Georgia Institute of Technology College of Sciences

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.

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.

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

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.