Tuesday, January 21
Speaker: Philip Foth, Department of Mathematics, University of Arizona
Title: Ag
Abstract: I will review basic facts about the moduli space Ag of principally polarized abelian varieties of dimension g. In particular, I will identify it with a quotient of Siegel upper-half space, put a Kaehler structure on it, review theta functions, outline proofs of Torelli theorem on the imbedding of Mg and Satake theorem on quasiprojectivity of Ag.

Tuesday, January 28
Speaker: Minhyong Kim, Department of Mathematics, University of Arizona
Title: Something Vague About Mg

Tuesday, February 4
Speaker: Minhyong Kim, Department of Mathematics, University of Arizona
Title: Hilbert Schemes and Construction of Mg

Tuesday, February 11
Speaker: Kirti Joshi, Department of Mathematics, University of Arizona
Title:  Subvarieties of Ag
Abstract: I will discuss certain natural subvarieties in the moduli space of principally polarized abelian varieties Ag.

Tuesday, February 18
Speaker: Nicholas Ercolani, Department of Mathematics, University of Arizona
Title: Geometric Applications of Random Matrix Partition Function Asymptotics
Abstract: Recently there have been some exciting applications of partition function asymptotics to study the asymptotic statistical behavior of certain extremal problems that arise in such diverse areas as random matrix theory, algebraic combinatorics and random growth models. All of these applications exploit the detailed knowledge of the leading order asymptotics of certain random matrix partition functions that have become available through new techniques of Reimann-Hilbert analysis. In work with Ken McLaughlin, we have established the full asymptotic expansion for the random matrix partition function for deformed unitary ensembles in a neighborhood of the Gaussian Unitary Ensemble (GUE). This talk will provide some background for these results but will principally focus on applications and potential applications to graphical enumeration on Riemann surfaces and the enumerative geometry of their moduli spaces.

Tuesday, February 25
Speaker: Kirti Joshi, Department of Mathematics, University of Arizona
Title: Subvarieties of Ag II
Abstract: This will continue my 2/11 talk about some natural subvarieties of the moduli space of principally polarized abelian varieties.

Tuesday, March 4
Speaker: Anatoly N. Kochubei of National Academy of Sciences of Ukraine
Title: Differential equations in positive characteristic
Abstract: Let K be the field of formal Laurent series with coefficients in F_q. We study equations with Carlitz derivatives for F_q-linear functions on K, which are natural function field counterparts of linear ordinary differential equations. It is shown that, in contrast to both classical and p-adic cases, formal power series solutions have positive radii of convergence near a singular point of an equation. We also consider equations with a regular singularity. In particular, an analog of the equation for the power function, the Fuchs and Euler type equations, and Thakur's hypergeometric equation are investigated. Some properties of the above equations are similar to the classical case while others are different. For example, a simple model equation shows a possibility of existence of a non-trivial continuous locally analytic F_q-linear solution which vanishes on an open neighbourhood of the initial point.

Tuesday, March 11
Speaker: Susan Tolman of University of Illinois at Urbana-Champaign
Title: Symplectic quotients and loop groups
Abstract: There is a natural notion of quotient for group actions on symplectic manifolds, known as the "symplectic quotient". Many interesting spaces arise as the symplectic quotients by compact Lie groups, and we have many techniques to compute their invariants. On the other hand, a large class of other spaces only arise as symplectic quotients by infinite dimensional groups, such as loop groups. We will discuss work with J. Weitsman and R. Bott to extend the finite dimensional results to this case.

