MINT Distinguished Lectures

Albert Cohen (Laboratoire Jacques-Louis Lions, Sorbonne Université)

Date Day Time Location Title
2022-05-25 Mon 12:15 Schreiber 309 From linear to nonlinear n-width: optimality in reduced modelling
2022-05-26 Wed 14:39 Porter Hall Optimal sampling in least-squares methods

Albert Cohen (Laboratoire Jacques-Louis Lions, Sorbonne Université)

Date Day Time Location Title
2022-05-25 Mon 12:15 Schreiber 309 From linear to nonlinear n-width: optimality in reduced modelling
2022-05-26 Wed 14:39 Porter Hall Optimal sampling in least-squares methods

Abstracts:

From linear to nonlinear n-width: optimality in reduced modelling

The concept of n-width has been introduced by Kolmogorov as a way of measuring the size of compact sets in terms of their approximability by linear spaces. From a numerical perspective it may be thought as a benchmark for the performance of algorithms based on linear approximation. In recent years this concept has proved to be highly meaningful in the analysis of reduced modeling strategies for complex physical problems described by parametric problems. This lecture will first review two significant results in this area that concern (i) the practical construction of optimal spaces by greedy algorithms and (ii) the preservation of the rate of decay of widths under certain holomorphic transformation. It will then focus on recent attempts to propose non-linear version of n-widths, how these notions relate to metric entropies, and how they could be relevant to practical applications.

Optimal sampling in least-squares methods

Recovering an unknown function from point samples is an ubiquitous task in various applicative settings: non-parametric regression, machine learning, reduced modeling, response surfaces in computer or physical experiments, data assimilation and inverse problems. In this lecture, I will present recent results that are relevant to the context where the user is allowed to select the measurement points, sometimes refered to as active learning. These results allow us to derive an optimal sampling point distribution when the approximation is searched in a linear space of finite dimension n and computed by weigted-least squares. Here optimal means both that the approximation is comparable to the best possible in this space, and that the sampling budget m barely exceeds n. The main involved tools are Christoffel functions, matrix concentration inequalities, reproducing kernel hilbert spaces.

Daniel Peralta-Salas (ICMAT, Madrid)

Date Day Time Location Title
2020-01-20 Mon 12:15 Schreiber 006 Emergence of topological structures in elliptic PDE
2020-01-22 Wed 14:00 Dan David 201 Selected topics on the dynamics of the steady Euler flows

Daniel Peralta-Salas (ICMAT, Madrid)

Date Day Time Location Title
2020-01-20 Mon 12:15 Schreiber 006 Emergence of topological structures in elliptic PDE
2020-01-22 Wed 14:00 Dan David 201 Selected topics on the dynamics of the steady Euler flows

Abstracts:

Emergence of topological structures in elliptic PDE

In recent years, new experimental methods and designs have enabled the observation and study of topological structures emerging from a wide range of physical phenomena, from fluid mechanics to electromagnetic theory. These structures take the form, e.g., of knotted vortex tubes in fluid flows, or knotted topological dislocations in optics and condensed matter theory. They provide a powerful visual tool to gain understanding of complex physical systems. Mathematically, these physical processes are described by a vector or scalar field which satisfies a system of partial differential equations, and the emerging topological structures are instances of invariant manifolds of these fields.

The goal of this lecture is to introduce the theory we have recently developed in collaboration with A. Enciso to address the study of these problems. Some of the questions that I will consider are: does there exist a steady Euler flow having stream lines of all knot and link types? Can the nodal components of the eigenfunctions of a Schrodinger operator exhibit arbitrary topology? I will focus on the aspects of the theory for elliptic (time independent) PDE.

Selected topics on the dynamics of the steady Euler flows

An inviscid and incompressible fluid in equilibrium on a Riemannian manifold is described by (an autonomous) vector field X that satisfies the stationary Euler equations. In this talk I will focus on the dynamics of steady Euler flows on compact manifolds, emphasizing the geometric aspects. The topics I will cover include a characterization of the volume-preserving vector fields that are Eulerizable and some rigidity results for steady Euler flows on the round sphere.

