# Mathematics Colloquium

### Previous Lectures

Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Their study originated in

the groundbreaking work of J.L. Taylor on noncommutative spectral theory in the 1970s, but it is mostly in the last decade that the theory established itself as a new and active research area. I will survey some aspects of these developments, including (if time permits) recent work on interpolation and extension problems. The talk will be aimed at a general mathematical audience and should be accessible for graduate students (or even advanced undergraduates).

Many properties of rational functions f arising from problems in number theory, dynamics, and complex analysis, can be studied by writing f as a composition f_1 o ... o f_r, where the f_i's are indecomposable rational functions, i.e. cannot be decomposed nontrivially further. However basic questions such as determining the relationship between two such decompositions of f remain unknown. We shall describe progress towards a description and its applications to various problems.

Bacterial swarming is a collective mode of motion in which cells migrate rapidly over surfaces. Swarming is typically characterized by densely packed groups moving in irregular, yet coherent patterns of whirls and flows.

Analysis of individual cell trajectories within dense swarms reviles that the interplay between the single cell motion and the collective flow results in chaotic dynamics. Moreover, trajectories are consistent with Lévy walks – random processes in which the Gaussian central limit theorem fails. A model suggests a new route in which Lévy walking can result from chaotic dynamics.

The talk will explain these observations – no prior knowledge is required. More generally, I will try to convey how the phenomenon of collective bacterial movement draws from and can contribute new ideas to a range of mathematical subjects such as stochastic processes, hydrodynamics and dynamical systems.

Joint work with Avraham Be'er (BGU) and Andy Reynolds (Rothamsted Research, UK).

A Hausdorff topological group $(G, \tau)$ is called minimal if there exists no Hausdorff group topology on $G$ which is strictly coarser than $\tau$.

We say that a topological group $G$ is hereditarily minimal, if every subgroup of $G$ is minimal.

By Prodanov's Theorem an infinite compact abelian group $K$ is isomorphic to $\Z_p$ (p-adic integers) for some prime $p$ if and only if $K$ is hereditarily minimal.

We study hereditarily minimal groups. The following theorem is one of our main results.

**Theorem**

Let $G$ be an infinite hereditarily minimal locally compact group that is either compact or locally solvable. Then $G$ is either center-free or isomorphic to $\Z_p$, for some prime $p$.

In particular,

**Corollary**

If $G$ is an infinite hereditarily minimal locally compact nilpotent group, then $G$ is isomorphic to $\Z_p$ for some prime $p$.

This is a joint work with D. Dikranjan, D. Toller and W. Xi.

Small cancellation groups and their generalizations are used for constructing groups with various exotic properties. The theory of small cancellation groups can be developed both geometrically (via van Kampen diagrams) and combinatorially. Van Kampen diagrams for small cancellation groups display negative curvature features. For the combinatorial approach we are able to develop its ring theoretic analog

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

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.

A Gaussian stationary process is a random function f:R-->R or f:C-->C,

whose distribution is invariant under real shifts, and whose evaluation at

any finite number of points is a centered Gaussian random vector.

The mathematical study of these random functions goes back at least 75 years,

with pioneering works by Kac, Rice and Wiener.

Nonetheless, many basic questions about them, such as the fluctuations of their number of zeroes,

or the probability of having no zeroes in a large region, remained unanswered for many years.

In this talk, we will provide an introduction to Gaussian stationary process and

describe how a new spectral perspective, combined with tools from harmonic, real and

complex analysis, yields new results about such long-lasting questions.

In 1687, Sir Isaac Newton established that the area cut off from an oval in $\mathbb R^2$

by a straight line never depends algebraically on the line (the question was motivated by

Kepler's law in celestial mechanics). In 1987, V. I. Arnold proposed to generalize Newton's

observation to higher dimensions and conjectured that all smooth bodies, with the exception

of ellipsoids in odd-dimensional spaces, have an analogous property. The talk is devoted to

the current status of the conjecture

Special Lecture Series on “Variations on a Theme of Borel” - 3rd Lecture

Over 60 years ago, Borel, on the basis of theorems of Mostow, conjectured a topological rigidity statement that has become central to topology. During these lectures, I will use his heuristic to suggest other statements, some true, some false, and some conjectural. Overall, this area is devoted to a profound influence that the fundamental group has on topology and geometry.

Over 60 years ago, Borel, on the basis of theorems of Mostow, conjectured a topological rigidity statement that has become central to topology. During these lectures, I will use his heuristic to suggest other statements, some true, some false, and some conjectural. Overall, this area is devoted to a profound influence that the fundamental group has on topology and geometry.

Over 60 years ago, Borel, on the basis of theorems of Mostow, conjectured

a topological rigidity statement that has become central to topology. During these lectures, I will use his heuristic to suggest other statements, some true, some false, and some conjectural. Overall, this area is devoted to a profound influence that the fundamental group has on topology and geometry.

Introductory lecture for Prof. Weinberger's lecture

Taking as departure point an article by Cameron, Gadouleau, Mitchell and Peresse on maximal

lengths of subsemigroup chains, we introduce the subsemigroup complex H(S) of a

nite semigroup

S as a (boolean representable) simplicial complex de

fined through chains in the lattice of subsemi-

groups of S. The rank of H(S) is the above maximal length minus one and H(S) provides other

useful invariants concerning the lattice of subsemigroups of S. We present a research program for

such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The

results include alternative characterizations of independence and bases, asymptotical estimates on

the number of bases, or establishing when the complex is pure or a matroid.

This is joint work with Stuart Margolis (Bar-Ilan University, Ramat Gan, Israel) and John

Rhodes (University of California, Berkeley, USA).

