University of Georgia
Department of Mathematics

Seminar Schedule
February 5 – February 9, 2007

All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.

MONDAY, February 5, 2007

Algebra
2:30pm, Room 410
Speaker: Bill Graham, University of Georgia
Title of talk: Positivity in Schubert calculus
Abstract: Positivity is known for the multiplication in several rings associated to the flag variety: cohomology, K-theory, and equivariant cohomology. It is conjectured to hold in equivariant K-theory, which would generalize the other cases. This talk will discuss positivity in several settings, including the equivariant K-theory of projective space (which is joint work with Shrawan Kumar).

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

Topology
2:30pm, Room 304
Speaker: Sa'ar Hersonsky, University of Georgia
Title of talk: Discrete Uniformization of triangulated planar domains
Abstract: The famous Finite Riemann Mapping Theorem (FRMT) that was proved independently by Cannon-Floyd-Parry and Schramm (mid 90's) asserts that given a triangulated planar topological quad. there exists a realization of it by a straight rectangle which is tilled by squares. Each square correspond to a vertex of the given triangulation. While (in general) degeneracies will occur the combinatorics of the triangulation is roughly preserved in the tilled rectangle. We will present the following

Theorem (Her 2006). Let W be a bounded planar triangulated domain whose boundary consist of a finite number of curves. Let E1 and E2 be disjoint and each a finite union of closed arcs or closed curves contained in the boundary of W. Then there exists a unique (up to scaling) geometric realization of { W, E1, E2}.

TUESDAY, February 6, 2007

VIGRE-Graduate Student Seminar
2:00pm, Room 304
No meeting this week

WEDNESDAY, February 7, 2007

Algebraic Geometry
2:30pm, Room 410
Speaker: Brendan Hassett, Rice University
Title of talk: Towards a canonical model for the moduli space of curves
Abstract. Consider the moduli space of pointed stable curves as a log-variety, with boundary \delta corresponding to the nodal curves. We seek to describe its log canonical model with respect to K + A\delta. When A=1, we recover the moduli space of stable curves; for A=0, this would be the canonical model of the moduli space, which is expected to exist for g>>0 after work of Eisenbud-Harris-Mumford and Farkas. For some intermediate values of A, the log canonical model can be constructed with Geometric Invariant Theory and other techniques. Examples of spaces that arise include D. Schubert's moduli spaces of pseudostable curves (with nodes and cusps), parameter spaces for bicanonical curves (allowing tacnodes as well), and moduli spaces of weighted pointed stable curves of genus zero. (work with D. Hyeon and M. Simpson)

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

Number Theory/Arithmetic Geometry
3:30pm, Room 304
Speaker: Pete Clark, University of Georgia
Title of talk: Abelian points on algebraic varieties
Abstract: Given a variety V defined over K and an extension field L of K, it is of course interesting to ask whether V has points rational over L. In fact the "converse" is also interesting: given a field extension L/K, one can ask what classes of K-varieties necessarily have L-rational points. The case in which K is a local or global field (e.g., Q) and L is the maximal abelian extension of K was the topic of an early paper of Gerhard Frey: he showed that there exist varieties -- specifically, torsors under abelian varieties -- defined over K without L-rational points. Frey's result is treated in the seminal "Field Arithmetic" text, but seems not to have been continued: in particular the result appears there in "ineffective" form: they do not exhibit a particular curve.

In this talk I shall present many further results on this problem and even more open questions. A sample result: for every odd d > 1, there exists a geometrically integral plane curve C over Q without abelian points. Intriguingly, the most natural examples seem to live in Kodaira dimension zero, and I can get many (but not all!) conceivable examples with non-negative Kodaira dimension by pulling-back from these. Over Q I have not been able to "cross the Calabi-Yau line," and indeed that this is impossible is suggested by a famous conjecture of Artin and partially supported by important work of (e.g.) Lang, Birch and Kanevsky. However, over a "more transcendental" field -- Q((t)) -- I can construct geometrically rational surfaces and Fano 4-folds without abelian points.

Note: The material may be of some interest to algebraic geometers.

VIGRE – Quantum Mechanics
4:00pm, Room 302

THURSDAY, February 8, 2007

VIGRE – ODE
2:00pm, Room 326

VIGRE – Moduli spaces
2:00pm, Room 304

VIGRE – Geometry
2:00pm, Room 410

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

Colloquium
3:30pm, Room 304
Speaker: Brendan Hassett, Rice University
Title of talk: Approximation results for varieties of low degree
Abstract: In the 1930's, C.C. Tsen showed that a homogeneous polynomial over the function field of a complex projective curve has a nontrivial solution provided the degree of the polynomial is less than than the number of variables. In 2001 Graber, Harris, and Starr generalized this result by proving that every rationally connected variety over the function field of a curve has a rational point. We can recast this in geometric terms: If f:X--->B is surjective map from a smooth projective variety to a curve with rationally connected fibers, then f admits a section. Once we know that a section exists, we can ask approximation questions about the sections: Can we find a section through a prescribed set of points? With prescribed Taylor series at those points? Our results depend on the singularities occuring in the fibers of f. (joint with Y. Tschinkel)

FRIDAY, February 9, 2007

Applied Math Seminar
(Pizza at 12:10pm)
12:20pm-1:10pm, Room 326
Speaker: Caner Kazanci, University of Georgia
Title of talk: Mathematical Biology and Ecological Systems
Abstract: In this talk, I present some of the mathematical problems arising in biological sciences in general. I will then focus on ecological applications, and introduce Ecological Network Analysis, a graph theoretical approach to study energy or material flow in ecosystems. I will discuss a new stochastic method that we developed, called Particle Tracking Algorithm. Similar to a microscope, this method enables us to observe each and individual energy (or mass) packets flow in the network; and we can simultaneously label and track all of their movements. Unlike agent or individual based models (ABM, IBM) where particles move according to a pre-defined algorithm, particle flow occurs ‘naturally’ (compatible with master equation) without human intervention. This brings the possibility of investigating all predefined ecological network properties, such as cycling index, residence time, dominance of indirect effects, and evaluate how well these algebraic definitions reflect their meaning.

Geometry
2:30pm, Room 410
Speaker: TBA
Talk title: TBA

VIGRE–Algebra
3:30pm, Room 304

VIGRE - Hodge Theoretic questions in Algebraic Geometry
3:30pm, Room 303