# Colloquium[ Edit ]

## Moderator: Ron Rosenthal

*Abstract:*

Please see the attached file.

*Abstract:*

The Choquet order on measures is used to establish that states on a function system always have a representing measure supported on the set of extreme points of the state space (in a technical sense). We introduce a new operator-theoretic order on measures, and prove that it is equivalent to the Choquet order. This leads to some improvements in the classical theory, but more importantly it leads to some new operator-theoretic consequences. In particular, we establish Arveson’s hyperrigidity conjecture for function systems. This yields a significant strengthening of the classical approximation theorems of Korovkin and Saskin. This is joint work with Matthew Kennedy.

The lecture will take place in Amado 233 (NOTE THE UNUSUAL ROOM).

*Abstract:*

In 1964, Arnold constructed an example of a nearly integrable deterministic system exhibiting instabilities. In the 1970s, Chirikov, a physicist, coined the term “Arnold diffusion” for this phenomenon, where diffusion refers to the stochastic nature of instability.One of the most famous examples of stochastic instabilities for nearly integrable systems,discovered numerically by Wisdom, an astronomer, is the dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. In the talk we will describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distributions given by a deterministic evolution of certain random initial conditions weakly converge to a diffusion process.This result is conceptually different from known mathematical results, where the existence of “diffusing orbits” is shown. This work is based on joint papers with Castejon, Guardia, J.Zhang, and K.Zhang.

*Abstract:*

Suppose that for each point x of a metric space X we are given a compact convex set K(x) in R^D. A "Lipschitz selection" for the family (K(x):x\in X} is a Lipschitz map F:X->R^D such that F(x) belongs to K(x) for each x in X.The talk explains how one can decide whether a Lipschitz selection exists. The result is joint work with P. Shvartsman.

*Abstract:*

Legendre duality is prominent in mathematics, physics, and elsewhere. In recent joint work with Berndtsson, Cordero-Erausquin, and Klartag, we introduce a complex analogue of the classical Legendre transform. This turns out to have ties to several foundational works in interpolation theory going back to Calderon, Coifman, Cwikel, Rochberg, Sagher, and Weiss, as well as in complex analysis/geometry going back to Alexander--Wermer, Slodkowski, Moriyon, Lempert, Mabuchi, Semmes, and Donaldson.

*Abstract:*

Let G be a group and let r(n,G) denote the number of isomorphism classes of n-dimensional complex irreducible representations of G. Representation growth is a branch of asymptotic group theory that studies the asymptotic and arithmetic properties of the sequence (r(n,G)). In 2008 Larsen and Lubotzky conjectured that all irreducible lattices in a high rank semisimple Lie group have the same polynomial growth rate. In this talk I will explain the conjecture and describe the ideas around the proof of a variant of the conjecture: if the lattices have polynomial representation growth (which is known to be true in most cases) then they have the same polynomial growth rate. This is a joint work with Nir Avni, Benjamin Klopsch and Christopher Voll.

*Abstract:*

The Hilbert scheme of points on the plane is one of the central objects of modern geometry. We will review some of the interesting connections of this space with representation theory and the theory of symmetric functions, and we will present some recent geometric results motivated by knot theory.

*Abstract:*

See the attached file.

*Abstract:*

Diffeology, introduced around 1980 by Jean-Marie Souriaufollowing earlier work of Kuo-Tsai Chen, gives a wayto generalize differential calculus beyond Euclidean spaces.Examples include (possibly non-Hausdorff) quotients of manifoldsand spaces of smooth mappings between (possibly non-compact) manifolds.A diffeology on a set declares which maps from open subsetsof Euclidean spaces to the set are "smooth". In spite of its simplicity, diffeology often captures surprisingly rich information.I will present the subject through a sample of examples, results,and questions.

