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Series: School of Mathematics Colloquium

The classical Weyl equidistribution theorem says that if v is a non-resonant
vector then the sequence v, 2v, 3v... is uniformly distributed on a
torus. In this
talk we discuss the rate of convergence to the uniform distribution.
This is a joint work with Bassam Fayad.

Series: School of Mathematics Colloquium

Statistical modeling amounts to specifying a set of candidates for what the probability distribution of an observed random quantity might be. Many models used in practice are of an algebraic nature in thatthey are defined in terms of a polynomial parametrization. The goal of this talk is to exemplify how techniques from computational algebraic geometry may be used to solve statistical problems thatconcern algebraic models. The focus will be on applications in hypothesis testing and parameter identification, for which we will survey some of the known results and open problems.

Series: School of Mathematics Colloquium

Random matrix theory is a fast developing topic with connections to so many areas of mathematics: probability, number theory, combinatorics, data analysis, mathematical physics, to mention a few.
The determinant is one of the most studied matrix functionals. In our talk, we are going to give a brief survey on the studies of this functional, dated back to Turan in the 1940s. The main focus will be on recent developments that establish the limiting law in various models.

Series: School of Mathematics Colloquium

Kickoff of the Tech Topology Conference from December 7-9, 2012.

This will be a Colloquium talk, aimed at a general audience. The topic is the curve complex, introduced by Harvey in 1974. It's a simplicial complex, and was introduced as a tool to study mapping class groups of surfaces. I will discuss recent joint work with Bill Menasco about new local pathology in the curve complex, namely that its geodesics can have dead ends and even double dead ends.

Series: School of Mathematics Colloquium

Populations, whether they be viral particles, bio-chemicals, plants or animals, are subject to intrinsic and extrinsic sources of stochasticity. This stochasticity in conjunction with nonlinear interactions between individuals determines to what extinct populations are able to persist in the long-term. Understanding the precise nature of these interactive effects is a central issue in population biology from theoretical, empirical, and applied perspectives.
For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable.
For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness
asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical work on Kansas prairies, acorn woodpecker populations, and microcosm experiments demonstrating these phenomena will be discussed.

Series: School of Mathematics Colloquium

Isoperimetric problems in Gaussian spaces have been studied since the 1970s. The study of these problems involve geometric measure theory, symmetrization techniques, spherical geometry and the study of diffusions associated with the heat equation. I will discuss some of the main ideas and results in this area along with some new results jointly with Joe Neeman.

Series: School of Mathematics Colloquium

Self-adjoint $n$-by-$n$ matrices have a natural partial ordering,
namely $ A \leq B $ if the matrix $ B - A$ is positive semi-definite.
In 1934 K. Loewner characterized functions that preserve this ordering;
these functions are called $n$-matrix monotone.
The condition depends on the dimension $n$, but if a function
is $n$-matrix monotone for all $n$, then it must extend analytically
to a function that maps the upper half-plane to itself.
I will describe Loewner's results, and then discuss what happens
if one wants to characterize functions $f$ of two (or more) variables that
are matrix monotone in the following sense:
If $ A = (A_1, A_2)$ and $B = (B_1,B_2)$ are pairs of commuting
self-adjoint
$n$-by-$n$ matrices, with $A_1 \leq B_1 $ and $A_2 \leq B_2$,
then $f(A) \leq f (B)$.
This talk is based on joint work with Jim Agler and Nicholas Young.

Series: School of Mathematics Colloquium

A basic strategy for linear optimization over a complicated convex set is
to try to express the set as the projection of a simpler convex set that
admits efficient algorithms. This philosophy underlies all
"lift-and-project" methods in
optimization which attempt to find polyhedral or spectrahedral lifts of
complicated sets. In this talk I will explain how the existence of a lift
is equivalent to the ability to factorize a certain operator associated to
the convex set through a cone.
This theorem extends a result of Yannakakis who showed that polyhedral
lifts of polytopes are controlled by the nonnegative factorizations of the
slack matrix of the polytope. The connection between cone lifts and cone
factorizations of convex sets yields a uniform framework within which to
view all lift-and-project methods, as well as offers new tools for
understanding convex sets. I will survey this evolving area and the main
results that have emerged thus far.

Series: School of Mathematics Colloquium

It is well-known that a deterministic dynamical system can exhibit
stochastic behavior that is due to the fact that instability along
typical trajectories of the system drives orbits apart, while
compactness of the phase space forces them back together. The consequent
unending dispersal and return of nearby trajectories is one of the
hallmarks of chaos.
The hyperbolic theory of dynamical systems provides a mathematical
foundation for the paradigm that is widely known as "deterministic
chaos" -- the appearance of irregular chaotic motions in purely
deterministic dynamical systems. This phenomenon is considered as one of
the most fundamental discoveries in the theory of dynamical systems in
the second part of the last century. The hyperbolic behavior can be
interpreted in various ways and the weakest one is associated with
dynamical systems with non-zero Lyapunov exponents.
I will discuss the still-open problem of whether dynamical systems with
non-zero Lyapunov exponents are typical. I will outline some recent
results in this direction. The genericity problem is closely related to
two other important problems in dynamics on whether systems with nonzero
Lyapunov exponents exist on any phase space and whether nonzero
exponents can coexist with zero exponents in a robust way.

Series: School of Mathematics Colloquium

The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing
the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators. The investigation of this problem combines combinatorial, algebraic and probabilistic tools. Several intriguing questions that remain open will be mentioned as well.