Algebras with Polynomial Identities is a well developed theory with strong Israeli roots. We will discuss a group theoretic analog of this theory.

. One of the primary goals of number theory is to understand the absolute Galois group of the rational numbers. An important goal of the Langlands Program, is to understand the finite dimensional representations of this group, using automorphic representations. In the other direction, passing from the Galois side to the automorphic setting, this can be regarded as an arithmetic parametrization of local and automorphic representations. This parametrization, which is conjectural in part, predicts a transfer or lifting of local and automorphic representations between two reductive algebraic groups.

The global lift has been established in the celebrated work of Cogdell, Kim, Piatetski-Shapiro and Shahidi for automorphic representations admitting a certain Fourier functional, using the Converse Theorem; and in general by Arthur and by Mok, using the trace formula.

In a joint work with Cai and Friedberg, we present a new proof of functoriality, using integral representations, which generalizes the work of Cogdell et al. to arbitrary automorphic representations. This proof is based on our recent collaboration with Ginzburg, where we generalized the classical doubling method. It is expected to have further applications to the problems of descent and to covering groups.

In this talk I will first explain the basic ideology behind geometric group theory: How and to what extend can we understand (finitely-generated) groups as geometric objects? I will discuss the classical Schwarz-Milnor lemma which provides a translation mechanism between groups and geometry. In particular I will discuss a certain class of isometric actions called geometric actions. I will then explain that the Schwarz-Milnor machinery not only applies to isometric actions, but also to quasi-isometric quasi-actions of groups, and try to convince you that this is actually the more natural context of modern geometric group theory.

In the final part of my talk, I will discuss some very recent developments which show that one can not only “quasify" the notion of an isometric action but also the notion of a group itself. This allows us to not only interpret groups, but also more general algebraic structures called approximate groups as geometric objects. Time permitting I will comment on various algebraic, geometric and analytic aspects of approximate groups. This final part is based on joint work with Michael Björklund and Matthew Cordes.

No prior knowledge of geometric group theory is required and large parts of the talk should be understandable to master and PhD students.

In this talk I will first explain the basic ideology behind geometric group theory: How and to what extend can we understand (finitely-generated) groups as geometric objects? I will discuss the classical Schwarz-Milnor lemma which provides a translation mechanism between groups and geometry. In particular I will discuss a certain class of isometric actions called geometric actions. I will then explain that the Schwarz-Milnor machinery not only applies to isometric actions, but also to quasi-isometric quasi-actions of groups, and try to convince you that this is actually the more natural context of modern geometric group theory.

In the final part of my talk, I will discuss some very recent developments which show that one can not only “quasify" the notion of an isometric action but also the notion of a group itself. This allows us to not only interpret groups, but also more general algebraic structures called approximate groups as geometric objects. Time permitting I will comment on various algebraic, geometric and analytic aspects of approximate groups. This final part is based on joint work with Michael Björklund and Matthew Cordes.

No prior knowledge of geometric group theory is required and large parts of the talk should be understandable to master and PhD students.

The study of affine crystallographic groups has a long history which goes back

to Hilbert's 18th problem. More precisely Hilbert (essentially) asked if there is only a finite number, up to conjugacy in \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.

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.

We will give an overview of questions one might ask about the first-order theory of free groups and related groups: how much information can first-order formulas convey about these groups or their elements, what algebraic interpretation can be given for model theoretic notions. It turns out that techniques from geometric group theory are very useful to tackle such problems. We will assume no special knowledge of model theory.

Recently considerable attention has been paid to the study of arithmetic sums of two planar sets A+G:={a+g: a in A, g in G}. We focus on the case when G is a piecewise C^2 curve, in particular when G is the unit circle. In this case there is a natural guess what the size (Hausdorff dimension, Lebesgue measure) of A+G should be. We verify it under some simple natural assumptions. We also address the more difficult question: under which condition does the set A+G have non-empty interior?

Translation invariant valuations which are continuous in the Hausdorff metric play a special role in the theory and its applications to integral geometry. Theory of such valuations is an active topic in convexity. In recent years it was realized that the space of such valuations admits rich structures, in particular the multiplicative structure. The latter turned out to be useful in integral geometry. First I will explain some of the classical background and examples. Then I will discuss more recent results mentioned above.

"Symbolic dynamics" is a powerful technique for describing the combinatorial structure of large collections of orbits of dynamical systems with "chaotic" behaviour. I will describe this technique, and will report on recent advances on the question what sort of "chaos" is needed to this method to succeed. The talk is meant for a general audience, including people with little or no background in dynamical systems.

Descents of permutations have been studied for more than a century. This concept was vastly generalized, in particular to standard Young tableaux (SYT). More recently, cyclic descents of permutations were introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding concept for SYT, Rhoades found a very elegant solution for rectangular shapes. In an attempt to extend the concept of cyclic descents, explicit combinatorial definitions for two-row and certain other shapes have been found, implying the Schur-positivity of various quasi-symmetric functions. In all cases, the cyclic descent set admits a cyclic group action and restricts to the usual descent set when the letter *n* is ignored. Consequently,

the existence of a cyclic descent set with these properties was conjectured for all shapes, even the skew ones. This talk will report on the surprising resolution of this conjecture: Cyclic descent sets exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies non-negativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants. We shall also comment on issues of uniqueness. Based on joint works with Sergi Elizalde, Vic Reiner, Yuval Roichman.