*Abstract:*

In this talk, we will study optimization problems with ambiguous stochastic constraints where only partial information consisting of means and dispersion measures of the underlying random parameter is available. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). The approach is based on the availability of tight upper and lower bounds on the expectation of a convex function of a random variable, first discovered in 1972. We then use these bounds to derive exact robust counterparts of expected feasibility of convex constraints and to construct new safe tractable approximations of chance constraints. We test the applicability of the theoretical results numerically on various practical problems in Operations Research and Engineering.

*Abstract:*

For almost every real number x, the inequality |x-p/q|<1/q^a has finitely many solutions if and only if a>2. By Roth's theorem, any irrational algebraic number x also satisfies this property, so that from that point of view, algebraic numbers and random numbers behave similarly.We will present some generalizations of this phenomenon, for which we will use ideas of Kleinbock and Margulis on analysis on the space of lattices in R^d, as well as Schmidt's subspace theorem.

*Abstract:*

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.

*Abstract:*

Abstract The Graph Isomorphism problem is the algorithmic problem to decide whether or not two given finite graphs are isomorphic. Recent work by the speaker has brought the worst-case complexity of this problem down from exp(\sqrt{n log n}) (Luks, 1983) to quasipolynomial (exp((log n)^c )), where n is the number of vertices.

In the first talk we state a core group theoretic lemma and sketch its role in the algorithm: the construction of global automorphisms out of local information.

The focus of the second and third talks will be the development of the main combinatorial “divide-and-conquer” tool, centered around the concept of coherent configurations. These highly regular structures, going back to Schur (1933), are a common generalization of strongly regular graphs and the more general distance-regular graphs and association schemes arising in the study of block designs on the one hand and the orbital structure of permutation groups on the other hand. Johnson graphs are examples of distance-regular graphs with a very high degree of symmetry.

Informally, the main combinatorial lemma says that any finite relational structure of small arity either has a measurable (say 10%) hidden irregularity or has a large degree of hidden symmetry manifested in a canonically embedded Johnson graph on more than 90% of the underlying set.

*Abstract:*

The purpose of this talk is to introduce a new concept, the "radius" of elements in arbitrary finite-dimensional power-associative algebras over the field of real or complex numbers. It is an extension of the well known notion of spectral radius.

As examples, we shall discuss this new kind of radius in the setting of matrix algebras, where it indeed reduces to the spectral radius, and then in the Cayley-Dickson algebras, where it is something quite different.

We shall also describe two applications of this new concept, which are related, respectively, to the Gelfand formula, and to the stability of norms and subnorms.

*Abstract:*

I will discuss isoperimetric problems and their generalizations and applications. The generalization will involve more global notions of boundary as well as partitions into more than 2 parts.

*Abstract:*

Metallic nanoparticles are optically extraordinary in that they support resonances at wavelengths that greatly exceed their own size. These “surface-plasmon” resonances are normally in the visible range, the (roughly scale-invariant) “colours” sensitively depending on material and shape. In creating the dichroic glass of the Lycurgus cup, the ancient Romans had exploited the phenomenon, probably unknowingly, already in the 4th Century. Nowadays, surface-plasmon resonance is fundamental to the field of nanophotonics, where the goal is to manipulate light on small scales below the so-called diffraction limit. Numerous emerging applications rely on the ability to design and realise compound nanostructures that support tunable and strongly localised resonances.

In this talk I will focus on the misleadingly simple-looking eigenvalue problem governing the colours of plasmonic nanostructures. I’ll present new asymptotic solutions that describe the resonances of the multiple-scale structures ubiquitous in applications: dimers of nearly touching nanowires (2D) and spheres (3D), elongated nano-rods, particles nearly touching a mirror etc. The plasmonic spectrum of these structures can be quite rich. For example, the spectrum of a sphere dimer is compound of three families of modes, each behaving differently in the near-contact limit; moreover, these asymptotic trends mutate at moderately high mode numbers (and again at yet larger mode numbers). This non-commutativity of limits will lead me to a discussion of the convergence in 2D and 3D of the spectrum to a universal accumulation point (the “surface-plasmon frequency”) as the mode number tends to infinity. Time permitting, I will also discuss the asymptotic renormalisation of the singular eigenvalues of closely separated dimer configurations owing to “nonlocal” effects (with Richard V. Craster, Vincenzo Giannini and Stefan A. Maier).