Kari Astala (Aalto University, Finland)

Date Day Time Location Title
2019-03-18 Mon 12:15 Schreiber 006 Random tilings and a non-linear Beltrami equation
2019-03-19 Tue 14:10 Schreiber 209 Geometry of chord-arc curves

Kari Astala (Aalto University, Finland)

Date Day Time Location Title
2019-03-18 Mon 12:15 Schreiber 006 Random tilings and a non-linear Beltrami equation
2019-03-19 Tue 14:10 Schreiber 209 Geometry of chord-arc curves

Abstracts:

Random tilings and a non-linear Beltrami equation

Scaling limits of random structures in two dimensions often posses some conformal invariance properties.
This allows different methods from geometric analysis for their study. Random tilings are typical examples, with inviting questions on the geometry of their ordered and disordered (or frozen and liquid) limit regions.

In this presentation, based on joint work with E.Duse, I.Prause and X.Zhong, I show how such liquid regions carry a natural complex structure, described by a specific Beltrami equation. This leads to an understanding and classifying the geometry of the limiting boundaries for different random tilings and other dimer models.

Geometry of chord-arc curves

Chord-arc curves are natural geometric objects, images of lines or circles under a bi-Lipschitz mapping of the plane, yet many of their analytic properties lead to interesting questions in harmonic analysis or PDE’s. In this talk I will discuss two such questions, deformations of chord-arc curves (joint with M.J. Gonzales) and second, the size of rotation or spiralling of a chord-arc curve (joint with T.Iwaniec, I. Prause and E. Saksman)

Rida Laraki (Université Paris Dauphine)

Date Day Time Location Title
2019-01-07 Mon 12:15 Schreiber 006 Majority judgment: a new voting method
2019-01-08 Tue 16:15 Schreiber 007 Acyclic Gambling Games

Rida Laraki (Université Paris Dauphine)

Date Day Time Location Title
2019-01-07 Mon 12:15 Schreiber 006 Majority judgment: a new voting method
2019-01-08 Tue 16:15 Schreiber 007 Acyclic Gambling Games

Abstracts:

Majority judgment: a new voting method

The traditional theory of social choice offers no good solution to the problems of how to elect, to judge, or to rank. We propose a more realistic model where voters evaluate the candidates in a common language of ordinal grades. This small change leads to a new theory and method « majority judgment ». It is at once meaningful, resists strategic manipulation, and is not subject to the paradoxes encountered in practice, notably Condorcet’s and Arrow’s. We offer theoretical, practical, and experimental evidence—from national elections to figure skating competitions—to support the arguments.

References:
Balinski M. and R. Laraki (2007), A Theory of Measuring, Electing and Ranking, PNAS, 104(2), 8720-8725.
B & L (2011), Majority Judgment: Measuring, Ranking, and Electing, MIT Press.
B & L (2014). Judge: Don’t vote. Operations Research, 28, 483-511.
B & L (2017), Majority Judgment vs Majority Rule, preprint.
B & L (2018), Majority Judgment vs Approval Voting, preprint.

Acyclic Gambling Games

Joint work with Jérome Renault (TSE, France)
We consider 2-player zero-sum stochastic games where each player controls his own state variable living in a compact metric space. The terminology comes from gambling problems where the state of a player represents its wealth in a casino. Under natural assumptions (such as continuous running payoff and non expansive transitions), we consider for each discount factor the value vλ of the λ-discounted stochastic game and investigate its limit when λ goes to 0 (players are more and more patient). We show that under a new acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if his opponent does not move, can reach the zone when the current payoff is at least as good than the limit value, without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens-Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (vλ) may fail.

Paper link:
https://bfi.uchicago.edu/sites/default/files/research/WP_2018-34.pdf

James Maynard (Oxford University)

Date Day Time Location Title
2018-12-10 Mon 12:15 Schreiber 006 Primes in arithmetic progressions: The Riemann Hypothesis - and beyond!
2018-12-13 Thu 14:00 Schreiber 209 Prime numbers, algebraic curves, and modular forms

James Maynard (Oxford University)

Date Day Time Location Title
2018-12-10 Mon 12:15 Schreiber 006 Primes in arithmetic progressions: The Riemann Hypothesis - and beyond!
2018-12-13 Thu 14:00 Schreiber 209 Prime numbers, algebraic curves, and modular forms

Abstracts:

Primes in arithmetic progressions: The Riemann Hypothesis - and beyond!

