University of Georgia
Department of Mathematics

Seminar Schedule
September 29 - October 3, 2003

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

MONDAY, September 29, 2003

Numerical Analysis
1:30p.m., Room 524
Speaker: Andrew Sornborger, University of Georgia
Title of talk: Estimating continuous functions from discretely sampled data. An inverse problem approach.

Geometry
1:30p.m., Room 410
Speaker: Amelia Reeves, University of Georgia (student)
Title of talk: Open and Closed Interlocked Linkages
Abstract: A linkage is a collection of polygonal chains in 3-space whose joints are free to move, but whose edges have fixed length. A linkage is locked if its components cannot be separated without cutting one of the edges. In this talk, a few linkages will be proved to be locked or unlocked. The proofs are inspired by papers by Erik Demaine et. al.

Topology
2:30p.m., Room 323
Speaker: Rafal Komendarczyk, Georgia Tech
Title of talk: Closed nodal line conjecture and overtwisted contact structures
Abstract: In the 3-dimensional Riemannian geometry, contact forms equipped with an adapted Riemannian metric, turn out to be divergence free eigenforms of the Laplace-Beltrami operator (or equivalently a curl operator). Probably the most useful consequence of this fact can be found in hydrodynamics, where for an arbitrary Riemannian manifold these objects happen to be "generic" solutions of steady Euler equations. In this talk, I will discuss another, possibly useful consequence of the metric adaptation. Namely, if for a given convex surface, in a contact manifold, a transverse contact field is orthogonal to the surface, dividing curves become nodal lines for eigenfunctions of the scalar Laplacian on the surface. Therefore information about dividing curves can become useful when studying nodal lines (and possibly vice versa but it hasn't been proven yet). As a possible application, I will discuss a version of Payne's conjecture for closed Riemannian surfaces, the conjecture posed in 1964 states that the second eigenfunction of Laplacian on an arbitrary domain in flat $R^2$ cannot posses a closed nodal line. So far it's been proven true for simply connected convex domains and false for non-simply connected domains, the situation for arbitrary non-convex domains in $R^2$ hasn't been resolved yet. I will also discuss, how this conjecture is related to the dichotomy tight-overtwisted, and how it impacts the variational principle for tightness proposed some time ago by J. Etnyre an R. Ghrist.

VIGRE - Algebra Seminar
2:30p.m., Room 410
Organizer: Daniel K. Nakano, University of Georgia
Activity: We will continue the calculations for the nilpotent varieties for fields of prime characteristic after
meeting for five minutes.

Faculty and Graduate Social
3:00p.m., Room 409
Coffee, Tea, Cookies

Lie Theory
3:30p.m., Room 303
Speaker: Brian Boe, University of Georgia
Title of talk: Representation type of the blocks of category ${\mathcal O}_S$.
Abstract: Continuation of Dan Nakano's talk of last week, with application to solving the problem of the title.

TUESDAY, September 30, 2003

VIGRE Graduate Student Seminar
2:00-3:15pm, Room 304
Speaker: Joe Fu, Univerisity of Georgia
Title of talk: From Buffon's needle to the symplectic sesquialgebra: the coming revolution in integral geometry
Abstract: Integral geometry is the study of how to recover geometric characteristics of a shape (maybe a curve, or a surface) by averaging procedures. For example, the classical formula of Poincare-Crofton states that the average over all possible relative positions of the number of intersections of two curves C_1 and C_2 in the unit two-sphere is 1/(2\pi^2) times the product of the lengths of C_1 and C_2. This formula is closely related (identical?) to the solution of the famous Buffon needle problem: given a plane ruled by infinitely many parallel lines 1 inch apart, what is the probability that a 1-inch long needle dropped at random on the plane will intersect one of the lines?

The solution to these problems may be viewed as a special case of an all-embracing theorem known as the Principal Kinematic Formula, due originally to Blaschke in the 1930s. Recent work of S. Alesker has shown that this formula possesses some striking extensions: the original Blaschke formula may be viewed as an instance, corresponding to the orthogonal group, of a more general theorem that applies also to the unitary and symplectic groups. (Alesker won the quadrennially awarded European Mathematical Society Prize for this work at the EMS Congress in Barcelona in July 2000.) Unfortunately much remains unknown in these latter two cases, but further work of Alesker shows that the problem possesses an amazing algebraic structure by means of which the complete picture may well be within reach.

Analysis
3:30p.m., Room 326
Speaker: TBA
Title of talk: TBA

WEDNESDAY, October 1, 2003

Group Representation & Cohomology
2:30p.m., Room 410
No Meeting due to colloquium

Algebraic Geometry
2:30pm, Room 303
Speaker: Fedor Bogomolov (Courant Institute, New York University)
Title of talk: Unramified correspondences
I want to discuss some results related to the following conjecture(or better say extremal hypothesis):

C-H : Let $C,C'$ be two curve of genus $ > 1$ defined over $\bar Q$ (algebraic numbers) Then there exists a nonramified covering of $C$ which surjects onto $C'$.

The main result I am going to talk about is the following theorem (jointly with Yuri Tschinkel):

Theorem For any hyperelliptic curve $X$ there is an unramified covering of degree 72 which has a surjective map of degree 4 on a curve of genus 2 given by equation $y^6 =x(x-1)$.

The above covering of $X$ is obtained as a sequence of abelian coverings with groups $Z_2 , Z_3 +Z_3,Z_2,Z_2$ respectively. The construction is elementary.

I will also discuss some of its implications (some form of effective Mordell estimates for hyperelliptic curves) and potential
generalizations.

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

Colloquium
3:30p.m., Room 303
Speaker: Srikanth Iyengar, University of Missouri
Titleof talk: Aspects of the interaction between commutative algebra and algebraic topology.
Abstract: Techniques developed in algebraic topology have proved efficacious in solving problems arising in classical commutative ring theory. In turn, many concepts and tools developed in commutative algebra have turned out to be of relevance and utility in algebraic topology. I would like to illustrate the interaction between these subjects by focusing on two areas of research that I have been involved in: homological criteria for detecting ring theoretic properties, and duality phenomenon in algebra and topology.

Number Theory/Arithmetic Geometry
3:30pm, Room 304
No Meeting due to colloquium

THURSDAY, October 2, 2003

VIGRE Quantum Mechanics Seminar
2:00p.m., Room 303
Speaker: Robert Varley, University of Georgia
Title of talk: The dynamics of a quantum particle on the line

VIGRE - Contact Topology
9:00a.m., Room 326

Student Number Theory
3:30p.m., Room 304
No Meeting this week

FRIDAY, October 3, 2003

CATS
1:25p.m., Room 306
Speaker: Michael Geller, Computer Science, University of Georgia
Title of talk: Quantum Computing with Superconductors, part III
Abstract: Superconducting qubit storage and entanglement. In the final lecture I will describe an architecture we have proposed to couple qubits together to make a viable quantum computer. The architecture makes use of high-frequency mechanical resonators as "bus" qubits to store and transfer the quantum state or wave function of a Josephson junction, and to carry out quantum logic operations.

Wavelet Analysis
2:30p.m., Room 524
Speaker: Alex Petukhov, University of Georgia
Title of talk: Wavelet Frames and Their Applications