- Dmitro Arinkin (Wisconsin)
- Arend Bayer (Connecticut)
- Kiran Kedlaya (San Diego)
- David Speyer (Michigan)
- Ravi Vakil (Stanford)

9:00-10:00 Breakfast catered by "Until we eat again"

10-12:00 David Speyer: DeRham cohomology of cluster algebras

1:30-3:30 Dmitro Arinkin: Singular support of coherent sheaves

4:00 -6:00 Arend Bayer: Birational geometry of moduli of sheaves on K3s via wall-crossing

6:30 Dinner, Ruby's Barbeque

8:00-9:00 Breakfast catered by "Until we eat again"

9-11: Kiran Kedlaya: Cluster algebras and p-adic floating point arithmetic

11:30-1:30 Ravi Vakil: Stabilization of discriminants in the Grothendieck ring

- Dmitro Arinkin: Singular support of coherent sheaves
Abstract: Some geometric objects can be studied `microlocally': instead of just working with their support (the set of points where an object is non-trivial), one can consider their `singular support', which remembers the `direction' of non-trivial behaviour. Examples include the wave front of a distribution, the singular support of a constructible sheaf, and the characteristic variety of a D-module. The goal of this talk is to sketch such `microlocal' theory for coherent sheaves. It turns out that the singular support could be non-trivial only for coherent sheaves on a singular variety. I will define singular support for coherent sheaves on a singular variety that is locally a complete intersection. The singular support measures the `imperfection' of a coherent sheaf: it equals zero if and only if the coherent sheaf has finite Tor dimension (i.e., the sheaf is perfect). In the second half of the talk, I will show how the singular support can be used to enlarge the derived category of a singular variety. In this way, a singular variety carries not one but many derived categories, which differ from each other by a kind of `renormalization'. I also hope to explain the importance of such `renormalized' derived categories for the geometric Langlands program.

- Arend Bayer: Birational geometry of moduli of sheaves on K3s via wall-crossing
I will explain joint work with Emanuele Macri, in which we systematically relate wall-crossing for Bridgeland stability conditions to the birational geometry of moduli spaces of stable sheaves on K3 surfaces. Our results include a description of the nef cone in terms of the lattice of the K3 surface, and a proof of a well-known conjecture on the existence of Lagrangian fibrations. These results are new even in the case of the Hilbert scheme.

- Kiran Kedlaya: Cluster algebras and p-adic floating point arithmetic
Just like real numbers, general p-adic numbers are not finitary objects, and so cannot be exactly specified on a computer. One instead works with them using some scheme of finite approximations. For ring computations (in which no division is involved) one may use fixed-point approximations (i.e., truncation modulo a power of p) but for more general computations it is best to use an analogue of real floating-point arithmetic. When using real or p-adic floating-point arithmetic, one generally sees errors that compound over the course of a computation. But in the p-adic case, certain computations carry enough hidden algebraic structure to cause massive cancellations and drastically reduce the loss of accuracy. One can exhibit many examples of this phenomenon using cluster algebras; as an example, we prove a weak version of a numerical conjecture of David Robbins concerning the numerical stability of the Dodgson condensation algorithm for computing determinants of matrices. Joint work with Joe Buhler (CCR La Jolla).

- David Speyer: DeRham cohomology of cluster algebras
I'll report on work on progress with Thomas Lam on computing the algebraic DeRham cohomology of cluster varieties, with its weight filtration. Full success in this goal would lead to progress in computing extensions of Verma modules and has connections to the work of Hamed et. al. on computing scattering amplitudes. In the first half of the talk, I'll explain the background on deRham cohomology, weight filtrations and cluster algebras. In the second half, I'll explain the partial results we have proved, and the fascinating patterns we have observed and can't prove.

- Ravi Vakil: Stabilization of discriminants in the Grothendieck ring
We consider the ``limiting behavior'' of {\em discriminants}, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the "ring of motives", as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization can be described in terms of the motivic zeta values. The results extend parallel results in both arithmetic and topology. I will also present a conjecture (on ``motivic stabilization of symmetric powers'') suggested by our work. Although it is true in important cases, Daniel Litt has shown that it contradicts other hoped-for statements. This is joint work with Melanie Wood.

A block of rooms has been reserved at La Quinta Inn Austin Captiol, 300 E. 11th St. Austin (512)476-1166.

Participants should contact the hotel themselves to make a reservation.