Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll focus on one particular application of this approach that yields strong invariants of knots. I'll discuss a mysterious connection between these knot invariants and string theory, as well as a recent result (joint with Tobias Ekholm and Vivek Shende) that the invariants completely determine the underlying knot.

Thomson's problem, which goes back to 1904, asks how N points will arrange themselves on the sphere so as to minimize their electrostatic potential. A more general problem asks what happens for other power law potentials. In spite of quite a bit of experimental evidence accumulated over the past century, and some spectacular results for values of N associated with highly symmetric polyhedra, there have been few rigorous results for the modest case N=5. In my talk I will explain my recent proof that, for N=5, the triangular bi-pyramid is the minimizer with respect to all power laws up to a constant S=15.04808..., and then the minimizer changes to a pyramid with square base. My talk will have some nice computer animations.

The asymmetric simple exclusion process (ASEP) is a Markov chain describing particles hopping on a 1-dimensional finite lattice. Particles can enter and exit the lattice at the left and right boundaries, and particles can hop left and right in the lattice, subject to the condition that there can be at most one particle per site. The ASEP has been cited as a model for traffic flow, protein synthesis, the nuclear pore complex, etc. In my talk I will discuss joint work with Corteel and with Corteel-Mandelshtam, in which we describe the stationary distribution of the ASEP and the 2-species ASEP using staircase tableaux and rhombic tilings. I will also discuss the link between these models and Askey-Wilson polynomials and Macdonald-Koornwinder polynomials.

Strominger, Yau, and Zaslow proposed a geometric explanation for mirror symmetry via a dualization procedure relating symplectic manifolds equipped with Lagrangian torus fibration with complex manifolds equipped with totally real torus fibrations. By considering the family of symplectic manifolds obtained by rescaling the symplectic form, one obtains a degenerating family of complex manifolds, which is expected to be the mirror.

Because of convergence problems with Floer theoretic constructions, it is difficult to make this procedure completely rigorous. Kontsevich and Soibelman thus proposed to consider the mirror as a rigid analytic space, defined over the field C((t)), equipped with the non-archimedean t-adic valuation, or more generally over the Novikov field. This is natural because the Floer theory of a symplectic manifold is defined over the Novikov field.

After explaining this background, I will give some indication of the tools that enter in the proof of homological mirror symmetry in the simplest class of examples which arise from these considerations, namely Lagrangian torus fibrations without singularities.

A basic open problem in symplectic topology is to classify Lagrangian submanifolds (up to, say, Hamiltonian isotopy) in a given symplectic manifold. In recent years, ideas from mirror symmetry have led to the realization that even the simplest symplectic manifolds (e.g. vector spaces or complex projective spaces) contain many more Lagrangian tori than previously thought. We will present some of the recent developments on this problem, and discuss some of the connections (established and conjectural) between Lagrangian tori, cluster mutations, and toric degenerations, that arise out of this story.

The talk is purely historical, starting with PDE's in the 19-th century (Fourier transform and the heat equation, Cauchy-Kowalewski Theorem, harmonic functions, Maxwell equations) to the 20-th through Hadamard's work, with particular attention to developments in the second half of the last century. I will try to explain the transition from linear differential operators to pseudodifferential operators and the successful application of the latter to the complete analysis of linear PDE with simple real characteristic (basic definitions will be provided under assumption that the audience knows little about the whole subject). I hope to have time to indicate some glaring open problems.

In 2009 V. Markovic and the speaker proved that there are ubiquitous nearly geodesic subgroups in the fundamental groups of closed hyperbolic 3-manifolds. Since then there have been many attempts (some successful) to extend these results to other settings, including lattices in other Lie groups, nonuniform lattices, delta-hyperbolic groups, and the mapping class group. After a review of the fundamental principles and methods, I will try to describe some of the successes, some of the difficulties, and some of the applications of these kinds of results.

Modeling of a wide range of physical phenomena leads to tracking fronts moving with curvature-dependent speed. A particularly natural example is where the speed is the mean curvature. If the movement is monotone inwards, then the arrival time function is the time when the front arrives at a given point. It has long been known that this function satisfies a natural differential equation in a weak sense but one wonders what is the regularity. It turns out that one can completely answer this question. It is always twice differentiable and the second derivative is only continuous in very rigid situations that have a simple geometric description. The proof weaves together analysis and geometry.

We approach the formalism of quantum mechanics from the logician point of view and treat the canonical commutation relations and the conventional calculus based on it as an algebraic syntax of quantum mechanics. We then aim to establish a geometric semantics of this syntax. This leads us to a geometric model, the space of states with the action of time evolution operators, which is a limit of finite models. The finitary nature of the space allows us to give a precise meaning and calculate various classical quantum mechanical quantities. This talk is based on my paper "The semantics of the canonical commutation relation" arxiv.org/abs/1604.07745

Buoyancy forces result from density variations, often due to temperature variations, in the presence of gravity. Buoyancy-driven fluid flows shape the weather, ocean dynamics and climate, and the structure of the earth and stars. In 1916 Lord Rayleigh published a paper entitled "On Convection Currents in a Horizontal Layer of Fluid, when the Higher Temperature is on the Under Side" that introduced a minimal mathematical model of buoyancy-driven fluid flows now known as "Rayleigh-Benard convection" that has served for a century as one of the primary paradigms for nonlinear science, dynamical pattern formation, chaos and turbulence. In this presentation, following an introduction to and history of Rayleigh's model and review of some applications of convection, we describe recent progress and open challenges for mathematical analysis in the strongly nonlinear regime of turbulent convection.

