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Series: School of Mathematics Colloquium

Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed:the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A natural question is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss a result showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!). The talk will not assume much background beyond basic probability and analysis; in particular, no prior knowledge of Dvoretzky's theorem is needed.

Series: School of Mathematics Colloquium

Kick-off of the <a href="http://ttc.gatech.edu">Tech Topology Conference</a>, December 5-7, 2014

In 1985, Barnsley and Harrington defined a "Mandelbrot Set" M
for pairs of similarities -- this is the set of complex numbers z
with norm less than 1 for which the limit set of the semigroup
generated by the similarities x -> zx and x -> z(x-1)+1 is
connected. Equivalently, M is the closure of the set of roots of
polynomials with coefficients in {-1,0,1}. Barnsley and Harrington
already noted the (numerically apparent) existence of infinitely
many small "holes" in M, and conjectured that these holes were
genuine. These holes are very interesting, since they are "exotic"
components of the space of (2 generator) Schottky semigroups. The
existence of at least one hole was rigorously confirmed by Bandt in
2002, but his methods were not strong enough to show the existence
of infinitely many holes; one difficulty with his approach was that
he was not able to understand the interior points of M, and on the
basis of numerical evidence he conjectured that the interior points
are dense away from the real axis. We introduce the technique of
traps to construct and certify interior points of M, and use
them to prove Bandt's Conjecture. Furthermore, our techniques let
us certify the existence of infinitely many holes in M. This is
joint work with Sarah Koch and Alden Walker.

Series: School of Mathematics Colloquium

In this presentation I will show some of the surprising depth and complexity of elementary- and middle-grades mathematics, much of which has been revealed by detailed studies into how students think about mathematical ideas. In turn, research into students' thinking has led to the development of teaching-learning paths at the elementary grades, which are reflected in the Common Core State Standards for Mathematics. These teaching-learning paths are widely used in mathematically high-performing countries but are not well understood in this country. At the middle grades, ideas surrounding ratio and proportional relationships are critical and central to all STEM disciplines, but research is needed into how students and teachers can reason about these ideas. Although research in mathematics education is necessary, it is not sufficient for solving our educational problems. For the mathematics teaching profession to be strong, we need a system in which all of us who teach mathematics, at any level, take collective ownership of and responsibility for mathematics teaching.

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.