One of the oldest problems about prime numbers is asking how many primes there are of a given size in an arithmetic progression. Dirichlet’s famous theorem shows that there are large primes in the progression unless there is an obvious reason why not, but more refined questions lead quickly to statements equivalent to versions of the Riemann Hypothesis, which unfortunately remains unsolved.

Often in mathematics it is sufficient to show a statement is true ‘most of the time’, rather than ‘all of the time’. We know that the Riemann Hypothesis is true ‘most of the time’, which is often as good as the Riemann Hypothesis itself for applications. Moreover, some of the deepest results in Analytic Number Theory can go beyond this and say even stronger statements are true ‘most of the time’! One such statement was the key to Zhang’s breakthrough on bounded gaps between primes. I will describe these ideas, and some work in progress which extends them.

Prime numbers, algebraic curves, and modular forms

There are beautiful and deep connections between algebraic geometry, modular forms, and certain sums arising in analytic number theory. I will describe these connections, and how they can be combined with techniques for understanding the primes. In particular, they can be used to understand prime numbers in arithmetic progressions.

Benny Sudakov (ETH Zürich)

Date Day Time Location Title
2018-01-01 Mon 12:15 Schreiber 006 Rainbow structures, Latin squares & graph decompositions
2018-01-07 Sun 10:05 Schreiber 309 Unavoidable patterns in words

Benny Sudakov (ETH Zürich)

Date Day Time Location Title
2018-01-01 Mon 12:15 Schreiber 006 Rainbow structures, Latin squares & graph decompositions
2018-01-07 Sun 10:05 Schreiber 309 Unavoidable patterns in words

Abstracts:

Rainbow structures, Latin squares & graph decompositions

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares. Since then rainbow structures were the focus of extensive research and found applications in design theory and graph decompositions. In this talk we discuss how probabilistic reasoning can be used to attack several old problems in this area. In particular we show that well known conjectures of Ryser, Hahn, Ringel, and Graham-Sloane hold asymptotically.

Based on joint works with Alon, Montgomery, and Pokrovskiy.

Unavoidable patterns in words

A word $w$ is said to contain the pattern $P$ if there is a way to substitute a nonempty word for each letter in $P$ so that the resulting word is a subword of $w$. Bean, Ehrenfeucht and McNulty and, independently, Zimin proved Ramsey theorem for words. They characterised the patterns $P$ which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains $P$. Zimin’s characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by $Z_1 = x_1$ and $Z_n=Z_{n-1} x_n Z_{n-1}$.

We study the quantitative aspects of this theorem, showing that there are extremely long words (whose length is tower function) avoiding $Z_n$. Our results are asymptotically tight.

Joint work with David Conlon and Jacob Fox.

Sergey Fomin (University of Michigan)

Date Day Time Location Title
2017-12-13 Wed 14:10 Schreiber 309 Morsifications and mutations
2017-12-18 Mon 12:15 Schreiber 006 Computing without subtracting (and/or dividing)

Sergey Fomin (University of Michigan)

Date Day Time Location Title
2017-12-13 Wed 14:10 Schreiber 309 Morsifications and mutations
2017-12-18 Mon 12:15 Schreiber 006 Computing without subtracting (and/or dividing)

Abstracts:

Morsifications and mutations

I will discuss a surprising connection between singularity theory and cluster algebras, more specifically between (1) the topology of isolated singularities of plane curves and (2) the mutation equivalence of the quivers associated with their morsifications. Joint work with Pavlo Pylyavskyy and Eugenii Shustin.