The prediction that solutions of the Einstein equations in the interior of black holes must always terminate at a singularity was originally conceived by Penrose in 1969, under the name of "strong cosmic censorship hypothesis." The nature of this break-down (i.e. the asymptotic properties of the space-time metric as one approaches the terminal singularity) is not predicted, and remains a very hotly debated question to this day. One key question is the causal nature of the singularity (space-like, vs null for example). Another is the rate of blow-up of natural physical/geometric quantities at the singularity. Mutually contradicting predictions abound in this topic. Much work has been done under the assumption of spherical symmetry (for various matter models). We present recent developments (due to the speaker and G. Fournodavlos) which go well beyond this restrictive class. A key role is played by the axial symmetry reduction of the Einstein equations, where a wave map structure appears.

I will survey some applications of Donaldson's technique of quantitative transversality of "approximately holomorphic" functions in symplectic geometry. I will explain the basic terms and present the main ideas of the technique. Donaldson used it to show that the Poincare dual of any sufficiently large multiple of an integral symplectic form is represented by a symplectic submanifold. Another application is joint work with E. Giroux in which we prove the existence of Lefschetz fibrations on certain symplectic manifolds.

To each complete Riemannian manifold M is associated a dynamical system—the geodesic flow on the unit tangent bundle SM. To what extent are the dynamical properties of this flow wedded to the geometry of M? I will discuss some historical highlights, open questions and recent breakthroughs.

Symplectic topology is unique within geometry, in that the deeper structure of the spaces under consideration appears only after non-local "instanton corrections" have been taken into account. This is most readily apparent from a string theory motivation, but it also has a direct impact on classical problems from Hamiltonian mechanics. In the theory, the instanton corrections are set up as small perturbations, which corresponds to thinking of the target space as having infinitely large size (the "large volume limit"). Mirror symmetry suggests that it would be interesting to keep the size finite. Attempting to do that has seemingly paradoxical consequences, which one can sometimes get a handle on by changing the space involved. The talk will give an introduction to this problem, based on simple examples, and explain a little of what is known or expected.

A fundamental issue permeating the study of quasilinear hyperbolic PDEs is that, aside from equations with special structure, initially smooth solutions are expected to often form shocks in finite time. Roughly, a shock is a singularity such that the solution remains bounded but its derivatives blow up. Although many such results have been proved in one spatial dimension, there are very few rigorous results in higher dimensions. In this colloquium, I will provide an overview of recent progress on the formation of shocks in two and three spatial dimensions. I will start by describing prior contributions from many researchers including B. Riemann, P. Lax, F. John, S. Alinhac, and especially D. Christodoulou, whose remarkable 2007 monograph yielded a sharp description of shock formation in vorticity-free small-data solutions to the relativistic Euler equations in three spatial dimensions. I will then describe some of my recent work, some of it joint with J. Luk, G. Holzegel, S. Klainerman, and W. Wong, in which we extended Christodoulou's framework to prove similar results for general classes of equations and new types of initial conditions. I will especially focus on my work with J. Luk on the compressible Euler equations, in which we obtained the first constructive result on the long-time behavior of the vorticity up to the first singularity: for an open set of initial conditions, generic first derivatives of the velocity blow up but the vorticity remains bounded! The proof relies on a new formulation of the equations exhibiting surprisingly good structures, reminiscent of the type found in equations that admit global solutions. Remarkably, the good structures are a key ingredient in proving that a singularity forms. Throughout the talk, I will highlight some of the main ideas behind the analysis including the critical role played by geometric decompositions adapted to characteristic hypersurfaces.

Thanksgiving

I will explain how certain integrable structures give rise to meaningful probabilistic systems and methods to analyze them. Asymptotics reveal universal phenomena, such as the Kardar-Parisi-Zhang universality class. No prior knowledge will be assumed.

We will give an overview of the dynamics of polynomial diffeomorphisms of complex 2-space.

Recently, Balogh-Morris-Samotij and Saxton-Thomason developed a method of (approximately) counting independent sets in hypergraphs. This technique, now known as the “Container Method,” has already had many applications in extremal and probabilistic combinatorics, additive number theory and discrete geometry. For example it provides approaches to proving classical extremal results (e.g. the theorems of Szemeredi and Turan) in a random setting, and to asymptotic counting of discrete structures such as maximal triangle-free graphs and sum-free sets, and sets without k-term arithmetic progressions. I will give an overview of the area and sketch some sample applications of the technique.

