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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

Series: School of Mathematics Colloquium

An introduction for non-experts on real and finite Euler sums, also known as multiple zeta values.

Series: School of Mathematics Colloquium

Drinfeld's upper half-spaces over non-archimedean local fields are the founding examples of the theory of period domains. In this talk we consider analogs of Drinfeld's upper half-spaces over finite fields. They are open subvarieties of a projective space. We show that their automorphism group is the group of automorphisms of the ambient projective space. This is a problem in birational geometry, which we solve using tools in non-archimedean analytic geometry.

Series: School of Mathematics Colloquium

The emergence of the 2009 H1N1 influenza A strain and delays in
production of vaccine against it illustrate the importance of
optimizing vaccine allocation. We have developed computational
optimization models to determine optimal vaccination strategies with
regard to multiple objective functions: e.g.~deaths, years of life
lost, economic costs. Looking at single objectives, we have found that
vaccinating children, who transmit most, is robustly selected as the
optimal allocation. I will discuss ongoing extensions to this work to
incorporate multiple objectives and uncertainty.

Series: School of Mathematics Colloquium

Abstract: I will talk about two types of random processes -- the classical Sherrington-Kirkpatrick (SK) model of spin glasses and its diluted version. One of the main goals in these models is to find a formula for the maximum of the process, or the free energy, in the limit when the size of the system is getting large. The answer depends on understanding the structure of the Gibbs measure in a certain sense, and this structure is expected to be described by the so called Parisi solution in the SK model and Mézard-Parisi solution in the diluted SK model. I will explain what these are and mention some results in this direction.

Series: School of Mathematics Colloquium

Kickoff of the Tech Topology Conference from December 6-8, 2013. For complete details see

ttc.gatech.edu

We will start by defining the Jones polynomial of a knot and talking about some of its classical applications to knot theory. We will then define a fancier version ("categorification") of the Jones polynomial, called Khovanov homology and mention some of its applications. We will conclude by talking about a further refinement, a Khovanov homotopy type, sketch some of the ideas behind its construction, and mention some applications. (This last part is joint work with Sucharit Sarkar.) At least the first half of the talk should be accessible to non-topologists.

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.

Series: School of Mathematics Colloquium

In this talk, I shall sketch the study of the problem of Arnold diffusion from variational point of view. Arnold diffusion has been shown typical phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom:
$$
H(x,y)=h(y)+\epsilon P(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3.
$$
Under typical perturbation $\epsilon P$, the system admits ``connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\min\alpha$.

Series: School of Mathematics Colloquium

This talk deals with problems that are asymptotically related to best-packing and best-covering. In particular, we discuss how to efficiently generate N points on a d-dimensional manifold that have the desirable qualities of well-separation and optimal order covering radius, while asymptotically having a prescribed distribution. Even for certain small numbers of points like N=5, optimal arrangements with regard to energy and polarization can be a challenging problem.