Georgia Institute of Technology College of Sciences

 Around 1997, Shub and Smale proved that sufficiently good upper bounds
on the number of integer roots of polynomials in one variable --- as a function
of the input complexity --- imply a variant of P not equal to NP. Since then,
later work has tried to go half-way: Trying to prove that easier root counts
(over fields instead) still imply interesting separations of complexity
classes. Koiran, Portier, and Tavenas have found such statements over the real
numbers.

        We present an analogous implication involving p-adic valuations:    
If the integer roots of SPS polynomials (i.e., sums of products of sparse polynomials) of size s never yield more than s^{O(1)} distinct p-adic
valuations, then the permanents of n by n matrices cannot be computed by constant-free, division-free arithmetic circuits of size n^{O(1)}. (The
implication would be a new step toward separating VP from VNP.) We also show that this conjecture is often true, in a tropical geometric sense (paralleling a similar result over the real numbers by Briquel and Burgisser). Finally, we prove a special case of our conjectured valuation bound, providing a p-adic analogue of an earlier real root count for polynomial systems supported on circuits. This is joint work with Joshua Goldstein, Pascal Koiran, and Natacha Portier.

Loosely speaking, the maximum likelihood threshold of a statistical model is the fewest number of data points needed to fit the model using maximum likelihood estimation. In this talk, I will discuss combinatorial and algebraic-geometric approaches to studying this poorly understood quantity for a certain class of Gaussian models. This is based on joint work with Sean Dewar, Steven Gortler, Tony Nixon, Meera Sitharam, and Louis Theran

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.  

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher. 

I will highlight recent interplay between problems in extremal combinatorics and real algebraic geometry. This sheds a new light on undecidability of graph homomorphism density inequalities in extremal combinatorics, trace inequalities in linear algebra, and symmetric polynomial inequalities in real algebraic geometry. All of the necessary notions will be introduced in the talk. Joint work with Jose Acevedo, Sebastian Debus and Cordian Riener.