Computing without subtracting (and/or dividing)

Algebraic complexity of a rational function can be defined as the minimal number of arithmetic operations required to compute it. Can restricting the set of allowed arithmetic operations dramatically increase the complexity of a given function (assuming it is still computable in the restricted model)? In particular, what can happen if we disallow subtraction and/or division? Joint work with Dima Grigoriev and Gleb Koshevoy.

Gil Kalai (Hebrew University)

Date Day Time Location Title
2017-12-03 Sun 10:05 Schreiber 309 The combinatorics of convex polytopes via linear programming and beyond
2017-12-04 Mon 12:15 Schreiber 006 The combinatorics of convex polytopes and the simplex algorithm for linear programming

Gil Kalai (Hebrew University)

Date Day Time Location Title
2017-12-03 Sun 10:05 Schreiber 309 The combinatorics of convex polytopes via linear programming and beyond
2017-12-04 Mon 12:15 Schreiber 006 The combinatorics of convex polytopes and the simplex algorithm for linear programming

Abstracts:

The combinatorics of convex polytopes via linear programming and beyond

I will show how the basic properties of linear programming lead to deep and beautiful combinatorial theorems about convex polytopes. I will discuss especially the upper bound theorem and the reconstruction of the full combinatorics of simple polytopes from their graphs.

The combinatorics of convex polytopes and the simplex algorithm for linear programming

Linear programming is the problem of maximizing a linear function subject to a system of linear inequalities. The set of solutions for the linear inequalities is a convex polytope P (which can be unbounded). The simplex algorithm was developed by George Danzig. Geometrically it can be described by moving from one vertex to a neighboring vertex as to improve that value of the objective function. The precise rule for choosing the next vertex is called the “pivot rule”.

The simplex algorithm is one of the most successful mathematical algorithms. Understanding this success is an applied question, it is a vaguely stated, and it connects with computers. The problem has strong relations to the study of convex polytopes that mathematicians found fascinating from ancient times and was my own starting point.

After a brief overview of the history of the subject I will concentrate on diameter of graphs of polytopes and on randomized pivot rules.

Vadim Kaloshin (University of Maryland)

Date Day Time Location Title
2017-06-05 Mon 12:15 Schreiber 006 Can you hear the shape of a drum and deformational spectral rigidity of planar domains
2017-06-07 Wed 14:10 Schreiber 309 Birkhoff Conjecture for convex planar billiards

Vadim Kaloshin (University of Maryland)

Date Day Time Location Title
2017-06-05 Mon 12:15 Schreiber 006 Can you hear the shape of a drum and deformational spectral rigidity of planar domains
2017-06-07 Wed 14:10 Schreiber 309 Birkhoff Conjecture for convex planar billiards

Abstracts:

Can you hear the shape of a drum and deformational spectral rigidity of planar domain

M. Kac popularized the question `Can you hear the shape of a drum?’. Mathematically, consider a bounded planar domain $\Omega$ and the associated Dirichlet problem $\Delta u+\lambda^2 u=0, u|_{\partial \Omega}=0.$ The set of $\lambda$’s such that this equation has a solution, is called the Laplace spectrum of $\Omega$. Does Laplace spectrum determines $\Omega$? In general, the answer is negative.

Consider the billiard problem inside $\Omega$. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that any axis symmetric planar domain with sufficiently smooth boundary close to the disk is dynamically spectrally rigid, i.e. can’t be deformed without changing the length spectrum. This partially answers a question of P. Sarnak.

This is a joint work with J. De Simoi and Q. Wei.

Birkhoff Conjecture for convex planar billiards

G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the boundary is foliated by smooth closed curves and each billiard orbit near the boundary is tangent to one and only one such curve (in this particular case, a confocal ellipse). A famous conjecture by Birkhoff claims that ellipses are the only domains with this property. We show a local version of this conjecture - namely, that a small perturbation of an ellipse has this property only if it is itself an ellipse.

This is based on several papers with Avila, De Simoi, G.Huang, Sorrentino.