*Abstract:*

I will describe a new approach to chaotic flows in dimension three, using knot theory. I'll use this to show that one can get rid of the singularities in the famous Lorenz flow on R^3, and obtain a flow on a trefoil knot complement. The flow can then be related to the geodesic flow on the modular surface. When changing the parameters, we find other knots for the Lorenz system and so this uncovers certain topological phases in the Lorenz system.

*Abstract:*

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.

*Abstract:*

I will give a very personal overview of the evolution of mainstream applied mathematics from the early 60's onwards. This era started pre computer with mostly analytic techniques, followed by linear stability analysis for finite difference approximations, to shock waves, to image processing, to the motion of fronts and interfaces, to compressive sensing and the associated optimization challenges, to the use of sparsity in Schrodinger's equation and other PDE's, to overcoming the curse of dimensionality in parts of control theory and in solving the associated high dimensional Hamilton-Jacobi equations.

*Abstract:*

Studying the regular part and shock curves for the entropy solutions for scalar conservation laws is a major research in this field. Assume that the initial date is constant in the connected components outside a compact interval, T.P Liu and Dafermos-Shearer obtained an interesting criterion when the solution admits one shock after a finite time for the uniformly convex flux. In this talk I will talk about the same phenomena for the flux which has finitely many inflection points. The proof relies on the structure theorem for entropy solutions for convex flux.

*Abstract:*

Erdős-Ko-Rado type problems' have been widely studied in Combinatorics and Theoretical Computer Science over the last fifty years. In general, these ask for the maximum possible size of a family of objects, subject to the constraint that any two (or three…) of the objects `intersect' or `agree' in some way. A classical example is the so-called Erdős-Ko-Rado theorem, which gives the maximum possible size of a family of k-element subsets of an n-element set, subject to the constraint that any two sets in the family have nonempty intersection. As well as families of sets, one may consider families of more highly structured objects, such as graphs or permutations; one may also consider what happens when additional `symmetry' requirements are imposed on the families. A surprisingly rich variety of techniques from different areas of mathematics have been used successfully in this area: combinatorial, probabilistic, analytic and algebraic. For example, Fourier analysis and representation theory have recently proved useful. I will discuss some results and open problems in the area, some of the techniques used, and some links with other areas.

*Abstract:*

The concept of measurable entropy goes back to Kolmogorov and Sinai who in the late 50ies defined an isomorphism invariant for measure preserving Z-actions. While a similar theory can be developed in an analogous manner for abelian or even amenable groups, the situation gets far more complicated when dealing with groups which are "very" non-commutative, such as free groups. We start the talk with a warm-up about the classical Kolmogorov-Sinai entropy. Using the free group on two generators as an illustrative example, we show how to define cocycle entropy as a new isomorphism invariant for measure preserving actions of quite general countable groups. Further, we draw connections to other notions of entropy and to open problems in the field. We conclude the talk by clarifying pointwise almost sure approximation of cocycle entropy values. To this end, we present a first Shannon-McMillan-Breiman theorem for actions of non-amenable groups. Joint work with Amos Nevo.

*Abstract:*

The monodromy groups associated to differential equations (with regular singularities) have, associated to them a monodromy group. The monodromy group of differential equations associated to hypergeometric functions have generated a lot of interest recently; Peter Sarnak has raised the question of arithmeticity/thinness of these groups. We give a survey of results proved concerning this questions.

*Message:*

NOTE THE UNUSUAL DAY

*Abstract:*

The aim of the lecture is to show the importance of the knowledge of the set of decomposed places in a uniform pro-p extension of number fields for the mu-invariant of the Class group along the tower. The talk will be elementary and easily accessible. In particular, it will start with a presentation of the studied objects (Class group, Iwasawa Invariants, Cebotarev Density Theorem, etc.), of few general facts and open questions in the topic.