We will discuss veering triangulations associated to pseudo-Anosov mapping tori, and how they arise dynamically. We will survey some of the results obtained regarding these triangulations. Then we will discuss a new construction of these triangulations associated to certain pseudo-Anosov flows, which is joint work with Francois Gueritaud.

In this talk, I will explain some connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. I will begin with the theorem of Mordell and Weil from the 1920s, presented from a dynamical point of view. I will continue by describing a dynamical/geometric proof of a result of Masser and Zannier about torsion points on elliptic curves and “unlikely intersections.” Finally, I aim to explain the role of dynamical stability and bifurcations in finiteness statements.

Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as “chemotaxis” or “phototaxis.” Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important.

In this talk we will survey our recent results on mathematical models for phototaxis. We will start with a stochastic model, an interacting particle system, and a system of PDEs. Our main theorem establishes the system of PDEs as the limit dynamics of the particle system. We will then present another approach in which we develop particle, kinetic, and fluid models for phototaxis. We will conclude with describing our recent work on modeling selective local interactions with memory.

The main object of study of this talk are matrices whose entries are linear forms in a set of formal variables (over some field). The main problem is determining if a given such matrix is invertible or singular (over the appropriate field of rational functions).

As it happens, this problem has a dual life; when the underlying variables commute, and when they do not. Most of the talk will be devoted to explaining (some of) the many origins, motivations and interrelations of these two problems, in computational complexity, non-commutative algebra, (commutative) invariant theory, quantum information theory, optimization and more.

I will describe the state-of-art on the complexity of these problems. For the non-commutative version, where even decidability took decades to establish, we have recently found (with Garg, Gurvits and Olivera) a deterministic polynomial time algorithm (over the rationals). Strangely perhaps, for the commutative version, where decidability is nearly trivial, the best known deterministic algorithm requires exponential time. A probabilistic polynomial time algorithm is known, and making it deterministic is major open problem.

The Cherednik algebra \(B(c,n)\), generated by symmetric polynomials and the quantum Calogero-Moser Hamiltonian, appears in many areas of mathematics. It depends on two parameters - the coupling constant c and number of variables n. I will talk about representations of this algebra, and in particular about a mysterious isomorphism between the representations of \(B(m/n,n)\) and \(B(n/m,m)\) of minimal functional dimension. We explain the symmetry between m and n by showing that the characters of these representations can be expressed in terms of the colored HOMFLY polynomial of the torus knot \(T(m/d,n/d)\), where \(d=\mathrm{gcd}(m,n)\).

The talk is based on my joint work with E. Gorsky and I. Losev.

This talk will review conjectures and theorems about some simple dynamical models. These include random walk, edge-reinforced random walk, probabilistic cellular automata and random permutations. Although these models look quite classical some are closely related to quantum physics. Pictures will help illustrate the rich dynamics.

For the past century, mathematics has played a central role in the trends and fashions of education in this country. This talks tries to survey some of those trends, through wars and depressions, leading up to the current controversies about standards and curricula. This is a mathematician's view of mathematics education over the twentieth century.

We review various combinatorial problems with underlying classical or quantum integrable structures. We present a few mathematical problems or constructs, most of them combinatorial in nature, that either were introduced to explicitly solve or better understand physical questions (Mathematical Physics) or can be better understood in the light of physical interpretations (Physical Mathematics). The frontier between the two is subtle, and we will try to make this more concrete in a few examples.

Our main character is integrability, whether discrete, classical or quantum, which is a manifestation of the underlying symmetries of the problems at hand, and allows often for compact and elegant solutions. The objects we discuss are: Lorentzian triangulations, planar maps for 2D quantum gravity, and the representation-theoretic content of generalized Heisenberg quantum spin chains.

The structures encountered along the way are: paths and trees, discrete (non-commuting) integrable systems, Cluster Algebras, Macdonald operators and Double Affine Hecke Algebras.

(Based on joint works with J. Bouttier, E. Guitter, C. Kristjansen and R. Kedem.)

I will discuss recent work on the structure of black hole interiors for dynamical vacuum spacetimes (without any symmetry) and what this means for the question of the nature of generic singularities in general relativity and the celebrated strong cosmic censorship of Penrose. This is joint work with Jonathan Luk (Cambridge).

The goal of the lecture is to show interplay between supersymmetry and tensor categories. The main idea of supersymmetry is to work with \(Z_2\)-graded objects and modify usual identities by so called sign rule. Original motivation comes from physics and topology. For example, the complex of differential forms on a manifold is a supermanifold and the De Rham differential is a vector field on this supermanifold. One way to approach supersymmetry is via rigid symmetric tensor categories starting from the category of \(Z_2\)-graded vector spaces.