Carlos Kenig (University of Chicago)

Date Day Time Location Title
2017-03-27 Mon 12:15 Schreiber 006 The energy critical wave equation: An overview
2017-03-29 Wed 14:10 Schreiber 309 The energy critical wave equation: soliton decomposition along well chosen sequences of time in the non-radial case

Carlos Kenig (University of Chicago)

Date Day Time Location Title
2017-03-27 Mon 12:15 Schreiber 006 The energy critical wave equation: An overview
2017-03-29 Wed 14:10 Schreiber 309 The energy critical wave equation: soliton decomposition along well chosen sequences of time in the non-radial case

Abstracts:

The energy critical wave equation: An overview

We will describe the progress in the last 12 years on the understanding of the long-time dynamics for large solutions in the focusing case. Issues like blow-up, scattering and asymptotic decomposition into solitary waves will be discussed.

The energy critical wave equation: soliton decomposition along well chosen sequences of time in the non-radial case

We will discuss the recent work of Duyckaerts-Jia-Kenig-Merle establishing soliton resolution along well chosen sequences of times converging to the final time of existence, for non-radial solutions of the energy critical nonlinear wave equation which remain bounded in the energy space.

Paul Biran (ETH Zurich)

Date Day Time Location Title
2017-01-09 Mon 12:15 Schreiber 006 A Geometric Outlook on Fukaya Categories
2017-01-11 Wed 14:10 Schreiber 309 It takes Energy to Split Lagrangians

Paul Biran (ETH Zurich)

Date Day Time Location Title
2017-01-09 Mon 12:15 Schreiber 006 A Geometric Outlook on Fukaya Categories
2017-01-11 Wed 14:10 Schreiber 309 It takes Energy to Split Lagrangians

Abstracts:

A Geometric Outlook on Fukaya Categories

We will explain how the concept of Lagrangian cobordism can be used to approach in a geometric way the Fukaya category of a symplectic manifold (a rather abstract, yet central, object of study in modern symplectic topology). We will also explain how to perform geometric measurements in the Fukaya category via cobordisms. The talk is based on a series of joint works with Octav Cornea.

It takes Energy to Split Lagrangians

We will discuss a Hofer Geometry analog that naturally arises from Lagrangian cobordism theory, and see how it can be used to measure relations between Lagrangian submanifolds.

Nikolai Nadirashvili (Aix-Marseille University)

Date Day Time Location Title
2016-12-12 Mon 12:15 Schreiber 006 Inequalities for eigenvalues of Laplacian on surfaces
2016-12-14 Wed 14:10 Schreiber 309 Geometry of trajectories to Euler Equation and level sets of solutions to elliptic equations

Nikolai Nadirashvili (Aix-Marseille University)

Date Day Time Location Title
2016-12-12 Mon 12:15 Schreiber 006 Inequalities for eigenvalues of Laplacian on surfaces
2016-12-14 Wed 14:10 Schreiber 309 Geometry of trajectories to Euler Equation and level sets of solutions to elliptic equations

Abstracts:

Inequalities for eigenvalues of Laplacian on surfaces

We discuss isoperimetric inequalities for eigenvalues of the Laplacian on two-dimensional Riemannian manifolds. We survey known results, starting with classical inequalities of Hersch and Li–Yau for the first nonzero eigenvalue on a sphere and a projective plane and their recent generalization on the higher eigenvalues.

Geometry of trajectories to Euler Equation and level sets of solutions to elliptic equations

Abstract: We discuss Liouville theorems and some geometric properties of stationary flows of the ideal fluid in dimensions 2 and 3.

Gordon Slade (University of British Columbia)

Date Day Time Location Title
2016-12-05 Mon 12:15 Schreiber 006 Critical phenomena and renormalisation group
2016-12-05 Mon 14:15 Schreiber 006 Critical phenomena and renormalisation group - in more detail

Gordon Slade (University of British Columbia)

Date Day Time Location Title
2016-12-05 Mon 12:15 Schreiber 006 Critical phenomena and renormalisation group
2016-12-05 Mon 14:15 Schreiber 006 Critical phenomena and renormalisation group - in more detail

Abstract:

Critical phenomena and renormalisation group

The subject of phase transitions and critical phenomena in statistical mechanics is a rich source of interesting and difficult mathematical problems. There has been considerable success in solving such problems for systems in spatial dimension 2, or in high dimensions, but not in dimension 3. This lecture is intended to provide an introduction to recent work that employs a renormalisation group method to study spin systems and self-avoiding walk in dimension 4 (joint with Bauerschmidt and Brydges), as well as long-range versions of these models in dimensions 1, 2, 3 via an “epsilon expansion”.

