Geometry Seminar @ University of Georgia Fall 2004 - Spring 2005

Fridays, 2:30-3:30 pm, Boyd GSRC 326

September 2004	October 2004	November 2004	December 2004
January 2005	February 2005	March 2005	April 2005

January 2005

Date: Jan. 14, 2005
Organizational Meeting

Date: Jan. 21, 2005
Speaker: Jason Cantarella, U. of Georgia
Title: How hard is it to find the self-contacts of a tube or necklace?
Abstract: Suppose you are given a necklace of N tangent unit-radius spheres in 3-space, and you want to figure out whether or not any of these two spheres overlap. One obvious algorithm is to calculate the distances between each pair of sphere centers. Since there are N spheres, there are O(N^2) pairs, and this algorithm takes time O(N^2). Various computational-geometric methods have been proposed to accomplish this task faster-- for instance in (expected) time O(N log N) (but worst-case time O(N^2)). In this talk, we'll prove that for "well-behaved" necklaces, a class of partitioning methods is guaranteed to solve the problem in time O(N^{4/3}). (This is an expository talk.)
Date: Jan. 28, 2005 -- moved to room 222
Speaker: Sa'ar Hersonsky, Ben Gurion University
Title: Maximal cusps on boundaries of deformations of hyperbolic 3-manifolds
Abstract: Let $M$ be a compact 3-manifold with $\partial M=\neq\emptyset$. The space of allhyperbolic structures on the interior of $M$ (i.e. Riemannian metrics withcurvature equals -1), is well understood. It's boundary is a topic of much research. We show that a certain type of hyperbolic manifolds (which are easy to understand) are dense on the boundary. This is a joint work with Richard D. Canary (Ann-Arbor).

February 2005

Date: Feb. 4, 2005 Speaker: Jason Cantarella, U. of Georgia
Title: How hard is it to find the self-contacts of a tube or necklace, part II
Date: Feb. 11, 2005
Speaker: Malcolm Adams, U. of Georgia
Title: The Replicator Equations of Evolutionary Game Theory
Abstract: After a brief introduction to a nonlinear system of ordinary differential equations known as the replicator equations of evolutionary game theory, I will describe the solutions for a large class of such systems.
Date: Feb. 18, 2005
Speaker: Tomasso Pacini, Georgia Tech
Title: Area-minimizing cones in higher codimension and special Lagrangian submanifolds.
Abstract: In 1969, Bombieri-De Giorgi-Giusti discovered the first non-trivial examples of codim.-1 area-minimizing cones in R^n. In 1985, Hardt-Simon showed that any such cone admits smooth, area-minimizing "desingularizations": ie, complete hypersurfaces which are asymptotic to the cone at infinity.
In higher codimension, many examples of area-minimizing cones are known, but there are no general results concerning their desingularization. The goal of the seminar is to present "special Lagrangian" (SL) cones as a possibly fertile subclass in which to investigate this issue. In particular we show that, for SL cones in R^6, certain "soft" geometric obstructions to the desingularization problem vanish.
Date: Feb. 25, 2005
Speaker: Adam Fletcher, U. of Georgia
Title: Mathematicians Have Curves in All the Right Spaces
Abstract: It's back to the "basics" of differential and Riemannian geometry with a discussion of the ideas of curvature in the plane, in space, and on surfaces and manifolds. The talk will be aimed at first year graduate students and advanced undergraduates with some points of interest and refreshment for the seasoned geometer.

March 2005

Date: March 4, 2005
Speaker: Thomas Kephart, Vanderbilt University
Title: Applications of tight knots to physics
Abstract: We argue that systems of tightly knotted, linked, or braided quantized flux tubes will have a universal mass-energy spectrum, since the length of fixed radius flux tubes depend only on the topology of the configuration. We motivate the discussion with plasma physics examples, then concentrate on a particle physics model of glueballs as knotted QCD flux tubes. Other applications will also be discussed.
Date: March 11, 2005
No meeting due to the Southeast Geometry Conference, held March 11-12 in Columbia, SC.
Date: March 18, 2005
No meeting due to spring break.
Date: March 25, 2005
Speaker: Semyon Alesker, Tel Aviv University
Title: Valuations on convex sets, pluripotential theory, and non-commutative determinants

April 2005

Date: April 1, 2005
Speaker: Ted Ashton, U. of Georgia
Title: Passage to Pentagonia''
Abstract: All aboard for a sightseeing tour with very brief stops in Combinatorics, Matrix Algebra, Graph Theory, Complex Numbers, Geometry and Trigonometry.
Date: April 8, 2005
Speaker: Jason Parsley, U. of Georgia
Title: Calculating the writhe of a knot
Abstract: The writhe of a knot measures how much the curve wraps and coils around itself. We discuss Calugareanu's formula "Link = Twist + Writhe" with a demonstration. The majority of the talk will follow a 1995 paper by Aldinger, Klapper, and Tabor which calculates and estimates writhe through a variety of approaches. Most of this talk should be well-suited for graduate students.
Date: April 15, 2005
Speaker: Chad Mullikin, U. of Georgia
Title: What I know about Distortion
Abstract: I will discuss some recent results about Gromov's distortion problem for knots including some of my results regarding distortion criticality.
Date: April 22, 2005
Speaker: Ted Shifrin, U. of Georgia
Title: A surface with no twisting curves must be R^2 or S^2
Abstract: This is an informal talk on a theorem of Scherrer from 1940 that says that if all the closed curves on a surface have total twist zero, the surface must be a plane or a sphere. The proof I've figured out uses the first variation of the integral and moving frames and the structure equations. This should be accessible to grad students.
Date: April 29, 2005
There is no meeting of the seminar this week.