**Dr. Menachem Shlossberg invites you to a “haramat kosit” in celebration of obtaining a postdoctorate **

**at the University of Udine, Italy, at 1:30 PM next to the Colloquium Room**

**ד"ר מנחם שלוסברג מזמין אתכם להרמת כוסית לרגל קבלת משרת פוסט דוקטורט באונ' אודין באיטליה**

** בשעה 13:30 בחדר ע"י חדר המחלקה**

The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.

Model categories, introduced by Quillen, provide a very general context in which it is possible to set up the basic machinery of homotopy theory. In particalar they enable to define derived functors, homotopy limits and colimits, cohomology theories and spectral sequences to catculate them. However, the structure of a model category is usually hard to verify, and in some interesating cases even impossible to define. In this lecture I will define a much simpler notion then a model category, called a weak fibration category. By a theorem due to T. Schlank and myself, a weak fibration category gives rise in a natural way to a model category structure on its pro category, provided some technical assumptions are satisfied. This result can be used to construct new model structures in different mathematical fields, and thus to import the methods of homotopy theory to these situations. Examples will be given from the categories of simplicial presheaves, C*-algebras and complexes in Abelian categories. Applications will be discussed with each example.

The above encompasses joint work with Tomer M. Schlank, Yonatan Harpaz, Geoffroy Horel, Michael Joachim Snigdhayan Mahanta and Matan Prezma.

We present a new approach (joint with M. Bjorklund (Chalmers)) for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.

By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.

By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.

A plane curve is called nondegenerate if it has no inflection points.

How many classes of closed nondegenerate curves exist on a sphere?

We are going to see how this geometric problem, solved in 1970, reappeared along with its generalizations in the context of the Korteweg-de Vries and Boussinesq equations. Its discrete version is related to the 2D pentagram map defined by R. Schwartz in 1992.

We will also describe its generalizations, pentagram maps on polygons in any dimension and discuss their integrability properties.

While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis.

A simple example of an incidences problem: Given a set of n points and set of n lines, both in R^2, what is the maximum number of point-line pairs such that the point is on the line. Studying incidence problems often involves the uncovering of hidden structure and symmetries.

In this talk we introduce and survey the topic of geometric incidences, focusing on the recent polynomial techniques and results (some by the speaker). We will see how various algebraic and analysis tools can be used to solve such combinatorial problems.

In the talk I will discuss classical problems concerning the distribution of square-full numbers and their analogues over function fields. The results described are in the context of the ring Fq[T ] of polynomials over a finite field Fq of q elements, in the limit q → ∞.

I will also present some recent generalization of these kind of classical problems.

Hecke algebras H_q(W) of Coxeter groups W first emerged in the study of Chevalley groups in mid sixties and since then became central objects in Representation Theory of Coxteter groups and semisimple Lie groups over finite fields. In particular, as a one-parameter deformation of the group algebra kW of W, the Hecke algebra H_q(W) helps to classify representations of W and to equip each simple kW-module with the canonical Kazhdan-Lusztig basis.

Unfortunately, unlike the group algebra kW, the Hecke algebra H_q(W) lacks a Hopf algebra structure, that is, it is not clear how to tensor multiply H_q(W)-modules. Moreover, there is a general consensus that a naive Hopf structure on H_q(W), if exists, would essentially coincide with that on kW, so we would not gain any new information.

In my talk (based on joint work with D. Kazhdan) I suggest a roundabout: instead of forcing a naive Hopf structure on H_q(W), we find a ``reasonably small" Hopf algebra H(W) (we call it Hecke-Hopf algebra of W) that "naturally" contains H_q(W) as a coideal subalgebra.

The immediate benefit of this enlargement of H_q(W) is that each representation of H(W) and each representation of H_q(W) can be tensor multiplied into a new representation of H_q(W), thus allowing to create infinitely many new H_q(W)-modules out of a single one.

Hecke-Hopf algebras have some other applications, most spectacular of which is the construction of new infinite families of solutions to the quantum Yang-Baxter equation.

The theory of selection principles deals with the possibility of obtaining mathematically significant objects by selecting elements from sequences of sets. The studied properties mainly include covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially functions spaces. Often, the characterization of a mathematical property using selectionprinciple is a nontrivial task leading to new insights on the characterized property.

I will give an overview of this theory and, if time permits, present some resent results obtained jointly with Boaz Tsaban and Lyubomyr Zdomskyy.

We know by classical Fourier analysis that the unit cube in R^d has an orthogonal basis consisting of exponential functions. Which other domains admit such a basis? Fuglede conjectured (1974) that these so-called "spectral domains" could be characterized geometrically by their possibility to tile the space by translations. I will survey the subject and then discuss some recent results, joint with Rachel Greenfeld, where we focus on the conjecture for convex polytopes.

We explore results of Ramsey theory (also known as partition calculus) and show how they apply to cardinals, ordinals, trees, and arbitrary partial orders, leading up to the main result which is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem.

A full exposition of the results is contained in my PhD thesis, available at http://hdl.handle.net/1807/68124.

Hilbert’s fourteenth problem asks whether the algebra of invariants for an action of a linear algebraic group is finitely generated.

This is true for reductive groups and the problem is open for unipotent groups. We discuss the case of the adjoint action of a maximal unipotent subgroup U in GL_n(K) on the nilradical m of any parabolic subalgebra, where K is an algebraically closed field of zero characteristic. This action is extended to a representation in the algebra K[m]. I will show that the algebra of invariants K[m]^U is finitely generated. Besides, a set of algebraically independent invariants generating the field K(m)^U will be presented.

Abstract is attached.

We consider the issue of generalized stochastic processes, indexed by an abstract set of indices. What should the minimal required conditions on the indexing collection be, to study some of the usual properties of these processes, such as in- crement stationarity, martingale and Markov properties or integration question? The already known examples of processes indexed by functions or metric spaces can be addressed by this way.