David Conlon (University of Oxford)

Date Day Time Location Title
2016-03-20 Sun 10:05 Schreiber 309 Finite reflection groups and graph norms
2016-03-21 Mon 12:15 Schreiber 006 Recent developments in graph Ramsey theory

David Conlon (University of Oxford)

Date Day Time Location Title
2016-03-20 Sun 10:05 Schreiber 309 Finite reflection groups and graph norms
2016-03-21 Mon 12:15 Schreiber 006 Recent developments in graph Ramsey theory

Abstracts:

Finite reflection groups and graph norms

Given a graph $H$ and a symmetric function $f:[0,1]^2\rightarrow\mathbb{R}$, define the absolute $H$-norm of $f$ by

where $μ$ is the Lebesgue measure on $[0,1]$. We say that $H$ is weakly norming if the absolute $H$-norm is a norm on the vector space of symmetric functions. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, it is known that even cycles, complete bipartite graphs and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we demonstrate that any graph which is edge-transitive under the action of a certain natural family of automorphisms is weakly norming. This result includes all previous examples of weakly norming graphs and adds many more.

Joint work with Joonkyung Lee.

Recent developments in graph Ramsey theory

Given a graph $H$, the Ramsey number $r(H)$ is the smallest natural number $N$ such that any two-colouring of the edges of $K_N$ contains a monochromatic copy of $H$. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.

In this talk, I will describe some of this progress.

Rick Kenyon (Brown University)

Date Day Time Location Title
2015-11-15 Sun 10:00 Schreiber 309 A Variational Principle For Permutations
2015-11-16 Mon 12:15 Schreiber 006 Fixed-Energy Harmonic Functions

Rick Kenyon (Brown University)

Date Day Time Location Title
2015-11-15 Sun 10:00 Schreiber 309 A Variational Principle For Permutations
2015-11-16 Mon 12:15 Schreiber 006 Fixed-Energy Harmonic Functions

Abstracts:

A variational principle for permutations

We study scaling limits of large permutations (``permutons’’), constrained by fixing certain of pattern densities, for example, the density of inversions or `123’ patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed 12 density, with fixed 12 and 123 densities, with fixed 12 density and the sum of 123 and 213 densities. In some cases one can find interesting phase transitions as one varies the densities. To obtain our results, we also provide a description of permutons using a dynamic construction.

This is joint work with Daniel Kral, Charles Radin and Pete Winkler.

Fixed-energy harmonic functions

We study the map from conductances to edge energies for harmonic functions on graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies.

For rational energies and boundary data the Galois group of $\mathbb{Q}^{tr}$ (the totally real algebraic numbers) over $\mathbb{Q}$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations. For planar graphs one can associate tilings of planar regions with rectangles of prescribed areas. Connections with square ice and SLE_{12} (based on work with Angel, Miller, Sheffield, Wilson) will be briefly discussed.

This is joint work with Aaron Abrams.

Shmuel Weinberger (University of Chicago)

Date Day Time Location Title
2015-11-09 Mon 12:15 Schreiber 006 Topological Problems of Data Analysis, I
2015-11-11 Wed 14:00 Schreiber 209 Topological Problems of Data Analysis, II

Shmuel Weinberger (University of Chicago)

Date Day Time Location Title
2015-11-09 Mon 12:15 Schreiber 006 Topological Problems of Data Analysis, I
2015-11-11 Wed 14:00 Schreiber 209 Topological Problems of Data Analysis, II

Abstract

Topological Problems of Data Analysis, I, II.

