Seminar Schedule
January 21 -January 25, 2008
All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.
MONDAY, January 21, 2008
University Holiday
TUESDAY, January 22, 2008
VIGRE - Graduate Student Seminar
2:00pm, Room 304
Speaker: Ed Azoff, University of Georgia
Title of talk: Invariant Subspaces
Abstract: Let T be a linear transformation acting on a complex vector space. A linear subspace S of the underlying vector space is invariant under T if T(S) is contained in S. The more invariant subspaces of T one can find, the more one learns about the structure of T. The best behaved linear transformations from this point of view are called reflexive. We will examine reflexive (collections of) linear transformations in both finite and infinite-dimensional settings. Applications of analytic function theory will be highlighted, and we will discuss progress on the invariant subspace problem -- whether every bounded Hilbert space operator leaves some non-trivial subspace invariant. Most of the talk should be accessible to those who have taken undergraduate courses in linear algebra and complex variables.
Mathematical Physics
3:30pm, Room 303
Speaker: Justin Manning, University of Georgia
Title of talk: The Poincare recurrence theorem
WEDNESDAY, January 23, 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 24, 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: Daniel Krashen, University of Pennsylvania
Title of talk: The u-invariant of fields
Abstract: The u-invariant of a field is defined to be the maximal
dimension of a quadratic form which has no nontrivial zeros. Although there are some expectations for what u-invariants should be of most "naturally occuring" fields, these invariants are unknown in a great number of situations. For example, if $F$ is a nonreal number field, it is known that $u(F) = 4$, and it is expected that the u-invariant of the rational function field $F(t)$ should be $8$. At this point, however, there is no known bound for $u(F(t))$ (and no proof it is even finite).
Important progress on this type of problem was obtained by Parimala and Suresh late last year, who showed that the u-invariant of a rational function field $F(t)$ is $8$ when $F$ is $p$-adic ($p$ odd). In this talk I will describe joint work with David Harbater and Julia Hartmann in which we give an independent proof and a generalization of this result using the method of "field patching."
FRIDAY, January 25, 2008
VIGRE-Algebra
2:30pm, Room 322
Applied Math
2:45pm--3:45pm, Room 302
Speaker: Tin Kong, University of Georgia
Title of talk: Trading Mean-Reverting Asset: Buying, Selling and Shorting
Abstract: This is an oral examination for Tin Kong.
Geometry
2:30pm, Room 410
Speaker: Joe Fu, University of Georgia
Title of talk: Hermitiian integral geometry
Abstract: This is the final version of joint work with A. Bernig on the integral geometry of C^n under the action of the affine unitary group. I'll describe several canonical bases for the space of unitary-invariant convex valuations \mu on C^n, and characterize in these terms the cones of nonnegative (\mu(K) >= 0 for all convex K) and monotone (K \subset L => \mu(K) \le \mu(L)) valuations. Then I'll state explicitly the principal kinematic formula in this setting, and show how it leads to a simple calculus for a variety of problems in the geometric probability of the unitary group. To illustrate, I'll compute explicitly the expected length of the curve of intersection of two real submanifolds of dimensions 4 and 5 placed in random positions in CP4. A transform of this last formula also gives an expression for the expected 7-dimensional volume of the Minkowski sum of a 3-disk and a 4-disk placed in random (unitary) positions in C4.
Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea, Cookies
Colloquium
3:30pm, Room 304
Speaker: Angela Gibney, University of Pennsylvania
Title of talk: Fulton's conjecture and upper and lower bounds for Mori cones
of algebraic varieties
Abstract: The Mori cone is a fundamental, often illusive, invariant of an algebraic variety and is the central object of study in higher dimensional algebraic geometry. In this talk I will explain Fulton's conjecture, which predicts a very simple description of the Mori cone of the moduli space of curves. I'll show how one can naturally obtain upper and lower bounds for the Mori cone of a large class of varieties. In the case of the moduli space of curves, the upper bound is the cone described by Fulton's conjecture. In particular, this gives a new possibility for the Mori cone and a new perspective on Fulton's conjecture.