After elementary introduction to supersymmetry and tensor categories I illustrate how both theories enrich each other on the following examples:

- Theorem of Deligne that any rigid symmetric tensor category satisfying certain finiteness conditions is in fact the category of representations of a supergroup;
- Mixed Schur Weyl duality in supercase
- Construction of universal symmetric tensor categories as abelian envelopes of the Deligne' category \(\mathrm{Rep}(GL(t))\).

The proof of the prime number theorem (for the Riemann zeta function) is based on an old idea of de la Valee Poussin. The Poussin method has been generalized to all L-functions associated to automorphic forms.

Given two L-functions with Dirichlet coefficients a(n), b(n),
respectively, the Rankin-Selberg L-function is defined to have Dirichlet
coefficients given by the product of a(n) and the complex conjugate of
b(n). Although it is conjectured that Rankin-Selberg L-functions on GL(n)
should be automorphic for GL(n^{2}) this has not been proved yet for n>2,
and because of this (up until now) it has not been possible to obtain a
strong version of the prime number theorem for Rankin-Selberg L-functions
in general. I shall explain (in an elementary manner) a new approach
(based on an idea of Sarnak) for obtaining a strong version of the prime
number theorem for Rankin-Selberg L-functions on GL(n) with n > 2.
This talk is based on joint work with Xiaoqing Li.

There are various notions of an “optimal position” for a knot K in the 3-sphere. For example, Schubert introduced the idea of bridge number for a knot, in which the knot is described as some number of trivial arcs (bridges) in two 3-balls glued together along their boundaries. In a refinement of this, one can place a knot to lie with minimal complexity relative to a family of parallel planes; this is the now-standard notion of thin position for a knot. I'll explain some of the reasons why this has been a very useful idea in knot theory and 3-manifolds. In joint work with Hass and Rubinstein we explored replacing the family of parallel planes in thin position by an n-parameter family of surfaces. I'll discuss some of these results for n=2, and some natural (so far unanswered) questions when n is 3 or higher.

I will discuss some elementary constructions inspired by the Almost Abelian Anabelian geometry program of F. Bogomolov. This is a joint work with F. Bogomolov.

A standard way to study groups is to look for big abelian subgroups. For finite groups this is difficult and only moderately successful, but for compact connected Lie groups it works wonderfully. Every compact connected Lie group G has a maximal torus that's unique up to conjugation, and G can be completely described by a simple combinatorial structure (root datum) related to such a torus.

Many compact Lie groups have maximal abelian subgroups that are not tori. (By an old theorem of Borel and Serre, this is related to torsion in the cohomology of G.) I'll describe (familiar and unfamiliar) examples of these subgroups, and work with Gang Han aimed at a "root datum" description of G in terms of such subgroups.

Each compact Riemannian manifold has geodesic flow and the set of eigen-functions of Laplace operator.The connectiom between them is a very subtle issue. In the integrable case geodesic flows have discrete spectrum and localized eigen-functions (in most cases).

If geodesic flows are ergodic we have Schnirelman theorem (proven also by Zelditch and Colin de Verdier) according to which the majority of eigen-function have in the limit uniform distribution. Rudnick and Sarnak raised a question about the existence of unbounded sets of non-uniformly distributed eigen-funstions. N.Anantharaman proved a remarkable theorem giving an estimate from below of the entropy of any sequence of eigen-fuctions in the case of Anosov flows.

Recently together with Ilya Vinogradov we constructed infinite sequences of eigen-functions of Laplacians in some two-dimensionaal domains in which the billiard flow is hyperbolc. The properties of these eigen-functions will be discussed in the talk.

The Kakeya problem is a geometry problem about the way
cylindrical tubes overlap in Euclidean space. A Kakeya set of thickness
delta is a set K in Euclidean space R^{n} which contains a cylinder of length
1 and radius delta pointing in every direction. How small can such a set
be? The known upper and lower bounds essentialy match in two dimensions,
but they are far apart when n is at least 3. This geometry problem is
connected to open problems in Fourier analysis, and it has been studied
intensively for several decades, but it remains wide open. In this talk, I
will introduce the problem and explain why it is difficult. Recent work
suggests that the problem is connected with algebraic geometry, and I will
explain this perspective.

Cluster algebra is a commutative algebra with distinguished set of generators forming a nice combinatorial structure. They were introduced by Fomin and Zelevinsky in 2001 as a tool to describe canonical basis by Lusztig. Cluster algebras are equipped with compatible Poisson structure. We will discuss few cases when this construction lead to discrete integrable dynamical systems.

(joint work with M.Gekhtman, S.Tabachnikov, and A.Vainshtein)

Plane Curve Shortening describes the motion of plane curves by their curvature. Classical theorems of Gage, Hamilton, Grayson describe the long time behaviour of compact solutions, and recent work of Daskalopoulos, Hamilton, and Sesum describe all ancient convex compact solutions (i.e. solutions that are defined for all negative times). In this talk I will present a survey of various existence theorems for new Ancient Solutions to Curve Shortening as well as new solutions to the initial value problem.

A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. We discuss the history of the problem. Some dramatic recent progress uses a new tool in this context: the Chow group of algebraic cycles.