Abstract: Dealing with large and multidimensional data is a ubiquitous problem in many scientific and financial settings. In recent years, geometric and topological methods have become increasingly popular. They suggest problems that are at the interface of probability, differential geometry, topology, and analysis. The first lecture will be devoted to an overview of these ideas, while the second will focus on some topological problems that arise in trying to make some heuristics rigorous and/or practical.

Van Vu (Yale University)

Date Day Time Location Title
2015-06-08 Mon 12:15 Schreiber 006 Random matrices: Gaps between eigenvalues
2015-06-09 Tue 14:10 Schreiber 209 Roots of random polynomials with general coefficients

Van Vu (Yale University)

Date Day Time Location Title
2015-06-08 Mon 12:15 Schreiber 006 Random matrices: Gaps between eigenvalues
2015-06-09 Tue 14:10 Schreiber 209 Roots of random polynomials with general coefficients

Abstracts

Random matrices: Gaps between eigenvalues

Gaps (or spacings) between consecutive eigenvalues of a random hermitian matrix plays a central role in random matrix theory. When Wigner used random matrix as a model in physics, it was the behavior of the gaps that captured his attention. Another famous example is the Dyson-Montgomery discussion connecting these gaps with gaps between zeroes of the zeta function.

On the other hand, we still do not understand these gaps very well at the microscopic level. For instance, it has only been proved very recently by Tao and the speaker that for a random +-1 matrix, with high probability all gaps are positive.

In this talk, we will discuss some recent progress in bounding these gaps, based on purely combinatorial techniques, for very general classes of random matrices. We will also present applications in both computer science and numerical analysis (among others, solutions to questions by Babai, and by Deker-Lee-Linial).

Joint work with H. Nguyen (OSU) and T. Tao (UCLA).

Roots of random polynomials with general coefficients.

Roots of polynomials with random coefficients have been studied for a long time, starting with fundamental works of Littlewood and Offord in the 1940s, who showed that random polynomials with iid coefficients have very few real roots. Polynomials with iid coefficients are frequently referred to as Kac polynomials.

The number of real roots of Kac polynomials has been computed asymptotically thanks to a remarkable sequence of works by Kac, Erdos-Offord, and Ibragimov-Maslova. However, we know very little about all other cases, where the coefficients are independent, but can have different distributions (different variances, for instance).

In this talk, we are going to give a review of these classical works, focusing on the main ideas behind them, and then present a recent development that provides a good understanding for polynomials with general coefficients with moderate growth (these coefficients can even have implicit variances). As a corollary, we refine and extend Ibragimov-Maslova result for hyperbolic polynomials. The talk is self-contained and several open questions will also be discussed.

Joint work with O. Nguyen (Yale) and Y. Do (UoV).

Grigori Olshanski (IITP and Higher School of Economics, Moscow)

Date Day Time Location Title
2015-04-14 Tue 14:10 Schreiber 209 Markov processes of algebraic origin
2015-04-15 Wed 14:10 Schreiber 309 What are infinite random permutations?

Grigori Olshanski (IITP and Higher School of Economics, Moscow)

Date Day Time Location Title
2015-04-14 Tue 14:10 Schreiber 209 Markov processes of algebraic origin
2015-04-15 Wed 14:10 Schreiber 309 What are infinite random permutations?

Abstracts:

Markov processes of algebraic origin

The Brownian motion on the group U(N) of unitary matrices gives rise to a Markov process on the matrix eigenvalues, that is, on N-particle configurations on the circle (Freeman Dyson, 1962). Heuristic arguments show that there should exist an analogous process with infinitely many particles. However, its rigorous construction is a difficult problem, and success in this direction has been achieved only in recent years. I will describe a different model of continuous time Markov dynamics, which is dual to Dyson’s model in the sense that it is formulated in terms of representations of the groups U(N) rather than eigenvalues. It turns out that in the “dual” model, the transition to N=infinity can be carried out by essentially algebraic tools. This is in contrast with Dyson’s model whose study in the N=infinity case requires hard analytic work.

What are infinite random permutations?

Random permutations can be viewed as a combinatorial analog of random matrices. In random matrix theory, people study the asymptotic behavior of spectra of large random matrices. Likewise, the literature in combinatorial probability contains many works on the limiting behavior of various characteristics of random permutations of large size. But are there reasonable models of random permutations of actually infinite size? I will describe some positive results in this direction.

