Talk Titles and Abstracts:

Speaker: Ana-Maria Castravet
Title: "Extremal divisors and rigid curves on the moduli space of stable rational curves" Abstract: I will describe some recent joint work with Jenia Tevelev on how to construct many examples of extremal divisors, rigid curves, and birational morphisms with unexpected properties for the Grothendieck Knudsen moduli space $\bar M_{0,n}$ of stable rational curves. The basic tool is an isomorphism between M_{0,n} and the Brill-Noether locus of a very special reducible curve corresponding to a hypergraph.

Speaker: Renzo Cavalieri
Title: "Wall crossing for double Hurwitz numbers" Abstract: We discuss a graph theoretic way of computing double Hurwitz numbers that sheds light on the piecewise polynomial structure shown by Goulden-Jackson-Vakil, re-proves the genus 0 wall crossing formula of Shadrin-Shapiro-Vainshtein and allows to obtain a general genus wall crossing formula. This is joint work with Paul Johnson and Hannah Markwig.

Speaker: Brent Doran
Title: "Fun with A^1: playing with the affine line in quotients, cohomologies, and connectivities" Abstract: If X x A^1 = A^n, what is X? For smooth manifolds the complete answer is known and involves the Poincare conjecture. The analogous problem in birational and in biregular geometry (Zariski Cancellation) is famously subtle. Upon reinterpretation it touches upon a number of fundamental issues, in particular regarding what "topological" (e.g., motivic) notions can tell us about genuinely algebro-geometric phenomena. En route we encounter non-reductive group actions and Hilbert's 14th problem (finite generation of rings), surprising facts about moduli of vector bundles, algebraic spaces that aren't schemes, a host of interesting stably rational varieties, a curious re-encoding of affine hypersurfaces, and quadrics behaving an awfully lot like spheres. Time permitting, the talk will give a gentle introduction to a few basic ideas, some quite recent, of A^1-homotopy hiding in the background, and their relevance for notions of connectivity in algebraic geometry. The material is mostly drawn from joint work with Aravind Asok and with Frances Kirwan.

Speaker: Jordan Ellenberg
Title: "Stable topology of Hurwitz spaces and arithmetic counting problems" Abstract: Hurwitz spaces are moduli spaces of finite branched covers of P^1. We will discuss the stabilization of the cohomology of these spaces as the number of branch points grows, with the Galois group of the cover being fixed; this can be thought of as a "Harer theorem" for this family of moduli space. It turns out that the function field analogues of many popular conjectures in analytic number theory (due to Cohen-Lenstra, Bhargava, etc.) reduce to topological questions about Hurwitz spaces. We will discuss the arithmetic consequences of the stabilization theorem, and of a geometrically natural conjecture about the stable cohomology classes of Hurwitz spaces. (joint work with Akshay Venkatesh and Craig Westerland)

Speaker: Christian Haesemeyer
Title: "Du Bois invariants and a question from algebraic K-theory" Abstract: Some 40 years ago, Bass asked whether NK_n(R) = 0 implies N^2K_n(R) = 0 for any ring R and any integer n. In this talk, we explain what the notation means and how a calculation of du Bois invariants of a certain semi qausihomogeneous singularity can lead to an answer.

Speaker: Max Lieblich
Title: "A counter counter-example example" Abstract: I will discuss an interesting division algebra over Q(t) which seems to violate what its discoverers called "a `Hasse principle'" with respect to a quantity known as the Faddeev index. Contemplating the example leads one to believe that it should be easy to write down a variety for which the Hasse principle corresponds to this "a `Hasse principle'". Surprisingly, this turns out to be maximally wrong. In particular, I will show that there is a canonically associated moduli problem over Q (in fact, a form of the moduli space of parabolic bundles of rank 2 on the projective line with parabolic structure at five points) that lacks local points at two places. This algebra fits into a family of such examples, all of which have locally obstructed canonical moduli problems. This gives yet another example how the use of geometric methods illuminates a seemingly purely algebraic phenomenon. All of the necessary non-commutative algebra will be (vaguely) explained!

Speaker: David Rydh
Title: "Compactification of Deligne-Mumford stacks" Abstract: Nagata's compactification theorem states that any separated scheme of finite type can be embedded into a proper scheme. In this talk I will describe how every separated Deligne-Mumford stack in characteristic zero can be embedded into a proper Deligne-Mumford stack. The main technique is a sufficiently good description of modifications of (tame) stacks in terms of "stacky blow-ups". This relies on a general extension theorem of finite \'etale covers which is proved via Riemann-Zariski spaces and desingularization.