Most of the cryptographic protocols used in everyday life are based on number theoretic problems such as integer factoring. We will give an introduction to lattice-based cryptography, a relatively recent form of cryptography offering many advantages over the traditional number-theoretic-based one. The talk will mainly focus on the so-called Learning with Errors (LWE) problem. This problem has turned out to be an amazingly versatile basis for cryptographic constructions, with tens of applications, including recent breakthrough work on fully homomorphic encryption by Gentry and others. In addition to applications, we will also mention very recent work on using algebraic number theory for making cryptographic constructions more efficient. The talk does not assume any prior knowledge in cryptography or lattices.

Non local diffusion processses appear in many circumstances: the quasi geostrophic equation, porous media with potential pressure and memory, the master equation. I will discuss some results and research directions.

Two-point symmetrizations are simple rearrangements that have been used to prove isoperimetric inequalities on the sphere. For each unit vector u, there is a two-point symmetrization that pushes mass towards u across the normal hyperplane. A key point is that the only sets invariant under all two-point symmetrizations are the entire sphere and the empty set; if the directions u are restricted to the positive hemisphere, then the polar caps are invariant as well.

I will discuss work with Greg Chambers and Anne Dranovski, in the context recent and classical symmetrization results.

How can full rotational symmetry be recovered from partial information? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d); in particular, a continuous function that is symmetric under these reflections must be radial. How many two-point symmetrizations are needed to verify that a function which increases under these symmetrizations is radial? I will show that d+1 such symmetrizations suffice, and will discuss the ergodicity of the random walk generated by the corresponding folding maps on the sphere.

In this talk we give an overview of recent developments and new directions of manifolds which satisfy the Einstein equation \( Rc=cg \), or more generally just manifolds with bounded Ricci curvature \( \left|Rc\right| < C \). We will discuss the solution of the codimension four conjecture, which roughly says that Gromov-Hausdorff limits \( (M^n_i,g_i)\to(X,d) \) of Einstein manifolds are smooth away from a set of codimension four. In a very different direction, in this lecture we will also explain how Einstein manifolds may be characterized by the behavior of the analysis on path space P(M) of the manifold. That is, we will see a Riemannian manifold is Einstein if and only if certain gradient estimates for functions on P(M) hold. One can view this as an infinite dimensional generalization of the Bakry-Emery estimates.

The work of Thurston has shown that for many finite volume hyperbolic 3-manifolds, there is a particularly interesting component of the SL(2,C)-character variety that carries a great deal of information about the topology of the manifold. For example, in the case when the manifold is a hyperbolic knot complement in S^3, this component is a curve. This talk will describe efforts to understand "which curves arise" and how algebro-geometric properties of this curve relate to the manifold.

An old idea for counting solutions of non-linear elliptic PDEs leads naturely to holomorphic maps, and to problems considered by string theorists. Following this path, I will explain a 1998 conjecture of physicists R. Gopakumar and C. Vafa about spaces called "Calabi-Yau 3-folds", and give a geometric interpretation. At the end, I will outline a geometric proof, with pictures, of the GV conjecture (joint work with E. Ionel) based on assessing the contributions of "clusters" of curves.

In this talk, which aims at a general mathematical audience, I shall describe progress on several questions concerned with global properties of low-dimensional dynamical systems, which seemed completely intractable only a few years ago.

I will discuss proper holomorphic mappings between balls in complex Euclidean spaces of different dimensions. One of the main points will be comparing the notions of spherical and homotopy equivalence. Rather than trying to prove any deep theorems, I will focus on the many links with other areas of mathematics. In particular I will show how studying group invariant maps lead to an irrigidity result for maps between hyperquadrics. The talk will be accessible to a general audience, including graduate students.

Airplane boarding is an operations research problem with high customer visibility. OR departments and management in airlines constantly experiment with boarding policies in an attempt to optimize the procedure. Recently, a couple of airlines experimented with the boarding policy which allows priority boarding to passengers with no carry on luggage to be placed in the overhead bins. We will try to analyze this policy. The policy is somewhat reminiscient of express line queues in supermarkets which also attempt to separate “fast” from “slow” customers. We will show that this superficial analogy can be made more substantial. In particular, we will examine the role of modular symmetries in express line queues and airplane boarding and the role of the weight and conductor in understanding the behavior of such queues. We will also provide some evidence that Murphy's law holds in queueing theoretic settings and if time permits we will explain that the motion of the earth around the sun is uniformly random. The talk will be self contained, experience with airplane boarding or supermarket checkout counters will be helpful.

In all dimensions other than four, there are at most finitely many diffeomorphism classes of smooth, simply-connected, closed manifolds within a homeomorphism class. Since the pioneering work of Donaldson, Taubes, Friedman-Morgan, Fintushel-Stern and many others, we have known that there is an infinite variety of smooth four-manifolds within many homeomorphism classes. The Seiberg-Witten invariants have been our chief tool for detecting this complexity and thus the number of Seiberg-Witten basic classes on a four-manifold varies greatly within a homeomorphism class.

