This seminar will happen on Tuesdays, roughly monthly, alternating between Stanford and Berkeley. There will be two lectures, with a break in between, and plenty of time for informal discussions. There will also be a dinner afterwards (subsidized for students). Drives and parking will be organized. For more information, please contact David Eisenbud (de@msri.org) or Tom Graber (graber@math.berkeley.edu) at Berkeley, or Jun Li (jli@math.stanford.edu) or Ravi Vakil (vakil@math.stanford.edu) at Stanford. Here is a map of Stanford's campus, with parking indicated.

Click here for the Stanford Algebraic Geometry Seminar, and here for the Berkeley Commutative Algebra and Algebraic Geometry Seminar.

Tuesday, November 12, at Stanford: (poster)

Mark Gross (San Diego): Affine structures, log structures, and mirror symmetry (3:00-4:00, Rm. 380-383N)

Abstract: I will discuss an algebro-geometric version of the Strominger-Yau-Zaslow mirror symmetry conjecture. The original conjecture seeks to explain mirror symmetry via dual special Lagrangian torus fibrations on mirror pairs of Calabi-Yau manifolds. The algebro-geometric version takes the base of such fibrations as the fundamental object governing mirror symmetry, and constructs degenerations of Calabi-Yau manifolds from certain combinatorial structures on this base. Mirror symmetry is then given by a combinatorial duality, the discrete Legendre transform, between these combinatorial structures. I will try to explain the basic outline of this construction.

Dave Benson (Georgia): Local cohomology and computation of group cohomology (4:30-5:30, Rm. 380-380W)

Abstract: I shall talk about the following problem. Suppose that you want to compute the cohomology of a finite group. You might begin a projective resolution, and compute the cohomology in the first few degrees, and compute products in the Yoneda fashion by composing maps on resolutions. But how do you know when you have found all the generators and relations? Based on some ideas of Jon Carlson, I shall describe a method for telling when you can stop. The method is closely connected with local cohomology and Koszul complexes. It is related to a conjecture about the Castelnuovo-Mumford regularity of the cohomology ring, but does not rely on verifying this conjecture.

Tuesday, December 10, at Berkeley: (poster version 1, 2)

Craig Huneke (Kansas): Free resolutions and initial ideals (location TBA)

Abstract: This talk will discuss two fundamental objects in commutative algebra: the minimal free graded resolution of a homogeneous ideal in a polynomial ring and initial ideals of homogeneous ideals in polynomial rings. The main question in this talk will be: what is the relationship, if any, between these two objects? We will survey some known results, and then talk about some recent progress made this semester at MSRI.

Karen Smith (Michigan): Jumping Numbers and Multiplier Ideals

Abstract: Jumping numbers and multiplier ideals can be described as new invariants of singularities. I'll try to present a snapshot of an ongoing joint project with Lawrence Ein and Rob Lazarsfeld in which we apply multiplier ideals to prove uniform results in commutative algebra.

Tuesday, February 11, at Stanford.

Aaron Bertram (Utah): Quantum aspects of classical (temperate zone) Grassmannians (3:00-4:00, Rm. 380-383N)

Abstract: I'll talk about joint work with Ciocan-Fontanine and Kim proving a conjecture of Hori-Vafa on the nature of the J-function of the complex Grassmannian. This result can be thought of as a ``quantum'' analogue of a very pretty result due to Shaun Martin relating intersection numbers of the Grassmannian G(r,n) (or any symplectic quotient X//G of a Hamiltonian G-manifold) with intersection numbers of the r-fold product of projective spaces (P^{n-1})^r (or the symplectic quotient X//T by a maximal torus of G).

Bernd Sturmfels (Berkeley): Tropical Grassmannians (4:30-5:30, Rm. 380-383N)

Abstract: In tropical algebraic geometry, the zero sets of polynomial equations are piecewise-linear. Any tropical variety is a polyhedral complex derived from the Grobner fan of its defining equations. It was first introduced by Bergman in 1971 under the name "logarithmic limit set". After introducing the basics on amoebas and tropical geometry, we present joint work work with David Speyer on the geometry and combinatorics of tropical Grassmannians. As a running example, we demonstrate that lines in tropical projective space are trees, and we prove that the tropical Grassmannian of lines is precisely the space of phylogenetic trees studied by Billera, Holmes and Vogtman.

Tuesday, March 11, at Berkeley. (poster)

Brendan Hassett (Rice): Towards a canonical model for the moduli space of curves (3:45-4:45, Rm. 344 Evans Hall)

Abstract: (Joint with D. Hyeon) Consider the moduli space of stable curves as a log-variety, with boundary Delta corresponding to the nodal curves. We seek to describe its log canonical model with respect to K + A Delta. When A=1, we recover the moduli space of stable curves; for A=0, this would be the canonical model of the moduli space, which is expected to exist for g>23 by work of Eisenbud, Harris, and Mumford. For intermediate values of A, the log canonical model can be constructed with Geometric Invariant Theory. As A decreases, the log canonical model parametrizes curves with increasingly complicated singularities: cusps, tacnodes, and worse.

Persi Diaconis (Stanford): Upper Triangular Matrices??? (5:00-6:00, Rm. 344 Evans Hall)

Abstract: The group of n x n upper-triangular matricies with elements in a finite field is a simple, friendly-sounding object but it is a nightmare for representation theorists. You can prove that the conjugacy classes and characters are indescribable. Recently, Andre, Carter, and Yan have developed a super-class theorem which has elegant combinatorics and solves some problems we usually use character theory for. I will explain and illustrate on a simple random walk problem. This is joint work with Ery Arias-Castro and Richard Stanley.