We show how the set-indexed framework of Ivanoff-Merzbach allows to study these generalized processes.

Some set-indexed processes can be considered as random measures on some δ- ring. Some generalized processes can be defined as an integral with respect to some measure on the indexing collection. The example of set-indexed Lévy processes is considered. The links with function-indexed processes could be discussed.

If time permits, we could also discuss regularity issue : continuity or Hölder regularity.

This talk is based on works in collaboration with Ely Merzbach and Alexandre Richard.

Recall that the real line is that unique separable, dense linear ordering with no endpoints in which every bounded set has a least upper bound.

Around the year of 1920, Souslin asked whether the term *separable* in the above characterization may be weakened to *ccc*. (A linear order is said to be separable if it has a countable dense subset. It is ccc if every pairwise-disjoint family of open intervals is countable.)

Amazingly enough, the resolution of this single problem led to many key discoveries in set theory. Also, consistent counterexamples to this problem play a prominent role in infinite combinatorics.

In this talk, we shall tell the story of the Souslin problem, and report on an advance recently obtained after 40 years of waiting.

Jordan algebras J of charateristic not 2 sometimes contain

a set of idempotents (e^2=e) that generate J such that their adjoint

map ad_e: u \mapsto ue (u\in J) has the minimal polynomial

x(x-1)(x-1/2), and with additional restrictions on products

of elements in the eigenspaces of ad_e (for each e).

Generalizing these properties (not only of such Jordan

algebras) Hall, Rehren, Shpectorov (HRS) introduced ``Axial algebras

of Jordan type''. In my talk I will present structural results

on Axial algebras of Jordan type 1/2 (a case which was not

dealt with in HRS), I will discuss their idempotents e, the corresponding

``Miyamoto involutions'' \tau(e) and the group that these involutions

generate.

This is joint work with J. Hall, S. Shpectorov.

Corson (1961) started a systematic study of certaintopological properties of the weak topology w of Banach spaces E. This

line of research provided more general classes such as reflexive

Banach spaces, Weakly Compactly Generated Banach spaces and the class

of weakly K-analytic and weakly K-countably determined Banach spaces.

On the other hand, various topological properties generalizing

metrizability have been studied intensively by topologists and

analysts. Let us mention, for example, the first countability,

Frechet-Urysohn property, sequentiality, k-space property, and

countable tightness. Each property (apart the countable tightness)

forces a Banach space E to be finite-dimensional, whenever E with the

weak topology w is assumed to be a space of the above type. This is a

simple consequence of a theorem of Schluchtermann and Wheeler that an

infinite-dimensional Banach space is never a k-space in the weak

topology. These results show also that the question when a Banach

space endowed with the weak topology is homeomorphic to a certain

fixed model space from the infinite-dimensional topology is very

restrictive and motivated specialists to detect the above properties

only for some natural classes of subsets of E, e.g., balls or bounded

subsets of E. We collect some classical and recent results of this

type, and characterize those Banach spaces E whose unit ball B_w is

k_R-space or even has the Ascoli property. Some basic concepts from

probability theory and measure theoretic properties of the space l_1

will be used.

A finitely generated group $G$ has only a finite number, say $a_n(G)$, of subgroups of any given index $n$. The study of subgroup growth, i.e. of the behavior of this sequence, has been an active area of research for several decades. A variant problem investigates the sequence $a_n^\wedge (G)$ counting subgroups of index $n$ whose profinite completion is isomorphic to that of the original group $G$, and in particular the analytic properties of the Dirichlet series derived from this sequence.

The expansion condition in Hall's marriage theorem can be extended to an unbiased 2-sided one.

This enables an alternative (and simpler) proof of Evans' (proven) Conjecture:

A partial nxn Latin square with n-1 dictated entries admits a completion to a full Latin square.

PMs are used to successively fill the square by rows, columns or diagonals. Latin square tables correspond to quasi-groups; the ones corresponding to groups are only a tiny fraction of them, as n grows. However, for Sudoku tables of order mnxmn, the completion (say by diagonals) usually fails, even if there are no dictated entries, unless they are conjugates of a twisted product of two groups, of orders n and m.

We shall discuss the Chirikov standard map, an area-preserving map of the torus to itself in which quasi-periodic and chaotic dynamics are believed to coexist. We shall describe how the problem can be related to the spectral properties of a one-dimensional discrete Schroedinger operator, and present a recent result.

Based on joint work with T. Spencer.

We define refined tropical enumerative invariants counting plane tropical curves of a given degree and a given positive genus and having marked points on edges and at vertices. This extends Block-Goettsche and Goettsche-Schroeter refined tropical invariants. As a consequence we obtain tropical (complex) descendant invariants and (real) broccoli invariants of positive genus.

(Joint work with F. Schroeter.)