*Abstract:*

I will describe a long line of research around the asymptotic density of rational points in transcendental varieties, starting from the work of Bombieri-Pila on analytic curves in the late eighties and on to its vast generalization in the work of Pila-Wilkie about ten years ago. The latter sits at the crossroads between analysis, logic and diophantine geometry, and has attracted considerable attention in the last decade after playing the key role in new proofs of several conjectures on unlikely arithmetic intersections, including the first proof of the Andre-Oort conjecture (by Pila). I will give a taste of the philosophy of these applications in an elementary example. Finally I will discuss one of the main open problems of the area, the Wilkie conjecture, and describe some recent progress obtained in a joint work with Dmitry Novikov.

*Abstract:*

For a finite group G one can consider important structures such as: Expander Graphs, Random Walks, Word Maps, etc. Many properties of these structures can be approached using “Fourier type” sums over the characters of representations of G.

A serious obstacle in applying these Fourier sums, seems to be a lack of control over the dimensions of representations of G.

In my talk, for the sake of clarity, I will discuss only the case of the finite special linear group G=SL(2,F_q). I will show how one can solve several interesting problems by ordering and constructing the representations of G according to their “size”.

This talk is an example from a joint project with Roger Howe (Yale), where we introduce a language to define the “size" of representations, and develop a method to construct representations of finite classical groups according to their “size".

The lecture is accessible to advanced undergraduate students.

*Abstract:*

What does a random planar triangulation on n vertices looks like? More precisely, what does the local neighbourhood of a fixed vertex in such a triangulation looks like? When n goes to infinity, the resulting object is a random rooted graph called the Uniform Infinite Planar Triangulation (UIPT). Angel, Benjamini and Schramm conjectured that the UIPT and similar objects are recurrent, that is, a simple random walk on the UIPT returns to its starting vertex almost surely. In a joint work with Ori Gurel-Gurevich we prove this conjecture. The proof uses the electrical network theory of random walks and the celebrated Koebe-Andreev-Thurston circle packing theorem. We will give an outline of the proof and explain the connection between the circle packing of a graph and the behaviour of a random walk on that graph.

(Please note the unusal location. )

*Abstract:*

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group G, let m(G) be the minimal integer m for which there exists a Galois extension N/Q that is ramified at exactly m primes (including the infinite one). So, the problem is to compute or to bound m(G). In this paper, we bound the ramification of extensions N/Q obtained as a specialization of a branched covering φ: C → P^1(Q) . This leads to novel upper bounds on m(G), for finite groups G that are realizable as the Galois group of a branched covering. Some instances of our general results are: 1 ≤ m(S_m) ≤ 4 and n ≤ m(S^n_m) ≤ n + 4, for all n, m > 0. Here S_m denotes the symmetric group on m letters, and S^n_m is the direct product of n copies of S_m. We also get the correct asymptotic of m(G^n), as n → ∞ for a certain class of groups G. Our methods are based on sieve theory results, in particular on the Green-Tao-Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory. Joint work with Lior Bary-Soroker.

*Abstract:*

The goal of this talk is to convince you that you have been unknowingly using bounded cohomology all your life and to encourage you to come out and use it more openly. To this end we will explain how the natural desire to count leads to bounded cohomology, and how Eudoxus used bounded cohomology to define the ordered field of real numbers around 230 BC. Slightly more recent developments in bounded cohomology and its interactions with geometry, algebra, probability and combinatorics will also be discussed. We will also explain the special relationship between bounded cohomology and the Technion, which goes back if not to ancient times then at least to the 1980s. We will state a number of open problems which can be understood by a first year student, but whose solution might be a challenge even for professional researchers. Throughout the talk we will focus on the second bounded cohomology and its combinatorial description through quasimorphisms.

*Abstract:*

Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

*Abstract:*

I will describe how we can exploit the locality of a maximal independent set (MIS) to the extreme, by showing how to update an MIS in a dynamic distributed setting within only a single adjustment in expectation. The approach is surprisingly simple and is based on a novel analysis of the sequential random greedy algorithm.

