University of Georgia
Department of Mathematics
Seminar Schedule
October 10– October 14, 2005
All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.

MONDAY, October 10, 2005

Topology/Geometry
2:30-4:30pm, Room 303
Speaker: Joe Fu, University of Georgia
Title of talk: Valuations on manifolds, after Alesker and Bernig-Brocker
Abstract: I'll discuss recent work of these authors generalizing the notion of a convex valuation (= a certain kind of functional on the space of convex subsets of a finite-dimensional real vector space) to that of a "smooth valuation" (= a certain kind of functional on certain special subsets of a smooth manifold). A smooth valuation can be represented (via a subtle but striking fact called the Casselman-Wallach theorem) in several different but natural ways--- for example, Bernig and Brocker have recently described a natural isomorphism between the space of smooth valuations and a certain quotient of the space of differential forms of the sphere bundle SM of M--- yielding different and natural operations on them. Perhaps the most important is the existence of a natural product, which most emphatically does not correspond to the wedge product of forms on SM.

Algebra
2:30-3:30pm., Room 410
Speaker: Jon Kujawa, University of Georgia
Title: Cohomology and support varieties made easy.
Abstract: This talk continues last week's algebra seminar. We continue our look at what invariant theory can do for us as we study cohomology for lie superalgebras. In particular, we will introduce support varieties for Lie superalgebras and calculate some examples. Again, invariant theory comes to the rescue. We will discuss a recent result which allows us to sidestep what would otherwise be a gruesome calculation.

Faculty and Graduate Social
3:00pm, Room 409
Coffee, Cookies, Tea

TUESDAY, October 11, 2005

VIGRE-Graduate Student Seminar
2:00p.m., Room 303
Speaker: Amod Agashe, Florida State University
Title of talk: Diophantine equations
Abstract: Consider a polynomial equation in two or more variables with coefficients that are rational numbers. Such equations are called Diophantine equations, and one can ask for a description of solutions to such equations whose coordinates are also rational numbers. This problem has been studied with interest for centuries. As an example, the rational solutions to the Diophantine equation x-squared + y-squared = 1 correspond to Pythagorean triples; for example, x=3/5, y=4/5 is a solution coming from the triple (3,4,5). In this talk we will study solutions to Diophantine equations in two variables, with emphasis on quadratic equations and certain cubic equations, where the local-to-global principle and the Birch and Swinnerton-Dyer conjecture provide a nice answer (respectively).

Number Theory
3:30-5:00pm., Room 304
Speaker: Amod Agashe, Florida State University
Title of talk: The Birch and Swinnerton-Dyer formula
Abstract: For an elliptic curve, the second part of the Birch and Swinnerton-Dyer conjecture relates the quotient of the special L-value (assumed non-zero) by the real period to certain arithmetic invariants of the elliptic curve, in particular the orders of the Shafarevich-Tate group, the torsion subgroup, and the component groups. We will give a formula that expresses this ratio as a rational number. A factor of this ratio can be related to the order of the Shafarevich-Tate group using the notion of visibility, and moreover, this formula indicates significant cancellation between the orders of the torsion subgroups and the component groups in the Birch and Swinnerton-Dyer formula. We will mention what is expected (based partly on calculations of Cremona and Stein) and what we can prove.

WEDNESDAY, October 12, 2005

Algebraic Geometry
2:30-3:45pm, Room 410
Speaker: Bill Graham, University of Georgia
Title of talk: Nonabelian localization in equivariant K-theory
Abstract: Let $G$ be a linear algebraic group acting on a smooth space $X$, and let $K_G(X)$ denote the equivariant K-theory of $X$. If $h \in G$, let $X^h$ denote the fixed point locus of $h$ in $X$. If $G$ is a diagonalizable group, the localization theorem states that the restriction map $K_G(X) \to K_G(X^h)$ is an isomorphism after a suitable localization. Much of the power of this theorem comes from the fact that there is an explicit inverse to this isomorphism, which can be used, for instance, to give easy proofs of character formulas such as Weyl's. Vezzosi and Vistoli generalized the fact about restriction maps to arbitrary $G$. This talk will discuss the construction of the explicit inverse. One application of this result is a version of the Kawasaki-Riemann-Roch theorem. This is joint work with Dan Edidin.

Faculty and Graduate Social
3:00pm, Room 409
Coffee, Cookies, Tea

VIGRE-Algebra Group
3:45p.m., Room 302

THURSDAY, October 13, 2005

VIGRE-Feynman Diagrams
2:00pm, Room 326

VIGRE – Cardiac Physiology
2:00p.m., Room 640

VIGRE Algebraic Geometry Group
3:30 pm in Room 323

FRIDAY, October 14, 2005

Probability Theory
2:30pm, Room 303
Speaker: Lirong Yu, University of Georgia
Title of talk: Term structure and defaultable bonds (cont)

Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea, Cookies

Colloquium
3:30pm, Room 304
Speaker: Yuesheng Xu, Dept. of Math., Syracuse University
Title of talk: Convergence Theorems for Multi-Parameter Regularization Methods for Solving Ill-Posed Operator Equations
Abstract: We consider in this talk solving ill-posed operator equations. Based on a decomposition for the solution space, we propose a multi-parameter regularization for solving the equations. We provide convergence theorems for the regularization methods to be convergent