Seminar Schedule
January 14 -January 18, 2008
All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.
MONDAY, January 14, 2008
Algebra
2:30pm, Room 410
Speaker: William Graham, University of Georgia
Title of talk: Positivity in equivariant K-theory: conjectures and results
Abstract: Let G be a complex semisimple group and let $B \supset T$ be a Borel subgroup and a maximal torus, respectively. Let $X=G/P$, where $P \supset B$; this is a partial flag variety. The $T$-equivariant $K$-theory of $X$, denoted $K_T(X)$, is a free module over the representation ring $R(T)$. One important basis consists of the classes of structure sheaves of Schubert varieties; another important basis, which has been used by Kostant and Kumar, is the dual basis to this basis. (These bases are essentially different, in contrast to the situation in equivariant cohomology where the dual basis to a Schubert basis is a Schubert basis for the opposite Borel subgroup.) We conjecture that the structure constants for the multiplication in $K_T(X)$, in the dual basis to structure sheaves, satisfy a certain "positivity" property (more precisely, they can be expressed as certain products with known signs). This property is similar to the positivity conjectured by Griffeth and Ram for the structure constants obtained using the basis of structure sheaves. As evidence for our conjecture we give an explicit formula for these constants for $X$ equal to projective space, which implies positivity. We can also prove the conjecture in some other special cases.
VIGRE – Algebraic Geometry
2:15pm, Room 222
Topology
2:30pm, Room 303
No meeting this week
Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea, Cookies
TUESDAY, January 15, 2008
VIGRE - Graduate Student Seminar
2:00pm, Room 304
Speaker: Maxim Arap, University of Georgia
Title of talk: An Introduction to Toric Varieties
Abstract: After a brief historical and motivational remark for the theory of toric varieties there will be a brief review of basic definitions in algebraic geometry followed by examples. Subsequently, a definition of a toric variety will be given. The remaining part of the presentation will be devoted to the construction of (affine) toric varieties from certain combinatorial data. The core of the presentation will be based on some of the lectures that I attended at CRM in May of 2007. The presentation will be aimed at the audience with no background in algebraic geometry and many examples will be provided.
Mathematical Physics
3:30pm, Room 303
Speaker: Emily Pritchett, University of Georgia
Title of talk: Entropy in statistical mechanics and the 2nd law in thermodynamics
WEDNESDAY, January 16, 2008
Algebraic Geometry
2:30pm, Room 326
No meeting this week
Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea, Cookies
Number Theory/Arithmetic Geometry
3:30pm, Room 304
Speaker: TBA
Title of talk: TBA
THURSDAY, January 17, 2008
VIGRE – Tropical Geometry
2:00pm, Room 304
VIGRE – Circle Packing
2:00pm, Room 326
Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea, Cookies
Colloquium
3:30pm, Room 304
Speaker: Howie Weiss, Georgia Tech
Title of talk: Can You Hear the Shape of a Potential?
Abstract: Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction on a lattice determines a lattice spin system with associated potential. The pressure and free energy of the potential are fundamental characteristics of the system. However, even for the simplest systems, the information about the (microscopic) potential that the (macroscopic) free energy captures is subtle and poorly understood.
We study whether, or to what extent, potentials are determined by their free energy. In particular, we show that for a one- dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy.
We show (and exploit) that this rigidity problem has striking analogies to inverse problems in spectral geometry that Kac summarized ``Can you hear the shape of a drum?", and to inverse problems in algebraic number theory.
No physics background will be needed to understand the talk.
VIGRE – Number Theory
3:30pm, Room 303
FRIDAY, January 18, 2008
VIGRE-Algebra
2:30pm, Room 322
Applied Math
2:30pm, Room 302
Speaker: Ming-Jun Lai, University of Georgia
Title of talk: Tight Wavelet Frames over the Sphere
Abstract: I will use trigonometric B-splines and polynomial B-splines to construct tight wavelet frames over the sphere.
Geometry
2:30pm, Room 410
Speaker: Jesse Ratzkin, University of Georgia
Title of talk: The singular Yamabe problem
Abstract: In 1960 H. Yamabe claimed to prove the following result: given a compact Riemannian manifold (M,g), there is a conformally related metric g* = (e^u)g with constant scalar curvature. However, it took another 25 (or so) years and many very smart people to complete Yamabe's proof. One reason for these difficulties is that the conformal factor can blow up, yielding the singular Yamabe problem: given a compact Riemannian manifold (M,g) and a closed set C, find a conformally related metric g* = (e^u)g which is complete on M\C and has constant scalar curvature. I will discuss some results in the singular Yamabe problem, emphasizing the case of positive curvature. Time allowing, I will indicate some related things I've been thinking about.