We consider both time series as well as spatial distributions (in 1-4 dimensions). In the first, we observe that time series for individual and independently deviating random variables can manifest pattern through the emergence of peak-to-peak sequences that are visible to the eye yet fail all Fourier analysis schemes and reveal a seeming periodicity of 3-events per cycle. We note that this can explain observations of apparent cycles in mammalian animal populations. We consider models, as well, based on the Langevin equation of kinetic theory and the Smolouchowski relation that present circumstances where the apparent period can vary from 3-4 and, for a special subclass of problems, to periods between 2 and 3. We explore how cataloged observational data from global earthquake catalogues, magnetospheric AL index observations, Old Faithful Geyser eruption data, and the performance of the Standard & Poor's 500 index (percent daily variation) manifest different degrees of statistical agreement with the theory we derived. We present a simple model for many mammalian population cycles whose underlying phenomenological basis has strong biological implications. We then employ directed graphs to explore nearest-neighbor relationships and isolate the character of spatial clustering in 1-4 dimension. We observe that the one-dimensional problem is formally equivalent to that presented by peak-to-peak sequences in time series and also demonstrates a mean number of points per cluster of 3 in one dimension. We then take the first moment of each of the clusters formed, and observed that they too form clusters. We observe the emergence of a hierarchy of clusters and the emergence of universal cluster numbers, analogous to branching ratios and, possibly, Feigenbaum numbers. These, in turn, are related to fractals as well as succularity and lacunarity, although the exact nature of this connection has not been identified. Finally, we show how hierarchical clustering emerging from random distributions may help provide an explanation for observations of hierarchical clustering in cosmology via the virial theorem and simulation results relating to the gravitational stabilization in a self-similar way of very large self-gravitating ensembles.

These symmetry patterns are described by the absolute Galois group of the field, whose

structure is in general still a mystery.

We will describe what is known about this symmetry group: classical facts, consequences

of the epochal work by Veovodsky and Rost, and very recent structural results and conjectures

related to higher cohomology operations and intersection theorems.

Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers' question. The second proof is direct and it has nice applications in combinatorics. The talk will be accessible to a general audience. [This is a joint work with Omri Solan and Barak Weiss].

In 1846, Arthur Cayley defined a correspondence between orthogonal

matrices of determinant one and skew-symmetric matrices. This

observation was a starting point of a long (and yet unfinished)

story. In the talk we will overview its highlights, with a focus on

the achievements obtained during the past decade and some open

problems.

Let mu_{m,n} be the canonical invariant measure on the Grassmann manifold

of m-dimensional subspaces in C^{m+n}; the flat coordinates on the Grassmann

manifold allow us to consider mu_{m,n} as a measure on the space Mat(m x n) of

complex matrices. By definition, the family of measures mu_{m,n} has

the property of consistency under natural projections

Mat((m + 1) n) ---> Mat(m n) ; Mat(m x (n + 1)) ---> Mat(m x n)

and consequently defines a probability measure on the space Mat of infinite

complex matrices. The measure mu is by definition unitarily-invariant and admits

a natural one-parameter family of unitarily-invariant deformations mu^(s), called

the Pickrell measures. The Pickrell measures are finite for s > -1 and infinite

for s < 0.

The first main result of the talk is the solution to the problem, posed by

Borodin and Olshanski in 2000, of the explicit description of the ergodic decomposition of infinite Pickrell measures. The decomposing measures are naturally identified with sigma-finite processes on the half-line R+ and can be viewed as sigma-finite analogues of determinantal point processes. For different values of the parameter s, these measures are mutually singular.

In the second part of the talk we will discuss absolute continuity and singularity of determinantal point processes. The main result here is that determinantal point processes on Z induced by integrable kernels are indeed quasi-invariant under the action of the in nite symmetric group. The Radon-Nikodym derivative is found explicitly. A key example is the discrete sine-process of Borodin, Okounkov and Olshanski. This result has a continuous counterpart: namely, that

determinantal point processes with integrable kernels on R, a class that includes processes arising in random matrix theory such as the sine-process, the process with the Bessel kernel or the Airy kernel, are quasi-invariant under the action of the group of di eomorphisms with compact support.

The first part of the talk is based on the preprint http://arxiv.org/abs/1312.3161;

the second part, on the preprint http://arxiv.org/abs/1409.2068.

In 1990 Helmut Hofer introduced a bi-invariant metric on symplectomorphism

groups which nowadays plays an important role in symplectic topology and Hamiltonian dynamics.

I will review some old, new and yet unproved results in this direction.

Although higher structures have been around for quite some time, they recently have come back into focus through renewed interest in higher categories. There are several reasons for this.

In geometry one is trying to interpret extended cobordism theories, where the higher structures are meant to mimic higher codimensions. An analogue in algebra is known to the 2-categorical level, the prime example being the 2-category of rings, bi-modules and bi-module morphisms. Beyond this there are many open questions of fundamental nature. The central problem is what type of coherence to require.

In physics higher structures naturally appear in two related fashions. The first is through the extended field theories and the second through field theories with defects. This is mathematically mimicked by cobordisms and defect lines and points abstractly interpreted as inclusions into higher dimensional objects.

The "truncated" versions of higher structures can be assembled into infinity up to a homotopy everything version. This is the setting of the influential program of Lurie which provides firm foundations to derived algebraic geometry, and, hopefully, to higher differential geometry which is not yet that well established.

Geometric and physical points of view combine in the constructions of string topology and in the proofs of the cobordism hypothesis. One approach to this, which will also be an integral part of this program, is the operadic/monadic point of view as many liigher categorical structures can be interpreted as actions of certain liigher dimensional operads/monads. The classical homotopy theory teaches us that this is the correct way to encode higher homotopies and homotopical algebra in general.

The complexity of higher dimensional structures and necessity to work with them efficiently has required reconsideration of the foundations of mathematics. A new theory called univalent foundations or homotopy type theory emerges in recent years which has a potential to become a common language for mathematicians working with higher categorical structures. We wish to include this theory as a supplement to our main topics, but also as a possible future direction of research.

The Cerny conjecture, concerned with the minimal length of a reset word in a finite automata, is considered one of the most longstanding open problem in the theory of finite automata. In this talk, we discuss the background of the conjecture, attempts at a proof, and partial results obtained so far by various researchers. In the second part, we present our recent results, which shade a light on the question of why the conjecture is so hard to prove.