Alex Furman (University of Illinois at Chicago)

Date Day Time Location Title
2015-01-12 Mon 12:15 Schreiber 006 Rigidity for groups with hidden symmetries
2015-01-14 Wed 14:10 Schreiber 007 Simplicity of the Lyapunov spectrum via boundary theory

Alex Furman (University of Illinois at Chicago)

Date Day Time Location Title
2015-01-12 Mon 12:15 Schreiber 006 Rigidity for groups with hidden symmetries
2015-01-14 Wed 14:10 Schreiber 007 Simplicity of the Lyapunov spectrum via boundary theory

Abstracts:

Simplicity of the Lyapunov spectrum via boundary theory

Consider products of matrices that are chosen using some ergodic stationary random process on $G=SL(d,R)$, e.g. a random walk on $G$. The Multiplicative Ergodic Theorem (Oseledets) asserts that asymptotically such products behave as $\exp(n\Lambda)$ where $\Lambda$ is a fixed diagonal traceless matrix, called the Lyapunov spectrum of the system. The spectrum $\Lambda$ depends on the system in a mysterious way, and is almost never known explicitly. The best understood case is that of random walks, where by the work of Furstenberg, Guivarch-Raugi, and Gol’dsheid-Margulis we know that the spectrum is simple (i.e. all values are distinct) provided the random walk is not trapped in a proper algebraic subgroup. Recently, Avila and Viana proved a conjecture of Kontsevich-Zorich that asserts simplicity of the Lyapunov spectrum for another system related to Teichmuller flow.

In the talk we shall describe an approach to proving simplicity of the spectrum based on ideas from boundary theory that were developed to prove rigidity of lattices.

Based on joint work with Uri Bader.

Rigidity for groups with hidden symmetries

Abstract: In the 1970s G.A.Margulis proved that linear representations of certain discrete subgroups (namely lattices) in such Lie groups as SL(3,R), are essentially determined by the ambient Lie group. This phenomenon, known as superrigidity, has far reaching applications and has inspired a large body of research in such areas as geometry, dynamics, descriptive set theory, operator algebras etc.

We shall try to explain the superrigidity of lattices and related groups by looking at some hidden symmetries (Weyl group) that they inherit from the ambient group.

The talk is based on a joint work with Uri Bader.

Henri Berestycki (EHESS, Paris)

Date Day Time Location Title
2014-12-01 Mon 12:15 Schreiber 006 Propagation in non homogeneous media and applications
2014-12-03 Wed 14:10 Schreiber 209 The effect of domain shape on propagation and blocking for reaction-diffusion equations

Henri Berestycki (EHESS, Paris)

Date Day Time Location Title
2014-12-01 Mon 12:15 Schreiber 006 Propagation in non homogeneous media and applications
2014-12-03 Wed 14:10 Schreiber 209 The effect of domain shape on propagation and blocking for reaction-diffusion equations

Abstracts:

Propagation in non homogeneous media and applications

The classical theory of the Fisher and Kolmogorov-Petrovsky-Piskunov equation derives the spreading properties for a reaction-diffusion equation in a homogeneous setting. A well known /invasion speed/ governs the asymptotic speed of propagation. This equation plays an important role in a variety of contexts in ecology, biology and physics. In this talk, I will first introduce reaction-diffusion equations, and describe some of their motivations as well as the underlying mechanism. I will then review the classical theory for homogeneous Fisher-KPP equations.

In applications, one often wishes to understand the effects of heterogeneity. This leads to challenging mathematical questions and I will describe some of them. I will then discuss more in detail the effect of inclusion of a line with fast diffusion on biological invasions in the plane. I will report on recent results on this question.

The effect of domain shape on propagation and blocking for reaction-diffusion equations

I will discuss reaction-diffusion equations motivated by biology and medicine for which the aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves.

I will first describe the motivations of these questions. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve qualitative results for some non-linear elliptic and parabolic partial differential equations.

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