No background in distributed computing will be assumed. The talk is based on joint work with Elad Haramaty and Zohar Karnin.

*Abstract:*

I will describe several mathematical models producing large random topological spaces and state results about topological properties of such spaces (their Betti numbers, fundamental groups etc).

*Abstract:*

Needle decomposition is a technique in convex geometry, which enables one to prove isoperimetric and spectral gap inequalities, by reducing an n-dimensional problem to a 1-dimensional one. This technique was promoted by Payne-Weinberger, Gromov-Milman and Kannan-Lovasz-Simonovits. In this lecture we will explain what needles are, what they are good for, and why the technique works under lower bounds on the Ricci curvature.

*Abstract:*

Let p:C^n --> C^m be a polynomial map. The first part of the talk will be about the relation between the singularities of the fibers of p and the analytic properties of push-forwards of smooth measures by p. The second part of the talk will be about applications to counting points on varieties, character sums, and random matrices.

*Abstract:*

In this talk, we will try to illustrate the potential of stochastic calculus as a tool for proving inequalities with a geometric nature. We'll do so by focusing on the proofs of two new bounds related to the Gaussian Ornstein-Uhlenbeck convolution operator, which heavily rely on the use of Ito calculus. The first bound is a sharp robust estimate for the Gaussian noise stability inequality of C. Borell (which is, in turn, a generalization of the Gaussian isoperimetric inequality). The second bound concerns with the regularization of $L_1$ functions under the convolution operator, and provides an affirmative answer to the Gaussian variant of a 1989 question of Talagrand. Based in part on a joint work with James Lee.

*Abstract:*

The inverse Galois problem - show that every finite group is the Galois group of a polynomial with rational coefficients - will be the common thread of the talk. Going back to the early stages of the problem, I will focus on the geometric approach, which is based on some specialization process. More than two hundred years after Galois, the problem remains largely open. I will explain some recent progress, point out some difficulties and indicate some new research lines.

*Abstract:*

This is not a mathematics talk but it is a talk for mathematicians.

Too often, we think of historical mathematicians as only names assigned to theorems.

With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.

*Abstract:*

I will review recent results concerning the Ginzburg - Landau equations. These equations were first developed to understand macroscopic behaviour of superconductors; later, together with their non-Abelian generalizations - the Yang-Mills-Higgs equations, they became a key part of the standard model in elementary particle physics. They also have found important applications in geometry and topology.

These equations have remarkable solutions, localized topological solitons, called the magnetic vortices in the superconductivity and the Nielsen-Olesen or Nambu strings in the particle physics, as well as extended ones, magnetic vortex lattices.

I will review the existence and stability theory of these solutions and how they relate to the modified theta functions appearing in number theory and algebraic geometry. Certain automorphic functions play a key role in the theory described in the talk.

*Abstract:*

This talk has to do with knots invariantswhich are elements of the Brauer groups and of the Tate Shafarevitchgroups of curves over number fields. Constructing theseinvariants involves a close analysis of the canonicalAzumaya algebra which is defined over an open dense subsetof Thurston's canonical curve in the representation variety of the knot group. This is joint work with Alan Reid and Matt Stover.

*Abstract:*

The two main geometric invariants of a rational function are its monodromy group and ramification type. I will explain the progress made during the past 125 years towards determining all possibilities for these invariants, including contributions by Hurwitz, Zariski, Thom, Guralnick, Thompson, and Aschbacher. I will also present applications to number theory, algebraic geometry, and complex analysis.

*Abstract:*

Nature and human society oﬀer many examples of self-organized behavior: ants form colonies, birds ﬂock together, mobile networks coordinate their rendezvous, and human opinions evolve into parties. These are simple examples for collective dynamics, in which local interactions tend to self-organize into large scale clusters of colonies, ﬂocks, parties, etc. We discuss the dynamics of such systems, driven by “social engagement” of agents with their neighbors.

