem of recognizing the 3-sphere lies in the class NP intersect coNP, assuming the Generalized Riemann Hypothesis. This is joint work with Greg Kuperberg. Dr. Nir Lev Dr. Nir Lev 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 Anton Khoroshkin Anton Khoroshkin ~~ 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). Baruch Barzel Baruch Barzel ~~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 Yuval Peres Yuval Peres ~~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 Chen Meiri Chen Meiri ~~ 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. Ami Viselter Ami Viselter ~~ 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. Daniel Hershkowitz Daniel Hershkowitz TBA Peter Sternberg Peter Sternberg : 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. Serge Itshak Lukasiewicz Serge Itshak Lukasiewicz 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 Assaf Rinot Assaf Rinot 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. Andrei Reznikov Andrei Reznikov 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) Prof. Luis Caffarelli, University of Texas at Austin Prof. Luis Caffarelli, University of Texas at Austin Wolf Prize Day 2012 Prof. Michael Aschbacher, Caltech Prof. Michael Aschbacher, Caltech Wolf Prize Day 2012 Gidi Amir Gidi Amir 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. Reuven Cohen Reuven Cohen 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. - Ronen Peretz, Ben Gurion University Ronen Peretz, Ben Gurion University - 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 Prof. E. Leuzinger, Institute for Algebra and Geometry KIT, Germany Prof. E. Leuzinger, Institute for Algebra and Geometry KIT, Germany 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.

Steve Gelbart, Weizmann Institute
Steve Gelbart, Weizmann Institute

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.