About a decade ago, Fintushel and Stern conjectured a lower bound on the number of Seiberg-Witten basic classes on a smooth, closed four-manifold of a given homeomorphism type. More recently, Marino, Moore, and Peradze, motivated by considerations in supersymmetric gauge theory, introduced the notion of superconformal simple type and proved that it implied the lower bound of Fintushel-Stern. Marino-Moore-Peradze conjectured that all smooth, closed, simply-connected four-manifolds have superconformal simple type.

In this talk, I will discuss how the SO(3) monopole cobordism formula and a topological analysis of the compactification of the moduli space of SO(3) monopoles implies this conjecture of Marino-Moore-Peradze and thus the lower bound on the number of Seiberg-Witten basic classes.

References: http://arxiv.org/abs/1408.5307 and http://arxiv.org/abs/1408.5085A control system is a dynamical system on which one can act by using controls. For these systems a fundamental problem is the stabilization issue: is it possible to stabilize a given unstable equilibrium by using suitable feedback laws? (Think to the classical experiment of an upturned broomstick on the tip of one's finger.) We present some pioneer devices and works (Ctesibius, Watt, Foucault, Maxwell, Lyapunov...), some recent mathematical results, and an application to the regulation of the rivers La Sambre and La Meuse.

According to the definition introduced by T. Gowers in 2008, a finite group G is called D-quasirandom for some parameter D, if all non-trivial unitary representations of G have dimension greater or equal to D. For example, the group \( \mathrm{SL}(2, \mathbb{F}_p) \) is \( (p-1)/2 \) quasirandom for any prime \( p \). Informally, a finite group is quasirandom if it is D-quasirandom for a large value of D. Answering a question posed by L. Babai and V. Sos, Gowers have shown that, in contrast with the more familiar "abelian" situation, quasirandom groups can not have large product-free subsets. One of the goals of this lecture is to discuss the connection between the combinatorial phenomena observed in quasirandom groups and the ergodic properties of the minimally almost periodic groups (these were introduced by J. von Neumann as groups which do not admit non-constant almost periodic functions). This connection will allow us to give simple explanation of some of the Gowers' results as well as of more recent results obtained in joint work with T. Tao, as well as in the more recent work of T. Austin, and in the joint work with D. Robertson and P. Zorin-Kranich.

The Lee-Yang Circle Theorem states that polynomials constructed in a certain way have their zeros on the unit circle. Further results of this type concern polynomial with purely negative or purely imaginary zeros. Such polynomials occur in statistical mechanics and also in graph theory. We shall indicate how these results are obtained and what they are good for.

The Strong Perfect Graph Theorem states that graphs with no induced odd cycle of length at least five, and no complements of one behave very well with respect to coloring. But what happens if only some induced cycles (and no complements) are excluded? Gyarfas made a number of conjectures on this topic, asserting that in many cases the chromatic number is bounded by a function of the clique number. In this talk we discuss recent progress on some of these conjectures. This is joint work with Alex Scott and Paul Seymour.

Let X be our favorite space of continuous functions on \(\mathbb{R}^n\), let E be a (possibly awful) subset of \(\mathbb{R}^n\), and let f be a function on E. How can we decide whether f extends to a function F in X? If an F exists, then how small can we take its norm? What can we say about its derivatives at a given point? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F with close-to-least-possible norm? How many computer operations does it take? What if we allow F to agree with f to a given accuracy on E, instead of demanding perfect agreement? What if we are allowed to discard a few points of E as "outliers"? Which points should we discard? The talk will present some old and some new results on these problems, including joint work with Arie Israel, Bo'az Klartag, and Garving (Kevin) Luli.

The topologies of the connected components of random real projective hypersurfaces of high degree follow a universal law of distribution. We explain this (and a more general phenomenon for random band limited functions), its source and some possible connections to percolation. Joint work with I.Wigman.

Knowledge generation in many disciplines follows roughly the following kind of trajectory:

Exploration → discovery → conjecture → seeking/finding warrants → certification.

Of course this linear image is an oversimplification, and, in fact
there is often a lot of feedback, and even fractal-like structure. And
the details of this process are of course discipline-specific. What
most distinguishes mathematics is the nature of its warrants:
(deductive) proof. While mathematical proving is a powerful and
complex practice it is neither intuitive nor natural -- it must be
learned. And proving, being a complex practice rather than a body of
knowledge, must be learned developmentally, over time. Yet we often
isolate the learning of proving to a ritualized version in a geometry
course, or to a single "bridge course" that serves as a kind of
border crossing into a restricted land of mathematical doing and
thinking.

I will discuss some proving related skills that I have observed, in teaching a proving-intensive course, to be particularly challenging, even for mathematically proficient students: making mathematical connections, reasoning from definitions, and "disarming" intuition. In each case, I shall describe some task designs intended to intervene on these challenges.