We will focus on two natural questions which arise in this context. First, what is the large time behavior of such systems? The underlying issue is how different rules of engagement influence the formation of large scale patterns such as clusters, and in particular, the emergence of “consensus”. We propose an alternative paradigm based on the tendency of agents “to move ahead” which leads to the emergence of trails and leaders.

Second, what is the group behavior of systems which involve a large number of agents? Here one is interested in the qualitative behavior of the group rather than tracing the dynamics of each of its agents. Agent-based models lend themselves to kinetic and hydrodynamic descriptions. It is known that smooth solutions of “social hydrodynamics”, if they exist, must ﬂock. But alignment-based models reﬂect the competition on resources, and left unchecked, may lead to ﬁnite-time singularities. We discuss the global regularity of such solutions for sub-critical initial configurations.

*Abstract:*

The Merkurjev-Suslin theorem asserts 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 is the minimal number of symbols k for which every n-torsion class is a product of k symbols.

We will survey some of the known results and (if time permits) give the idea behind the proof of a bound on the minimal number of symbols in a geometric situation (when the base field contains an algebraically closed field).

*Abstract:*

Quenched invariance principle (convergence in law to Brownian motion) for random walks on infinite percolation clusters and among i.i.d. random conductances in $\mathbb{Z}^d$ were proved during the last two decades. The proofs of these results strongly rely on the i.i.d. structure of the models and some stochastic domination with respect to super-critical Bernoulli percolation. Many important models in probability theory and in statistical mechanics, in particular, models which come from real world phenomena, exhibit long range correlations. In this talk I will present a new quenched invariance principle, for simple random walk on the unique infinite percolation cluster for a general class of percolation models on $\mathbb{Z}^d$, $d\geq 2$ with long-range correlations. This gives new results for random interlacements in dimension $d\geq 3$ at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field. An essential ingredient of the proof is a new isoperimetric inequality for correlated percolation models.

Based on a joint work with Eviatar Procaccia and Artëm Sapozhnikov

*Abstract:*

--- SPECIAL COLLOQUIUM: NOTE THE SPECIAL DATE AND TIME ---

An amazing discovery of physicists is that there are many seemingly quite different quantum field theories that lead to the same observable predictions. Such theories are said to be related by dualities. A duality leads to interesting mathematical consequences; for example, certain K-theory groups on the two spacetime manifolds have to be isomorphic. We will explain how some of these K-theory isomorphisms predicted by physics correspond to certain cases of the Baum-Connes Conjecture, which originally was introduced for totally different reasons.

*Abstract:*

James Cannon conjectured that a torsion free hyperbolicgroup whose boundary at infinity is a 2-sphere is the fundamental group of a closed hyperbolic 3-manifold. My first goal is to explain what all these words mean and why one could hope that this be true. I will then explain how this fits in with other problems, including another much older conjecture of Cannon's (that is known to be"slightly false") that gave a criterion for a topological space to be a topological manifold.

*Abstract:*

Classical Teichmüller theory is concerned with the study of (marked) Riemann surfaces. Due to the uniformization theorem, the Teichmüller space of a surface can be also described as the space of (marked) hyperbolic structures on a given topological surface S. A hyperbolic structure on S is governed by a discrete embedding of the fundamental group of S into the Lie group PSL(2,R). Higher Teichmüller theory concerns the study of more complicated geometric structures on surfaces, which are governed by discrete embeddings of the fundamental group of S into more general Lie groups, such as PSL(n,R) or Sp(2n,R).

I will give an introduction to higher Teichmüller theory, review some of the recent results and discuss current and future challenges.

*Abstract:*

We recall some of the theory of modular forms starting with results of Jacobi on the theta function and sum of four squares. We will look at Ramanujan's discriminant function, Hecke's operators and the theory of new forms of Atkin and Lehner. We will finish the talk with a new result about the characterization of the space of new forms. We do not assume any knowledge of modular forms.

*Abstract:*

Zeros of vibrational modes have been fascinating physicists forseveral centuries. Mathematical study of zeros of eigenfunctions goesback at least to Sturm, who showed that, in dimension d=1, the n-theigenfunction has n-1 zeros. Courant showed that in higher dimensionsonly half of this is true, namely zero curves of the n-th eigenfunction ofthe Laplace operator on a compact domain partition the domain into atmost n parts (which are called "nodal domains").