Suppose a light source is placed in a polygonal hall of mirrors (so light can bounce off the walls). Does every point in the room get illuminated? This elementary geometrical question was open from the 1950s until Tokarsky (1995) found an example of a polygonal room in which there are two points which do not illuminate each other. Resolving a conjecture of Hubert-Schmoll-Troubetzkoy, in joint work with Lelievre and Monteil we prove that if the angles between walls is rational, every point illuminates all but at most finitely many other points. The proof is based on recent work by Eskin, Mirzakhani and Mohammadi in the ergodic theory of the SL(2,R) action on the moduli space of translation surfaces. The talk will serve as a gentle introduction to the amazing results of Eskin, Mirzakhani and Mohammadi.

Non-Archimedean analytic geometry is an analog of complex analytic geometry over non-Archimedean (e.g., p-adic) fields. In the talk, I'll explain what non-Archimedean analytic spaces are, list basic facts about them, and tell about their applications

Hyperbolic groups can be defined through the geometry of Cayley graphs, viewed as geodesic metric spaces. One important feature of hyperbolic groups is the concept of boundary, which can be defined through the topological completion for an appropriate metric (such as the visual metrics), and has the advantages of compactness. An endomorphism of a hyperbolic group admits a continuous extension to the boundary if and only if it is uniformly continuous with respect to a visual metric, and a Hölder condition is a particularly nice way of achieving uniform continuity. In joint work with Vítor Araújo (Universidade Federal da Bahia), we have proved that an endomorphism of a hyperbolic group satisfies a Hölder condition with respect to a visual metric if and only if it is virtually injective and its image is a quasi-convex subgroup. Moreover, if the group is virtually free or torsion-free co-hopfian, then the endomorphism is uniformly continuous if and only if it satisfies a Hölder condition if and only if it is virtually injective. However, this stronger claim does not necessarily hold for arbitrary hyperbolic groups.

Cooperative interactions, their stability and evolution, provide an interesting context in which to study the interface between cellular and population levels of organization. Such interactions also open the way for the discovery of new population dynamics mechanisms.

We have studied a version of the public goods model relevant to microorganism populations actively extracting a growth resource from their environment. Cells can display one of two phenotypes – a productive phenotype that extracts the resources at a cost, and a non-productive phenotype that only consumes the same resource. We analyze the continuous differential equation model as well as simulate stochastically the full dynamics. It is found that the two sub-populations, which cannot coexist in a well-mixed environment, develop spatio-temporal patterns that enable long-term coexistence in the shared environment. These patterns are solely fluctuation-driven, since the continuous system does not display Turing instability. The average stability of the coexistence patterns derives from a dynamic mechanism in which one sub-population holds the environmental resource close to an extinction transition of the other, causing it to constantly hover around its critical transition point, forming a mechanism reminiscent of selforganized criticality. Accordingly, power-law distributions and long-range correlations are found.

When a time scale separation occurs between two dynamic parameters is defined, a structurally unstable point emerges and any small perturbation of the dynamics with additive noise leads to an equilibrium distribution in which both species coexist in context of additive but not multiplicative noise.

For three quarters of a century Linear Programming (LP) was the main tool for solving resource allocation problems (RAP)- one of the main problem in economics.

In 1975 L. V. Kantorovich and T. C. Koopmans shared the Nobel Prize in Economics **Nonlinear Equilibrium vs. Linear Programming for resource allocation problems.**“for their contributions to the theory of optimum allocation of limited resources."

When LP is used for RAP the prices for goods and the resource availability are given a priori and independent on the production output and prices for the resources. It often leads to solutions, which are not practical, because they contradict to the basic market law of supply and demand.

We consider an alternative to LP approach to RAP, which is based on Nonlinear Equilibrium (NE). The NE is a generalisation of Walras-Wald equilibrium, which is equivalent to J Nash equilibrium in n-person concave game.

NE eliminates the basic drawbacks of LP. Finding NE is equivalent to solving a variation inequality (VI) on the Cartesian product of the primal and dual non negative octants, projection on which is a very simple operation. For solving the VI we consider two methods: projected pseudo-gradient (PPG) and extra pseudo-gradient (EPG), for which projection is the main operation at each step.

We established convergence, proved global Q-linear rate and estimated complexity of both methods under various assumptions on the input data.

Both PPG and EPG can be viewed as pricing mechanisms for establishing economic equilibrium.

Let f be a power series (in several variables) or a C^\infty-smooth function. In many cases just a finite part of Taylor expansion is enough to determine f up to the change of coordinates. Alternatively, the deformations of f by terms of high enough orders are trivial. This phenomenon is called the finite determinacy.

An immediate application is the algebraization: f has a polynomial representative.

More generally, for maps of smooth spaces the finite determinacy (under various group-actions) has been intensively studied for about 50 years (by Mather, Tougeron, Arnol'd, Wall and many others).

**The chip ring game** Bjorner, Lovasz and Shor (BLS) introduced the following game in 1991: N chips are placed on the vertices on a n-vertex graph and at every turn, the solitaire player chooses a vertex i of degree di which has at least di chips on it and "fires" i by shifting a chip from i to each of i's neighbours.

**The game duration problem** BLS have proved the remarkable result that whenever this game terminates, it always does so in the same number of moves, irrespective of gameplay! (I will explain the background for this). They also gave an elegant upper bound on the number of moves. However, computer simulation reveals that the game actually ends in far fewer moves than the BLS bound in all examined cases.

**The new results** I will show a new approach to obtaining upper bounds on the game duration, based on a re nement of the classic BLS analysis together with a simple but potent new observation.