The rings studied by students in most first-year algebra courses turn out to have what's known as the "Invariant Basis Number" property: for every pair of positive integers \(m\) and \(n\), if the free left \(R\)-modules \(_RR^m\) and \(_RR^n\) are isomorphic, then \(m = n\). For instance, the IBN property in the context of fields boils down to the statement that any two bases of a vector space must have the same cardinality. Similarly, the IBN property for the ring of integers is a consequence of the Fundamental Theorem for Finitely Generated Abelian Groups.

In seminal work completed the early 1960's, Bill Leavitt produced a specific, universal collection
of algebras which fail to have IBN. While it's fair to say that these algebras were initially viewed
as mere pathologies, it's just as fair to say that these now-so-called *Leavitt algebras* currently play
a central, fundamental role in numerous lines of research in both algebra and analysis.

More generally, from any directed graph \(E\) and any field \(K\) one can build the Leavitt path algebra \(L_K(E)\). In particular, the Leavitt algebras arise in this more general context as the algebras corresponding to the graphs consisting of a single vertex. The Leavitt path algebras were first defined in 2004; over the ensuing decade, the subject has matured well into adolescence, currently enjoying a seemingly constant opening of new lines of investigation, and the significant advancement of existing lines. I'll give an overview of some of the work on Leavitt path algebras which has occurred in their first ten years of existence, as well as mention some of the future directions and open questions in the subject.

There should be something for everyone in this presentation, including and especially algebraists, analysts, flow dynamicists, and graph theorists. We'll also present an elementary number theoretic observation which provides the foundation for one of the recent main results in Leavitt path algebras, a result which has had a number of important applications, including one in the theory of simple groups. The talk will be aimed at a general audience; for most of the presentation, a basic course in rings and modules will provide more-than-adequate background.

In 1963, while playing novel number-theoretical games with a computer, I dreamt up a curious recursive definition for a real-valued function, which I dubbed "\(\mathrm{INT}(x)\)". This function's behavior turned out to be very unpredictable. I spent a good deal of time exploring it computationally, and also invented and explored some variations on the theme. To my frustration, though, I was able to prove only the most basic facts about \(\mathrm{INT}(x)\), and many intriguing questions remained completely unanswered, no matter how hard I worked. Eventually, as often tends to happen when one has pushed oneself as far as one can go and has hit up against one's limits, \(\mathrm{INT}(x)\) slowly faded into the background of my life.

Many years later, my physicist doctoral advisor Gregory Wannier proposed, as my potential Ph.D. research, a canonical and very natural but unsolved problem concerning the mysterious energy spectrum of crystals in magnetic fields. I went for Gregory's bait hook, line, and sinker, but soon I found, to my great frustration, that unlike Gregory, who had made some small progress analytically, I was completely unable to prove anything analytically about the equation (known as "Harper's equation"). After a period of stagnation, finding myself in a box canyon with no escape route except using a computer to do experimental mathematics, I started exploring Harper's equation computationally, and to my astonishment, I found that good old \(\mathrm{INT}(x)\) came back into the picture front and center. This was a delightful surprise, and all at once, out of the blue, my long-ago number-theoretical explorations turned out to give me some deep insights into the physics problem. However, despite all the progress, a lot of mystery remained (and still remains). In this talk, I will mainly describe Gregory Wannier's and my collaborative discoveries.

A mean curvature flow is an evolving submanifold \(M_t\) whose velocity is equal to its mean curvature. Mean curvature flow is in some respects the most natural evolution equation for a moving submanifold: it is the gradient flow of the area functional, as well as the analog of the heat equation for submanifolds. The lecture will survey mean curvature flow for a general audience.

In October 1910, the Dutch physicist Hendrik
Antoon Lorentz delivered a series of six lectures (the Paul Wolfskehl
lectures) to the faculty of the University of Göttingen titled
"old and new problems in physics." During the fourth lecture, with
David Hilbert and his student Hermann Weyl in the audience, he
conjectured that the number of eigenvalues for the Laplacian for a
region D in three space not exceeding the positive number λ is
proportional (with a precise constant) to the volume of D times
λ^{3/2}, when λ is large. (The problem had been
raised a month earlier by Arnold Sommerfeld at a lecture in
Königsberg.) Hilbert (apparently) predicted that the conjecture
would not be proved in his lifetime. He was wrong by several
years. The conjecture was proved by Weyl in 1912.

Weyl's celebrated theorem, commonly referred to as Weyl's Law, has been extended and refined in many directions. In this talk we first give an overview of some of the classical results in the field and discuss the elegant connections to Brownian motion first explored by Mark Kac in the 50's and 60's. We will then discuss problems that arise when the Brownian motion, which "goes" with the classical Laplacian, is replaced by other Lévy processes. (Such processes share many important properties with Brownian motion.) In particular, we will look at the rotationally invariant stable processes that "go" with fractional powers of the Laplacian.