September 2004

Date: Sept. 3, 2004
Speaker: Jason Parsley, U. of Georgia
Title: The Biot-Savart operator and linking integrals on S³, part I
Date: Sept. 10, 2004
Speaker: Jason Parsley, U. of Georgia
Title: The Biot-Savart operator and linking integrals on S³, part II
Date: Sept. 17, 2004 -- Postponed due to Hurricane Ivan
Date: Sept. 24, 2004
Speaker: Jason Parsley, U. of Georgia
Title: The Biot-Savart operator and linking integrals on S³, part III
Date: Wednesday, Sept. 29, 2004 -- special meeting, Room: Boyd 303
Speaker: John Pelesko, U. of Delaware
Title: MEMS, Random Packing, and Self Assembly
* note the special day and room of this talk

Date: Oct. 1, 2004
Speaker: John McCuan, Georgia Tech/U. of Georgia
Title: Geometric expansion of curves and surfaces
Date: Oct. 8, 2004
Speaker: John McCuan, Georgia Tech/U. of Georgia
Title: Geometric expansion of curves and surfaces, part II
Date: Oct. 15, 2004
Speaker: Ted Shifrin, U. of Georgia
Title: Surfaces of Revolution: From Archimedes to Symplectic Geometry.
Abstract: This talk will be elementary and aimed at graduate students. Did you know that if we slice a sphere with parallel planes, the area of the resulting "zone" depends only on the distance between the planes? A modern interpretation of this leads to a characterization of the sphere and a beautiful way of interpreting the curvature of a surface of revolution, among other things.
In a follow-up talk next week, Malcolm Adams will show how these concepts fit more generally in the domain of symplectic geometry, which connects mathematics and physics in a powerful and elegant way.
Date: Oct. 22, 2004
Speaker: Malcolm Adams, U. of Georgia
Title: Completely integrable hamiltonian systems, action - angle variables, and the spherical pendulum
Date: Oct. 29, 2004
No meeting -- fall break

November 2004

Date: Nov. 5, 2004
Speaker: Jingzhi Tie, U. of Georgia
Title: Isoperimetric problem on the Heisenberg Group
Date: Nov. 12, 2004
Speaker: Jason Cantarella, U. of Georgia
Title: Symmetries and Integrals of Motion in Optimal Control.
Abstract: Noether's theorem, relating the symmetries of a dynamical system to conserved quantities or "integrals of motion" for the system is one of the cornerstones of classical mechanics. In this talk, I'll discuss some results of Hector Sussmann on the application of these ideas to control theory-- in particular to the problem of finding shortest paths of bounded curvature. The results should be of particular interest to those of us studying kinks in ropelength-minimizing knots.
Date: Nov. 19, 2004 Speaker: Jason Cantarella, U. of Georgia
Title: Symmetries and Integrals of Motion in Optimal Control, part II
Date: Nov. 26, 2004
No meeting -- Thanksgiving weekend

December 2004

Date: Dec. 3, 2004
Speaker: John McCuan
Title: More on symmetric surfaces
Abstract: I will talk about obtaining parameterizations/generalized meridian curves for immersions whose images are invariant under a group of symmetries. I hope also to give a fuller account of our classification result of rotationally symmetric surfaces in the three-sphere (from the colloquium of Nov. 4).
Date: Dec. 10, 2004, Time: 12:30 pm
Speaker: Alex Kasman, College of Charleston
Title: KP Generators and the Geometry of the Hirota Bilinear Difference Equation
Abstract: The KP hierarchy is an infinite collection of nonlinear PDEs with applications in mathematical physics. Sato's theorem associates solutions of the KP hierarchy to flows on a Grassmann manifold. To make that amazing theorem clear to non-experts, I will first show an analogous (but easily understood) example of a linear ODE and its solution from a flow on the xy-plane. In each case the flow is generated by a certain linear operator, but we can ask ``What other generators could have been used to generate solutions in the same way?'' Although the answer is well known in the ODE case, the question in the nonlinear case is the main result my new paper with Michael Gekhtman. I will state our result and discuss its significance, especially focusing on its relationship to the "rank one conditions" that often appear in the context of integrable systems. The proof will also be sketched and involves a geometric interpretation of the Hirota Bilinear Difference Equation (HBDE) as a statement about linear maps between Grassmann cones.
* note the different starting time

This schedule is subject to change. If you have questions or are interested in giving a talk, please email the organizer.