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Series: School of Mathematics Colloquium

In the first part of the talk I shall present a linear model based
on the Schrodinger equation with constant coefficient and periodic
boundary conditions that explains the so-called Talbot effect in optics.
In the second part I will make a connection
of this Talbot effect with turbulence through the Schrodinger map which
is a geometric non-linear partial differential equation.

Series: School of Mathematics Colloquium

Random matrix theory (RMT) is a very active area of research and a greatsource of exciting and challenging problems for specialists in manybranches of analysis, spectral theory, probability and mathematicalphysics. The analysis of the eigenvalue distribution of many random matrix ensembles leads naturally to the concepts of determinantal point processes and to their particular case, biorthogonal ensembles, when the main object to study, the correlation kernel, can be written explicitly in terms of two sequences of mutually orthogonal functions.Another source of determinantal point processes is a class of stochasticmodels of particles following non-intersecting paths. In fact, theconnection of these models with the RMT is very tight: the eigenvalues of the so-called Gaussian Unitary Ensemble (GUE) and the distribution ofrandom particles performing a Brownian motion, departing and ending at the origin under condition that their paths never collide are, roughlyspeaking, statistically identical.A great challenge is the description of the detailed asymptotics of these processes when the size of the matrices (or the number of particles) grows infinitely large. This is needed, for instance, for verification of different forms of "universality" in the behavior of these models. One of the rapidly developing tools, based on the matrix Riemann-Hilbert characterization of the correlation kernel, is the associated non-commutative steepest descent analysis of Deift and Zhou.Without going into technical details, some ideas behind this technique will be illustrated in the case of a model of squared Bessel nonintersectingpaths.

Series: School of Mathematics Colloquium

The speaker will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities are two peculiar identities which express two infinite product modular forms as number theoretic q-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent work with Griffin and Warnaar, the speaker has obtained a comprehensive framework of identities for infinite product modular forms in terms of Hall-Littlewood q-series. This work characterizes those integral units that arise from this theory. In a related direction, the speaker revisits the classical Moonshine Theorem which asserts that the coefficients of the modular j-functions are dimensions of virtual characters for the Monster, the largest of the simple sporadic groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine the distribution of these representations in Moonshine. In joint work with Griffin and Duncan, the speaker has obtained exact formulas for these distributions.

Series: School of Mathematics Colloquium

This is a joint work with Pierre Pageault.
For a homeomorphism h of a compact space, a Lyapunov function is a real
valued function that is non-increasing along orbits for h.
By looking at simple dynamical systems(=homeomorphisms) on the circle,
we will see that there are systems which are topologically conjugate and
have Lyapunov functions with various regularity.
This will lead us to define barriers analogous to the well known Peierls barrier or to the Maסי potential
in Lagrangian systems. That will produce by analogy to Mather's theory
of Lagrangian Systems an Aubry set which is the generalized recurrence
set introduced in the 60's by Joe Auslander (via transfinite induction)
and a Maסי set which is essentially Conley's chain recurrent set.
No serious knowledge of Dynamical Systems is necessary to follow the lecture.

Series: School of Mathematics Colloquium

The area was essentially originated by the general question: How many
zeros of a random polynomials are real? Kac showed that the expected
number of real zeros for a polynomial with i.i.d. Gaussian coefficients
is logarithmic in terms of the degree. Later, it was found that most of
zeros of random polynomials are asymptotically uniformly distributed
near the unit circumference (with probability one) under mild
assumptions on the coefficients.
Thus two main directions of research are related to the almost sure
limits of the zero counting measures, and to the quantitative results on
the expected number of zeros in various sets. We give estimates of the
expected discrepancy between the zero counting measure and the
normalized arclength on the unit circle. Similar results are established
for polynomials with random coefficients spanned by various bases,
e.g., by orthogonal polynomials. We show almost sure convergence of the
zero counting measures to the corresponding equilibrium measures for
associated sets in the plane, and quantify this convergence. Random
coefficients may be dependent and need not have identical distributions
in our results.

Series: School of Mathematics Colloquium

The talk will survey the main definitions and properties of patchy vector fields and patchy feedbacks, with
applications to asymptotic feedback stabilization and nearly optimal feedback control design.
Stability properties for discontinuous ODEs and robustness of patchy feedbacks will also be discussed.

Series: School of Mathematics Colloquium

Given some class of "geometric spaces", we can make a ring as follows. (i) (additive structure) When U is an open subset of such a space X, [X] = [U] + [(X \ U)] (ii) (multiplicative structure) [X x Y] = [X] [Y].In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. (This talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood.

Series: School of Mathematics Colloquium

The general problem of extracting knowledge from
large and complex data sets is a fundamental one across all areas of the
natural and social sciences, as well as in most areas of commerce and
government. Much progress has been made on methods for capturing and
storing such data, but the problem of translating it into knowledge is more
difficult. I will discuss one approach to this problem, via the study of
the shape of the data sets, suitably defined. The use of shape as an
organizing problems permits one to bring to bear the methods of topology,
which is the mathematical field which deals with shape. We will discuss
some different topological methods, with examples.

Series: School of Mathematics Colloquium

In this age of high-dimensional data, many challenging questions take the following shape: can you check whether the data has a certain desired property by checking that property for many, but low-dimensional data fragments? In recent years, such questions have inspired new, exciting research in algebra, especially relevant when the property is highly symmetric and expressible through systems of polynomial equations. I will discuss three concrete questions of this kind that we have settled in the affirmative: Gaussian factor analysis from an algebraic perspective, high-dimensional tensors of bounded rank, and higher secant varieties of Grassmannians. The theory developed for these examples deals with group actions on infinite-dimensional algebraic varieties, and applies to problems from many areas. In particular, I will sketch its (potential) relation to the fantastic Matroid Minor Theorem.

Series: School of Mathematics Colloquium