University of Chicago Algebraic Geometry Seminar

February 13
Wieslawa Niziol (University of Utah):    On uniqueness of p-adic period morphisms.
Abstract: P-adic period morphisms relate p-adic etale cohomology and de Rham cohomology of algebraic varieties defined over p-adic local fields. We have three general constructions of such morphisms: the syntomic one due to Bloch, Kato, Fontaine-Messing, Tsuji; the almost etale one due to Faltings; and the motivic one due to myself. I will present uniqueness criteria for p-adic period morphisms and show that they imply equality of the above morphisms.

February 20
Dmitry Logachev (Simon Bolivar University, Caracas, Venezuela):   Duality for Anderson T-motives.

March 12
Paulo Cascini (UC Santa Barbara):   On the Minimal Model Program.
Abstract: The aim of the minimal model program is to generalize the classification of complex projective surfaces, known in the early 20th century, to higher dimensional varieties. Besides providing a historical introduction, we will discuss some recent results and new aspects of the Program.

April 2nd
Matt Kerr (Durham):   The Abel-Jacobi map on the Eisenstein symbol.
Abstract: In this talk we consider two different constructions of motivic cohomology classes on families of toric hypersurfaces and on Kuga varieties. Under suitable modularity conditions on the former we say how the constructions "coincide", obtaining a complete explanation of a phenomenon observed by Villegas, Stienstra, and Bertin in the context of Mahler measure. (This is where the AJ computation on the Kuga varieties, done using our formula with J. Lewis and S. Mueller-Stach, will be summarized.) We also look at an application of the toric construction in the non-modular case, to limits of normal functions for families of Calabi-Yau 3-folds.

April 9
Leonid Rybnikov (IAS):   Shift of argument subalgebras in Poisson algebras and their quantization. (joint work with B. Feigin and E. Frenkel)
Abstract: The symmetric algebra S(g) of a Lie algebra g carries a natural Poisson bracket. Shift of argument subalgebras (introduced by Fomenko and Mishchenko in 1978) form a family of maximal Poisson-commutative subalgebras in S(g) for semisimple g. This family is parametrized by regular elements of the dual space g*. I will discuss the quantization problem for shift of argument subalgebras, namely, how to lift these subalgebras to commutative subalgebras in the universal enveloping algebra U(g), and how to describe the spectra of the "quantum shift of argument subalgebras" of U(g) on (finite-dimensional) g-modules. These questions are related to the classical representation theory, in particular, it was observed by Vinberg, that the Gelfand-Tsetlin subalgebra in U(gl_n) is a certain limit of quantum shift of argument subalgebras, and hence the spectra of quantum shift of argument subalgebras on a finite-dimensional g-module can be regarded as a deformation of the corresponding Gelfand-Tsetlin polytope.

Special Seminar
April 15
Time: 1:30 to 3
Venue: E206
Speaker: Ignacio Sols, University of Madrid
Title: Restriction theorems for stable tensors.
Abstract: Geometric objects such as stable pairs, conic bundles or principal bundles on a projective variety are particular cases of a vector bundle equipped with a tensor, and their notions of (semi)stability leading to moduli spaces of such objects are also particular cases of a properly defined notion of (semi)stability for a tensor. In the case of (semi) stable vector bundles, theorems of preservation of this notion under general restrictions -such as Metha Ramanathan's theorem- have been shown to be very usefull to actual descriptions of such moduli. However, there have been problems to bring them into the context of tensors. These we will discuss, and will show which restriction theorems can still be proved in this context.

April 16
Harry Tamvakis:   A Giambelli formula for isotropic Grassmannians.
Abstract: The Schubert calculus in the cohomology ring of the usual Grassmannian GL_n/P has been studied extensively for well over a century, but the corresponding questions for quotients of the symplectic or orthogonal group are rather unexplored. After giving an overview of the relevant history, I will discuss my work with Buch and Kresch on a Giambelli formula for (non maximal) isotropic Grassmannians. We also introduce theta polynomials, a family of combinatorially explicit polynomials whose algebra coincides with the Schubert calculus on these spaces.

April 30
Henri Darmon (McGill University):   Algebraic cycles and Euler Systems for CM elliptic curves.
Abstract: I will present an ongoing project (in collaboration with Massimo Bertolini and Kartik Prasanna) whose goal is to construct and study new Euler systems of rational points on CM elliptic curves. The main novelty stems from the fact that these points arise from algebraic cycles on certain higher-dimensional Shimura varieties.

May 7
Eva Viehmann (Bonn, visiting U of C) :   Equidimensionality of some affine Deligne-Lusztig varieties.
Abstract: Affine Deligne-Lusztig varieties are analogues in the affine Grassmannian of classical Deligne-Lusztig varieties. They are also related to the reduction modulo p of Shimura varieties. In this talk (based on joint work with Urs Hartl) I explain how the Newton stratification on deformations of local shtuka can be used to prove equidimensionality of these varieties in the basic case and to describe their closures in the affine Grassmannian.

May 14
James Borger:   Witt vectors, Lambda-rings, and absolute algebraic geometry.
Abstract: I will explain how the closely related concepts of Witt vector and Lambda-ring give rise to a kind of algebraic geometry that is, in a precise sense, over a deeper base than the ring of integers. I will show that it conforms to several predictions about absolute geometry. I will also say something brief about how it relates to other issues in arithmetic algebraic geometry, such as crystalline cohomology, Buium's theory of arithmetic jet spaces, and explicit class field theory.

May 21
George Pappas (MSU):   TBA.

May 28
Francis Brown :   TBA.