It recently transpired (first on graphs with a subsequentgeneralization to manifolds) that the difference between this upperbound and the actual value can be interpreted as an index ofinstability of a certain energy functional with respect to suitablychosen perturbations. We will discuss two examples of thisphenomenon: (1) stability of the nodal partitions with respect to aperturbation of the partition boundaries and (2) stability of aneigenvalue with respect to a perturbation by magnetic field. In bothcases, the "nodal defect" of the eigenfunction coincides with theMorse index of the energy functional at the corresponding criticalpoint.

Based on joint work with R. Band, P.Kuchment, H. Raz, U.Smilansky andT. Weyand.

*Abstract:*

Suppose that one is given n points z_1, ... z_n in the unit disc and n complex numbers w_1, ..., w_n. It is always possible (and easy) to find an analytic function that interpolates this data, meaning that f(z_i) = w_i for all i. A more difficult problem is to determine whether or not there is an analytic function which interpolates the data and, in addition, is bounded on the disc by some given constant, say 1.

In 1916, G. Pick solved this problem, and gave an effective necessary and sufficient condition for the existence of such an interpolating function. Pick's theorem turns out to be best understood in the setting of Hilbert function spaces, and has been of great interest to mathematicians as well as engineers.

A couple of decades ago the question "for which algebras (other than the algebra of bounded analytic functions on the disc) does a theorem like Pick's theorem hold?" was raised, and eventually found a complete solution by Quiggin, McCullough and Agler & McCarthy. These algebras sometimes go under the name "Pick algebras". Subsequent work has clarified the structure of these algebras, and the classification of these algebras up to isomorphism has been one of my favourite problems in the last five years or so.

In my talk I will describe Pick's theorem, how it fits in the framework of Hilbert function spaces, what Pick algebras look like and what we know about the classification of these algebras.

The bottom line will be that every Pick algebra can be viewed as an algebra of bounded analytic functions on some analytic variety, and that the analytic varieties provide geometric invariants for the algebraic and operator algebraic structure of Pick algebras.

*Abstract:*

Mapping class groups (and their cousins - automorphism groups of free groups) are some of the most ubiquitous groups in mathematics. Their representation theory is known to be very rich, but still remains very mysterious and many very basic questions remain unanswered. We will describe some of what is known about this representation theory, and discuss a recent result answering one such basic question which has been open for some time: given an infinite order mapping class, is there a representation in which its image has infinite order?

*Abstract:*

We study the stability and instability of equilibrium solutions of a reaction-diffusion equationwith different types of boundary conditions. Stable solutions play an important role in thestudy of pattern formation in biology. Unstable solutions lack of physical significance. it turnsout that the stability depends on the curvature of the manifold and the geometric propertiesof the boundary. The main tool is the Bochner-Weitzenböck formula. The talk is intended to givea flavor of how to apply this formula and does not require any knowledge of geometric analysis.

*Abstract:*

Renormalization is a central idea of contemporary Dynamical Systems Theory, It allows one to control small scale structure of certain classes of systems, which leads to universal features of the phase and parameter spaces. We will review several occurrences of Renormalization in Holomorphic Dynamics: for quadratic-like, Siegel, and parabolic maps that enlighten the structure of many Julia sets and the Mandelbrot set. In particular, these ideas helped to construct examples of Julia sets of positive area (resolving a classical problem in this field). First examples were constructed by Buff and Cheritat about 10 years ago, and more recently a different class, with some interesting new features, was produced by Avila and the author. In the talk, we will describe these developments.

*Abstract:*

A number of topics in the qualitative spectral analysis of the Schr\"odinger operator $-\Delta + V$ are surveyed. In particular, results concerning the positivity and semiboundedness of this operator. he attention is focused on conditions both necessary and sufficient, as well as on their sharp corollaries.