Mathur: The relationship between analytic and algebraic geometry (GAGA) is a rich area of study. For example, Grothendieck's existence theorem states that if X is proper over a complete local Noetherian ring A, then a coherent sheaf on the formal scheme X^, a priori only definable using formal power series, is actually algebraic. Such GAGA-type results are now standard tools for studying varieties and their families. One sample consequence: a compatible system of line bundles L_n on the thickenings X_n of the special fiber of X/A uniquely algebraizes to a line bundle L on X. In this talk, we answer a question Grothendieck posed in the 1960s about Brauer classes: can a Brauer class be determined from a compatible system of classes on the X_n? This is joint work with Andrew Kresch.

Sawant: The category of strictly \A^1-invariant sheaves is central to the development of \A^1-algebraic topology over a field. It is a nontrivial fact that this is an abelian, symmetric monoidal category. In this talk based on joint work with Fabien Morel, I will outline a fundamental computation in this category, namely, that of the tensor product of the Milnor-Witt K-theory sheaf with the so-called free strictly \A^1-invariant sheaf on a point, defined over a finite separable field extension of the base field. The computation is made possible by cellular \A^1-homology introduced in our earlier joint work.

Soprunov: Fix a lattice polytope P and consider the family of polynomials whose monomials have exponents in P and whose coefficients are arbitrary elements of a finite field F_q. Our goal is to estimate the maximum number of F_q-zeros of the polynomials in such a family in terms of geometric/combinatorial invariants of P. When q is large, the maximum is given by polynomials with the largest number of factors. On the combinatorial side, this corresponds to "maximal Minkowski decompositions" inside P. I will first explain this connection and review the previously known case of bivariate polynomials (i.e. dim P=2). I will then focus on the case of trivariate polynomials (i.e. dim P=3) where the combinatorics is much richer and new phenomena arise. This problem has a direct application to minimum distance estimation for a class of linear error-correcting codes, called "toric codes". The talk is based on a joint work with Kyle Meyer and Jenya Soprunova.

Patel: TBA