Tuesday, March 18
Spring break, no seminar

Tuesday, March 25
Speaker: Paul Bressler, Department of Mathematics, University of Arizona
Title: Singularities of theta-divisors
Abstract: I will describe a certain invariant of Morse-theoretic nature associated to a subvariety of a complex manifold and calculalte it for the theta-divisor of a compact Riemann surface. (Joint work with Jean-Luc Brylinski)

Tuedsay, April 1
Speaker: Paul Bressler, Department of Mathematics, University of Arizona
Title: Levels and modularity
Abstract: For a discrete group G a level can be specified as a 3-cocycle on G with values in the multiplicative group. I will give my version of a level and associate to it a vector space with a linear action of SL(2,Z). Time permitting, I will also associate to a level a ``representation theory'' such that the ``dimension of a representation'' naturally takes values in the aformentioned vector space. The talk will be of elementary nature and suitable for a wide audience.

Tuesday, April 8
Speaker: Douglas Ulmer, Department of Mathematics, University of Arizona
Title: A counterexample to the Harris-Morrison slope conjecture.
Abstract: In 1990, Joe Harris and Ian Morrison made a conjecture about the "slope" of the moduli space Mg.  This number in some sense measures how effective divisors may meet the boundary of Mg and it is related to the possibility of writing down "the general curve of genus g."  This conjecture was recently disproved by  Farkas and Popa and the offending divisor turns out to be one defined by me and Fernando Cukierman in connection with a completely different problem related to K3 surfaces.  In the talk I will review some basics about divisors on Mg, describe the geometric content of the conjecture, and sketch the counterexample.


Tuesday, April 15
Speaker: Hermann Flaschka, Department of Mathematics, University of Arizona
Title: Random Ramanujan graphs.
Abstract: Ramanujan graphs are graphs with a certain extremal property. They are fairly typical among random graphs, but hard to construct explicitly. In next week's Number Theory Seminar, Minhyong Kim will explain the number-theoretic ideas needed for the construction of Ramanujan graphs. My talk will present basic background and results of computer experiments on a simple model. There is (almost) no geometry, and the ratio of theorems to pictures is about .01  .


Tuesday, April 22
Speaker:  Arlo Caine, Department of Mathematics, University of Arizona
Title: Penrose Tilings and Non-commutative Geometry.
Abstract: Penrose Tilings are a certain class of aperiodic tilings of the plane by isoceles triangles. This space of all such tilings admits a parametrization as space of sequences satisfying a grammar rule, modulo a countable equivalence relation. This quotient is a horrific space, it is not even Hausdorff! Connes and others have use methods of non-commutative geometry to analyze this space and explain certain remarkable features of Penrose Tilings. In this talk I will introduce Penrose Tilings and their basic properties. I will then derive the parameterization and introduce the non-commutative point of view.


Tuedsay, April 29
Speaker: Aissa Wade, Pennsylvania State University
Title: Locally conformal Dirac bundles.
Abstract: The notion of a locally conformal Dirac bundle provides a unifying framework for the study of both Poisson and locally conformal symplectic manifolds. In this talk, we will discuss some basic properties of locally conformal Dirac bundles. Every locally conformal Dirac bundle gives rise to two fundamental Lie algebras: the Lie algebra of admissible functions and that of infinitesimal automorphisms. We show that, under certain assumptions, there is a Lie homomorphism between these two Lie algebras.


Tuesday, May 6
Speaker:  Mikhail Kogan, Northeastern University
Title: Schubert varieties and Gelfand-Cetlin polytopes.
Abstract: Schubert polynomials Sw represent cohomology classes of Schubert varieties Xw in the flag manifold. Billey-Jockusch-Stanley and Fomin-Kirillov expressed Schubert polynomials as positive sums of monomials indexed by combinatorial objects called rc-graphs. We discuss the geometry underlying these monomial expressions. In particular, we construct a flat degeneration of the flag manifold to the toric variety Y associated to the Gelfand-Cetlin polytope. Every Schubert variety Xw degenerates to a reduced union of toric subvarieties of Y. The components of the degeneration of Xw correspond to certain faces of the Gelfand-Cetlin polytope. Moreover, these components are in bijection with rc-graphs which enter the monomial expression for Sw. This is joint work with Ezra Miller.