The new bounds are always at least as good as the BLS bound and in some cases the improvement is dramatic. For example, for the strongly regular graphs BLS reduces to O(nN) while the new bound reduces to O(n+N). For dense regular graphs BLS reduces to O(N) while the new bound reduces to O(n) (for such it holds that n = O(N)).

The proof technique involves a careful analysis of the pseudo-inverse of the graph's discrete Laplacian.

**The wider context** Time permitting, I will also discuss the appearance of chip ring (and its very close relative, the sandpile model) in diverse mathematical and scientific contexts.

The Inverse Galois Problem, asking which groups can be realizable as

Galois groups of fields, is a major problem in Galois theory.

For example the fact that there is no general formula for the roots of

a polynomial of degree five follows from the fact that

the symmetric group S_5, which is not solvable, is realizable as a

Galois group of a field.

Minac and Tan conjectured that if G is the Galois group of a field,

then G has vanishing triple Massey products (to be defined in the lecture).

In the talk I will give some general background on this new property

and its relation to the inverse Galois problem via a work of Dwyer, and try to give a

flavor of my proof of the Minac-Tan conjecture.

Ergodic theory studies actions of a group G by measure preserving transformations on a probability space. Usually the focus is on "essentially free" actions, namely actions for which almost all stabilizers are tirival. Classically the methods are analitic and combinatorial.

Recently it becomes more and more clear that in the study of non essentially free actions - sophisticated group theoretic tools also come into the picture. I will try to demonstrate this by an array of recent results due to Bader-Lacreux-Duchnese, Tucker-Drob, as well as some joint papers with Abert and Virag and myself.

בשעה 11:15 יערך טכס חלוקת הפרסים לזוכים בתחרות בר-אילן במתמטיקה לסטודנטים

Light refreshments will be served at 11:45 AM next to the Colloquium Room

כיבוד קל יוגש לפני ההרצאה בשעה 11:45 בחדר ע"י חדר המחלקה

The subject of this talk is the analysis of pure point distributions that have a pure point spectrum.

It will be discussed in the framework of “quasi-crystals" inspired by the experimental discovery in the middle of the '80s s of non-periodic atomic structures with diffraction patterns consisting of spots.

Based on joint work with Alexander Olevskii

~~ It is well known that the generating function of a generic finitely presented algebra is rational.

The purpose of this talk is to present an answer on a similar question in the case of algebraic operads.

Namely, I will show that the generating series of a generic nonsymmetric operad is an algebraic function

and the generating series of a generic symmetric operad is differentially algebraic.

Despite the motivation coming from the operad theory, a substantial part of the talk will only deal with the avoidance problems for (labeled) rooted trees hence will be accessible to nonspecialists.

based on joint work with D.Piontkovsky (arXiv:1202.5170).

~~One of the major achievements of statistical mechanics is the development of theoretical tools to bridge between the microscopic description of a system and its observed macroscopic behavior, tracking the emergence of large-scale phenomena from the mechanistic description of the system’s interacting components. A key factor in determining this emergent behavior is associated with the underlying geometry of the system’s interactions - a natural notion when treating structured systems, yet difficult to generalize when approaching complex systems. Indeed, social, biological and technological systems feature highly random and non-localized interaction patterns, which challenge the classical connection between structure, dimensionality and dynamics, and hence confront us with a potentially new class of dynamical behaviors. To observe these behaviors we developed a perturbative formalism that enables us to predict an array of pertinent macroscopic functions directly form the microscopic model describing the system’s dynamics. We find that while microscopically complex systems follow diverse rules of interaction, their macroscopic behavior condenses into a discrete set of dynamical universality classes.

Relevant papers:

Universality in network dynamics. Nature Physics 9, 673–681 (2013) doi:10.1038/nphys2741

Network link prediction by global silencing of indirect correlations. Nature Biotechnology 31, 720–725 (2013) doi:10.1038/nbt.2601

~~We compare several growth models on the two dimensional lattice. In some models, like internal DLA and rotor-router aggregation, the scaling limits are universal; in particular, starting from a point source yields a disk. In the abelian sandpile, particles are added at the origin and whenever a site has four particles or more, the top four particles topple, with one going to each neighbor. Despite similarities to other models, for the sandpile, the intriguing pattern that arises is not circular and depends on the particular lattice. A scaling limit exists for the sandpile, as was recently shown by Pegden and Smart, but it is not universal and still mysterious. This research has been greatly influenced by pictures of the relevant sets, which I will show in the talk. They suggest a connection to conformal mapping which has not been established yet.

Talk based on joint works with Lionel Levine

~~

In this talk we will present a model based on random walks for the density of subsets in finitely generated groups.

The main focus will be on the group large sieve method which is a tool for estimating the density by investigating the finite quotients of the group. We will describe applications of this method for linear groups as well as for mapping class groups.

~~

The concept of group duality is fundamental in the analysis of locally compact abelian groups.

The theory of (analytic) quantum groups was developed in order to provide a framework for duality of general locally compact groups.

The simple set of axioms describing "locally compact quantum groups" (LCQGs) introduced in '00 by Kustermans and Vaes is built on preceding, deep works of Kac and Vainerman, Enock and Schwartz, Woronowicz, Baaj and Skandalis, Masuda and Nakagami and many others.

LCQGs have an intriguing structure theory, and numerous results on locally compact groups have already been generalized to LCQGs.

In this talk we will motivate and introduce the definition of LCQGs, explain and exemplify how they are constructed and mention some of their applications.

Afterwards, we shall describe a generalization of recent work on aspects of ergodic theory of semigroup actions on von Neumann algebras to the context of quantum semigroups.

These results give a Jacobs-de Leeuw-Glicksberg splitting at the von Neumann algebra level.

