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Series: School of Mathematics Colloquium

The long term behavior of dynamical systems can be understood by studying invariant manifolds that act as landmarks that anchor the orbits. It is important to understand which invariant manifolds persist under modifications of the system. A deep mathematical theory, developed in the 70's shows that invariant manifolds which persist under changes are those that have sharp expansion (in the future or in the past) in the the normal directions. A deep question is what happens in the boundary of these theorems of persistence. This question requires to understand the interplay between the geometric properties and the functional analysis of the functional equations involved.In this talk we present several mechanisms in which properties of normal hyperbolicity degenerate, so leading to the breakdown of the invariant manifold. Numerical studies lead to surprising conjectures relating the breakdown to phenomena in phase transitions. The results have been obtained combining numerical exploration and rigorous reasoning.

Series: School of Mathematics Colloquium

This
is not a mathematics talk but it is a talk for mathematicians. Too
often, we think of historical mathematicians as only names assigned to
theorems. With vignettes and anecdotes, I'll convince you they were also
human beings and that, as the Chinese say, "May you live in interesting
times" really is a curse. More tales following up on the talk I gave
at GaTech in Nov., 2013. It is not assumed listeners heard that earlier
talk.

Series: School of Mathematics Colloquium

Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

Series: School of Mathematics Colloquium

The probabilistic method, pioneered by P. Erdös, has been key in many proofs from asymptotic geometric analysis. This method allows one to take advantage of numerous tools and concepts from probability theory to prove theorems which are not necessarily a-priori related to probability. The objective of this talk is to demonstrate several recent results which take advantage of stochastic calculus to prove results of a geometric nature. We will mainly focus on a specific construction of a moment-generating process, which can be thought of as a stochastic version of the logarithmic Laplace transform. The method we introduce allows us to attain a different viewpoint on the method of semigroup proofs, namely a path-wise point of view. We will first discuss an application of this method to concentration inequalities on high dimensional convex sets. Then, we will briefly discuss an application to two new functional inequalities on Gaussian space; an L1 version of hypercontractivity of the convolution operator related to a conjecture of Talagrand (joint with J. Lee) and a robustness estimate for the Gaussian noise-stability inequality of C.Borell (improving a result of Mossel and Neeman).

Series: School of Mathematics Colloquium

The
classic 1935 paper of Erdos and Szekeres entitled ``A combinatorial
problem in geometry" was a starting point of a very rich discipline
within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres
studied the following geometric problem. For every integer n \geq 3,
determine the smallest integer ES(n) such that any set of ES(n) points
in the plane in general position contains n members in convex position,
that is, n points that form the vertex set of a convex polygon. Their main result showed
that ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}. In 1960, they
showed that ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal.
Despite the efforts of many researchers, no improvement in the order of
magnitude has been made on the upper bound over the last 81 years. In
this talk, we will sketch a proof showing that ES(n) =2^{n +o(n)}.

Series: School of Mathematics Colloquium

The evolution, through spatially periodic linear dispersion, of rough
initial data leads to surprising quantized structures at rational times,
and fractal, non-differentiable profiles at irrational times. The
Talbot effect, named after an optical experiment by one of the founders
of photography, was first observed in optics and quantum mechanics, and
leads to intriguing connections with exponential sums arising in number
theory. Ramifications of these phenomena and recent progress on the
analysis, numerics, and extensions to nonlinear wave models will be
discussed.

Series: School of Mathematics Colloquium

A well-known theorem of Kuratowski (1930) in graph theory states
that a graph is planar if, and only if, it does not contain a
subdivision
of $K_5$ or $K_{3,3}$. Wagner (1937) gave a structural characterization of graphs containing no subdivision of $K_{3,3}$. Seymour in 1977 and, independently, Kelmans in 1979 conjectured that if a
graph does not contain
a subdivision of $K_5$ then it must be planar or contain a set of at most 4 vertices whose removal results in a disconnected
graph. In this talk, I will discuss additional background on this
conjecture (including connection to the Four Color Theorem), and outline
our recent proof of this conjecture (joint work with Dawei He and Yan
Wang).
I will also mention several problems that are related to this conjecture or related to our approach.

Series: School of Mathematics Colloquium

In this talk I will discuss some new applications of the
polynomial method to some classical problems in combinatorics, in
particular the Cap-Set Problem. The Cap-Set Problem is to determine the
size of the largest subset A of F_p^n having no three-term arithmetic
progressions, which are triples of vectors x,y,z satisfying x+y=2z. I will
discuss an analogue of this problem for Z_4^n and the recent progress on
it due to myself, Seva Lev and Peter Pach; and will discuss the work of
Ellenberg and Gijswijt, and of Tao, on the F_p^n version (the original
context of the problem).

Series: School of Mathematics Colloquium

Modern imaging data are often composed of several geometrically
distinct constituents. For instance, neurobiological images could
consist of a superposition of spines (pointlike objects) and
dendrites (curvelike objects) of a neuron. A neurobiologist might
then seek to extract both components to analyze their structure
separately for the study of Alzheimer specific characteristics.
However, this task seems impossible, since there are two unknowns
for every datum.
Compressed sensing is a novel research area, which was introduced in
2006, and since then has already become a key concept in various
areas of applied mathematics, computer science, and electrical
engineering. It surprisingly predicts that high-dimensional signals,
which allow a sparse representation by a suitable basis or, more
generally, a frame, can be recovered from what was previously
considered highly incomplete linear measurements, by using efficient
algorithms.
Utilizing the methodology of Compressed Sensing, the geometric
separation problem can indeed be solved both numerically and
theoretically. For the separation of point- and curvelike objects,
we choose a deliberately overcomplete representation system made of
wavelets (suited to pointlike structures) and shearlets (suited to
curvelike structures). The decomposition principle is to minimize
the $\ell_1$ norm of the representation coefficients. Our theoretical
results, which are based on microlocal analysis considerations, show
that at all sufficiently fine scales, nearly-perfect separation is
indeed achieved.
This project was done in collaboration with David Donoho (Stanford
University) and Wang-Q Lim (TU Berlin).

Series: School of Mathematics Colloquium

Effective bounds play a very important role in algebraic geometry with many applications. In this talk I will survey recent progress and open questions in the quantitative study ofreal varieties and semi-algebraic sets and their connections with other areas of mathematics -- in particular,connections to incidence geometry via the polynomial partitioning method. I will also discuss some results on the topological complexity of symmetric varieties which have a representation-theoretic flavor. Finally, if time permits I will sketch how some of these results extend to the category of constructible sheaves.