Prismatic cohomology is a recently discovered mechanism for producing certain 1-parameter deformations of de Rham cohomology in arithmetic geometry. In this talk, I'll give the examples motivating this theory, give an outline of its construction, and mention a couple of applications.

Combinatorial principles like inclusion-exclusion can be obtained by decategorifying natural structures on the category of finite sets, whose Grothendieck ring is the ring of integers, Z. In this talk, we explain what happens when similar structures are decategorified from the category of almost-finite cyclic sets, whose Grothendieck ring is the big Witt ring, W(Z). A natural example of an almost-finite cyclic set is the set of \overline{Fq}-points of an algebraic variety over Fq, and the corresponding element in W(Z) is the zeta function of the variety; using this, we explain how combinatorics in the ring of big Witt vectors can be applied to prove theorems in arithmetic and motivic statistics.

The symmetries of systems of polynomial equations can be be understood in terms of the geometry of the variety of zeroes (or solution set) of the polynomials. Roughly speaking, there are 3 kinds of geometries corresponding to positive, zero and negative curvature giving rise to 3 different kinds of symmetry groups. In this lecture, I will discuss recent advances in algebraic geometry that lead to very precise results on the structure of these symmetry groups.

The integral Hodge conjecture predicts which integral cohomology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. This statement is known to be false in general, as shown first by Atiyah and Hirzebruch. I'll give a short introduction to this conjecture, and present some new counterexamples. This is joint work with Olivier Benoist.

Algebraic varieties are geometric objects defined by polynomial equations. The minimal model program (MMP) is an ambitious program that aims to classify algebraic varieties. According to the MMP, there are 3 building blocks: Fano varieties, Calabi-Yau varieties and varieties of general type which are higher dimensional analogs of Riemann Surfaces of genus 0,1 or at least 2 respectively. In this talk I will recall the general features of the MMP and discuss recent advances in our understanding of Fano varieties and varieties of general type.

We will discuss a stable pair compactification of the moduli space of degree d surfaces in P^3.  These compactifications, introduced by Koll\'ar and Shepherd-Barron arising from ideas in the Minimal Model Program, are generally difficult to describe explicitly.  Motivated by work of Hacking, who obtained an explicit modular compactification for plane curves, we consider a surface S in P^3 as a pair (P^3, S).  Then, using a particular enlarged class of these pairs, we can compactify the moduli space of such surfaces.  Using this framework, for surfaces of odd degree d, we obtain a rough classification of the singular objects on the boundary of this moduli space and will discuss some divisorial components of the boundary.  We give a more detailed description for quintic surfaces. 

After recent spectacular progress in the classification of varieties over an algebraic closed field of characteristic 0 (e.g. the solution set of a system of polynomial equations defined by p1,...,prp1,...,pr in C[x1,...,xn]C[x1,...,xn]) it is natural to try and understand the geometry of varieties defined over an algebraically closed field of characteristic p>0p>0. Many technical difficulties arise in this context. Nevertheless, there has been much progress recently. In particular, the MMP was established for 3-folds in characteristic p>5p>5 by work of Birkar, Hacon, Xu and others. In this talk, we will explain some of the challenges and the recent progress in this active area of research.

This is a report on joint work in progress with Darmon, Rotger, and Venkatesh.    In a series of papers, Venkatesh and his collaborators have proposed a conjectural framework that incorporates the action of motivic cohomology groups on the cohomology of locally symmetric spaces.  The simplest case concerns the cohomology of the special fiber in characteristic p of the modular curve with coefficients in the sheaf of weight 1 forms. There the conjecture relates the action of derived Hecke operators, which only exist over coefficient rings in positive characteristic, to the (discrete) logarithms of certain units in the number fields cut out by two-dimensional complex representations of the absolute Galois group of Q.  Our main result is a proof of this conjecture as it pertains to weight one forms whose associated Galois representations are induced from Dirichlet characters of quadratic fields.  The proof is based on classical constructions in the theory of modular forms. If time permits, there will be a discussion of how Venkatesh's conjectures might fit into a (highly speculative) derived version of the Langlands program.

Madeleine Weinstein, UC Berkeley - 3:00pm - 4:00pm

Zvi Rosen, Florida Atlantic University - 4:00pm - 5:00pm