TBA

: I will introduce a variational problem that consists of the classic isoperimetric problem, i.e. minimization of perimeter subject to a volume constraint, perturbed by a nonlocal term modeling long-range interactions. This geometry problem is the focus of much activity these days and I will survey some results of my own and of others aimed at better understanding the rich energy landscape that emerges from the interplay between these two competing terms in the problem.

We start with a simple fact: the fundamental solutions of the Laplacian in Rn can be continued as multi-valued

analytic functions in Cn up to the complex bicharacteristic conoid. This extension ramies around the complex

isotropic cone: z2

1 + z2n

= 0 and has "moderate growth".

For an elliptic linear partial dierential operator of the second order with analytic coe-cients and simple complex characteristics in an open set Rn, this may be generalized: every fundamental solution can be continued at least locally as a multi-valued analytic function in Cn up to the complex bicharacteristic conoid.

This holomorphic extension is ramied around the bicharacteristic conoid and belongs to the so-called Nilsson

class ("moderate growth").

In fact, those results remain true for such operators with degree bigger than 4 , but the proofs are different due to the lack of natural geodesic distance associated to the operators

Those results may be connected with D-module theory, and more precisely with regular holonomic D-Modules.

We'll explain this link and state a general conjecture

Paul Cohen showed that the Continuum Hypothesis is independent of the usual axioms of set theory. His solution involved a new apparatus for constructing models of set theory - the method of *forcing*. As Cohen predicted, the method of forcing became very successful in establishing the independence of various statements from the usual axioms of set theory. What Cohen never imagined, is that forcing would be found useful in proving theorems.

In this talk, we shall present a few results in combinatorics whose proof uses the method of forcing, including our recent resolution of the infinite weak Hedetniemi conjecture.

The talk will be targeted to a general audience.

I will discuss recent progress in the problem of counting nodal domains of eigenfunctions of Laplacian (i.e., counting connected components of the complement to the zero set of real valued eigenfunctions). This in an old question taken up by Courant and his school. One of many intractable questions is under what conditions one have the number of nodal domains to be unbounded as the eigenvalue goes to infinity. The main difficulty in this problem is that it is known not to be a local property.

The example I will consider concerns with eigenfunctions of the Laplace-Beltrami operator on compact hyperbolic surfaces. The distinctive property of such a setup is its Quantum (Unique) Ergodicity (to be explained). I will discuss how this could be used in order to deduce strong bounds on eigenfunctions and how this forces the number of nodal domains to grow with the eigenvalue.

(Joint work with J. Bernstein, A. Gosh, P. Sarnak)

Wolf Prize Day 2012

Wolf Prize Day 2012

A central question in the theory of random walks on groups is how symmetries

of the underlying space gives rise to structure and rigidity of the random

walks. For example, for nilpotent groups, it is known that random walks have

diffusive behavior, namely that the rate of escape, defined

as the expected distance of the walk from the identity satisfies

E|Xn|~=n^{1/2}. On nonamenable groups, on the other hand we have E|Xn|

~= n. (~= meaning upto (multiplicative) constants )-

In this work, for every 3/4 <= \beta< 1 we construct a finitely generated

group so that the expected distance of the simple random walk from its

starting point after n steps is n^\beta (up to constants). This answers a

question of Vershik, Naor and Peres. In other examples, the speed exponent can

fluctuate between any two values in this interval.

Previous examples were only of exponents of the form 1-1/2^k or 1 , and were

based on lamplighter (wreath product) constructions.

(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups)

In this lecture we will describe how a variation of the lamplighter

construction, namely the permutational wreath product, can be used to get

precise bounds on the rate of escape in terms of return probabilities of the

random walk on some Schreier graphs. We will then show how groups of

automorphisms of rooted trees, related to automata groups , can then be

constructed and analyzed to get the desired rate of escape. This is joint

work with Balint Virag of the University of Toronto.

No previous knowledge of randopm walks, automaton groups or wreath products is

assumed.

Joint work with Sebastian Neumayer, Gil Zussman and Eytan Modiano

Communication networks are vulnerable to natural

disasters, such as earthquakes or floods, as well as to physical

attacks, such as an Electromagnetic Pulse (EMP) attack. Such

real-world events happen in specific geographical locations and

disrupt specific parts of the network. Therefore, the geographical

layout of the network determines the impact of such events on

the network's connectivity. Thus, it is desirable to assess

the vulnerability of geographical networks to such disasters.

I will discuss several algorithms, based on mixed

linear planning and computational geometry, to locate such

vulnerabilities, and present some case studies on real networks.

The Jacobian conjecture is a famous open problem in affine algebraic geometry which says

that a polynomial mapping in n complex variables with constant non zero determinant is injective

and surjective witha polynomial inverse mapping.

In this talk we will outline a proof of the surjectivity for the case of n=2 An abstract is attached

A group $\Gamma$ if of type $F_k$ if it admits an

Eilenberg MacLane complex with finite k-skeleton.

For such groups one can define the (k-1)-dimensional Dehn function,

which measures the difficulty to fill (k-1)-cycles by k-chains.

I will describe the optimal higher-dimensional Dehn functions for uniform

S-arithmetic subgroups of reductive groups over global fields.

I will also discuss a conjectural picture for non-uniform S-arithmetic

groups.

The Riemann Zeta-Function is simple to define but utterly

impossible to find all its zeros. Euler looked at it a century before

Riemann to study the prime numbers and the value of the zeta function at the

integers. By 1860, many far-reaching mysteries were uncovered.

We shall describe them, as well as today's Conjectures of Langlands

and Iwasawa which are built upon them.

- תאריך עדכון אחרון: 24/10/2017