Lucia Caporaso: Algebraic curves and their Jacobians from the point of view of tropical geometry


Dustin Cartwright: Dual complexes of uniruled varieties

The dual complex of a semistable degeneration records the combinatorics of the intersections in the special fiber. The dual complex is homotopy equivalent to the Berkovich analytification, which is a non-Archimedean analogue of the analytic topology on a complex algebraic variety. I will talk about the following relation between the geometry of the general fiber and the topology of the dual complex, and thus the analytification: in arbitrary characteristic, the dual complex of an n-dimensional uniruled variety has the homotopy type of an (n-1)-dimensional simplicial complex.

Antonella Grassi: Toric Noether-Lefschetz

The Noether-Lefschetz theorem states that a curve in a very general surface X  of degree  greater than 4 in the projective space is a restriction of a surface in the ambient space.The Noether-Lefschetz locus is the locus where the Picard number is greater than 1. I will discuss the analogues in  toric ambient spaces.

Brian Harbourne: Lines in P2---Open problems in combinatorics, commutative algebra and algebraic geometry

What arrangements of lines in the complex projective plane are there that have no simple crossings (i.e., no points where exactly two lines meet)? I will relate this open question to several open problems of recent interest, including the Bounded Negativity Conjecture, containment problems of symbolic powers in ordinary powers of ideals related to work of Ein-Lazarsfeld-Smith and Hochster-Huneke, and recent work extending the context of the SHGH Conjecture.

Daniel Krashen: Clifford algebras and the search for Ulrich bundles 

Abstract: The classical notion of the Clifford algebra of a quadratic form has been generalized to other types of higher degree forms by a number of authors. The representations of these generalized Clifford algebras turn out to correspond to “Ulrich bundles,” which are a very special class of vector bundle on a hypersurface. In this talk, I’ll describe joint work with Adam Chapman and Max Lieblich of a new construction, generalizing the previous ones, of a Clifford algebra of a finite morphism of proper schemes, I’ll discuss connections to the arithmetic of genus 1 curves, and I’ll present some new results on the existence of Ulrich bundles. 

Martin Olsson: A stronger derived Torelli theorem for K3 surfaces

In earlier work with Lieblich we introduced a notion of filtered derived equivalence,  and showed that if two K3 surfaces admit such an equivalence then they are isomorphic. In this talk I will discuss more refined aspects of filtered derived equivalences related to the action on the cohomological realizations of the Mukai motive.  This is joint work with Max Lieblich.

Alena Pirutka: Stable rationality and quadric bundles

We will discuss properties of stable rationality for families of quadric bundles over rational surfaces. This is a work in common with Brendan Hassett and Yuri Tschinkel.

Karen Smith: Frobenius in valuation rings

Abstract: I will explain some recent results regarding how the Frobenius map and F-singularities behave in the somewhat exotic setting of valuation rings of a function field k(X) of prime characteristic. For example, we will see that valuation rings are always F-pure but rarely Frobenius split. In this non-Noetherian world, the finiteness of Frobenius becomes a subtle and interesting feature. For example, the Frobenius map is finite if and only if the valuation ring is Abhyankar. For discrete valuations, this means the valuation is the valuation of some divisor. In the discrete valuation case, finiteness of Frobenius turns out also to be closely related to the property of excellence.