In 1932, von Neumann proposed classifying the statistical behavior of diffeomorphisms of manifolds. In modern language this means classifying diffeomorphisms that preserve a smooth volume element up to measure theoretic isomorphism. Despite important progress using entropy and spectral invariants, the general problem remained open. This talk proves that a complete classification is impossible in a rigorous sense—even on compact surfaces. The proof of the theorem involves producing new examples of diffeomorphisms with strong structural properties such as high distal rank.

This will be an introductory talk that explains the importance of embedding questions in symplectic geometry, and some recent results in this area.

Leonhard Euler (1707 –1783) is one of the towering figures from the history of mathematics. Here we look at two results that show how he acquired his lofty reputation. In a 1737 paper, Euler considered 1/2 + 1/3 + 1/5 + 1/7 + 1/11 +...—i.e., the sum of reciprocals of the primes—and established that this sum "is infinite." The proof rested upon his famous product-sum formula and required a host of analytic manipulations so typical of Euler's work. André Weil described this argument as "marking the birth of analytic number theory." The other result addressed 1 + 1/4 + 1/9 + 1/16 +...—i.e., the sum of reciprocals of the squares. Euler first evaluated this in 1734, and revisited it in 1741, but here we examine his 1755 argument that derived the sum by using l'Hospital's rule not once, not twice, but thrice! Euler has been described as "analysis incarnate." These two proofs should leave no doubt that such a characterization is apt. NOTE: This talk is accessible to anyone who has completed the calculus sequence.

Regularity is a numerical invariant that measures the complexity of the structure of homogeneous ideals in a polynomial ring. Papers of Bayer-Mumford and others give examples of families of ideals attaining doubly exponential regularity. In contrast, Bertram-Ein-Lazarsfeld, Chardin-Ulrich, and Mumford have proven that there are nice bounds on the regularity of the ideals of smooth (or nearly smooth) projective varieties. As discussed in an influential paper by Bayer and Mumford (1993), the biggest missing link between the general case and the smooth case is to obtain a decent bound on the regularity of all prime ideals (the ideals that define irreducible projective varieties). The long standing Eisenbud-Goto Regularity Conjecture (1984) predicts an elegant linear bound, in terms of the degree of the variety (also called multiplicity). The conjecture was proven for curves by Gruson-Lazarsfeld-Peskine, for smooth surfaces by Lazarsfeld and Pinkham, for most smooth 3-folds by Ran, and in many other special cases.

Recently, McCullough and I introduced two new techniques related to homogenization and blow-up algebras, and used them to provide many counterexamples to the Eisenbud-Goto conjecture. In fact we show that the regularity of prime ideals is not bounded by any polynomial function of the degree. I will explain the ideas behind these new techniques.

Supergeometry studies two kinds of objects. A super object is an object equipped with an algebra of functions that is (Z/2)-graded commutative. Examples: supermanifolds, superschemes, super Lie groups, super Lie algebras. A supersymmetric object, on the other hand, has a much tighter structure involving the action of a supergroup that mixes the even and odd directions. Roughly, the odd directions look like spinors over the even directions. The smallest example is a super Riemann surface: a supersymmetric space with 1 even and 1 odd, spinorial direction. Their theory is very rich. In particular, their moduli spaces furnish very interesting and non trivial examples of supermanifolds (or better, superstacks). Both these 'super moduli spaces' and their Deligne-Mumford compactifications play crucial roles in the foundations of perturbative superstring theory. I will explain and illustrate these basic notions, and if time allows might mention some related developments such as super toric varieties, super log structures and super Calabi-Yaus.

The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

We discuss how algebraic objects arise in limits of random interfaces, in particular in the "dimer model" and some of its generalizations.

Sarnak's Möbius disjointness conjecture speculates that the Möbius sequence is disjoint to all topological dynamical systems of zero topological entropy. We will survey the recent developments in this area, and discuss several special classes of dynamical systems of controlled complexity that satisfy this conjecture. Part of the talk is based on joint works with Wen Huang, Xiangdong Ye, and Guohua Zhang. No background knowledge in either dynamical systems or number theory will be assumed.

This talk will survey recent progress on a conjecture in number theory about the structure of class groups of number fields. Each number field has associated to it a finite abelian group, the class group, and as long ago as Gauss, deep questions arose about the distribution of class groups as the field varies over a family. Many of these questions remain unanswered. We will introduce one particular conjecture about p-torsion in class groups, and indicate how it is closely related to several other deep conjectures in number theory. Then we will present several contrasting ways we have recently made progress toward the p-torsion conjecture.

From any finite projective space, I will construct a ring that satisfies all the known properties of the cohomology ring of a smooth projective variety. I will indicate proofs of the hard Lefschetz theorem and the Hodge-Riemann relation in this context, and ask whether the ring is the cohomology ring of a geometric object.

This talk will illustrate some patterns in the homology of the space F_{k}(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces F_{k}(M) to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these configuration spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena—relationships between unstable homology classes in different degrees—established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius-Kupers-Randal-Williams.