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:
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(n2) 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 Rn 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 λ3/2, 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.