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Series: School of Mathematics Colloquium

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random particle simulations, flocks and mills, and their qualitative behavior. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. Compactly supported global minimizers determine the flock patterns whose existence is related to the classical H-stability in statistical mechanics and the classical obstacle problem for differential operators.

Series: School of Mathematics Colloquium

Charles Stein brought the method that now bears his name to life in a 1972 Berkeley symposium paper that presented a new way to obtain information on the quality of the normal approximation, justified by the Central Limit Theorem asymptotic, by operating directly on random variables. At the heart of the method is the seemingly harmless characterization that a random variable $W$ has the standard normal ${\cal N}(0,1)$ distribution if and only if E[Wf(W)]=E[f'(W)] for all functions $f$ for which these expressions exist. From its inception, it was clear that Stein's approach had the power to provide non-asymptotic bounds, and to handle various dependency structures. In the near half century since the appearance of this work for the normal, the `characterizing equation' approach driving Stein's method has been applied to roughly thirty additional distributions using variations of the basic techniques, coupling and distributional transformations among them. Further offshoots are connections to Malliavin calculus and the concentration of measure phenomenon, and applications to random graphs and permutations, statistics, stochastic integrals, molecular biology and physics.

Series: School of Mathematics Colloquium

We present algorithms for performing sparse univariate
polynomial interpolation with errors in the evaluations of
the polynomial. Our interpolation algorithms use as a
substep an algorithm that originally is by R. Prony from
the French Revolution (Year III, 1795) for interpolating
exponential sums and which is rediscovered to decode
digital error correcting BCH codes over finite fields (1960).
Since Prony's algorithm is quite simple, we will give
a complete description, as an alternative for Lagrange/Newton
interpolation for sparse polynomials. When very few errors
in the evaluations are permitted, multiple sparse interpolants
are possible over finite fields or the complex numbers,
but not over the real numbers. The problem is then a simple
example of list-decoding in the sense of Guruswami-Sudan.
Finally, we present a connection to the Erdoes-Turan Conjecture
(Szemeredi's Theorem).
This is joint work with Clement Pernet, Univ. Grenoble.

Series: School of Mathematics Colloquium

The piecewise linear objects appearing in tropical geometry are shadows, or skeletons, of nonarchimedean analytic spaces, in the sense of Berkovich, and often capture enough essential information about those spaces to resolve interesting questions about classical algebraic varieties. I will give an overview of tropical geometry as it relates to the study of algebraic curves, touching on applications to moduli spaces.

Series: School of Mathematics Colloquium

The talk is meant to be a gentle introduction to a part of combinatorial topology which studies randomly generated objects. It is a rapidly developing field which combines elements of topology, geometry, and probability with plethora of interesting ideas, results and problems which have their roots in combinatorics and linear algebra.

Series: School of Mathematics Colloquium

The multi-server queue with non-homogeneous Poisson arrivals and
customer abandonment is a fundamental dynamic rate queueing model for
large scale service systems such as call centers and hospitals. Scaling
the arrival rates and number of servers gives us fluid and diffusion
limits. The diffusion limit suggests a Gaussian approximation to the
stochastic behavior. The fluid mean and diffusion variance can form a
two-dimensional dynamical system that approximates the actual transient
mean and variance for the queueing process. Recent work showed that a
better approximation for mean and variance can be computed from a related
two-dimensional dynamical system. In this spirit, we introduce a new
three-dimensional dynamical system that estimates the mean, variance,
and third cumulant moment. This surpasses the previous two approaches by
fitting the number in the queue to a quadratic function of a Gaussian
random variable. This is based on a paper published in QUESTA and is
joint work with Jamol Pender of Cornell University.

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).