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.

We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al.. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph $G$ characterizing the relations of the type $A$ components of the queer crystal. We provide a counterexample to Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph $G$ on the type $A$ components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements.

I will describe how recent advances have made possible to study the birational geometry of hyperkaehler varieties of K3-type using the machinery of wall-crossing and stability conditions on derived categories as developed by Tom Bridgeland. In particular Bayer and Macrì relate birational transformations of the moduli space M of sheaves of a K3 surface X to wall-crossing in the space of Bridgeland stability conditions Stab(X). I will explain how it is possible to refine their analysis to give a precise description of the geometry of the exceptional locus of any birational contraction of M.

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

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

K-stability is the algebraic notion which is supposed to characterize whether a Fano variety admits a K\"ahler-Einstein metric. One important feature of the notion of K-stability is that it is supposed to give a nicely behaved moduli space. To construct the K-moduli space of Q-Fano varieties as an algebraic space, one important step is to prove the openness of K-(semi)stable locus in families. In this talk, I will explain the proof of openness of uniform K-stability in families of Q-Fano varieties. This is achieved via showing the lower semi-continuity of delta-invariant, an interesting invariant introduced by Fujita and Odaka as a variant of Tian's alpha-invariant. This is a joint work with Harold Blum.