Annual Southeast Geometry Conference
University of Georgia
Athens, Georgia

April 19--21, 2002


 

The SEGC is an annual gathering of geometers of all stripes from around the Southeast and elsewhere. All geometers are cordially invited.  If you would like to present your work at the conference,  please contact Joe Fu at fu@math.uga.edu.

Financial Support

Some financial support is available for graduate students and other unsupported researchers.
For more information please contact Jason Cantarella at cantarella@math.uga.edu.

Schedule of talks
All talks will be held in the Boyd Graduate Studies Research Center,  Room 328

Friday, April 19, 2002

Registration
1:30p.m., Room 409

2:00-2:45
Speaker: John McCuan, Georgia Tech
Title of talk: Symmetry for a coupled system of elliptic PDE
 Abstract:  I will discuss a system of coupled elliptic equations inspired by a problem involving an electrostatically deflected soap film.  Included will be aspects of the variational derivation, common approximate equations used in applications, and questions of symmetry.  The symmetry discussion involves a delicate argument
involving the Serrin-G-N-N corner lemma in dimensions 2 and 3.


3:00-3:45
Speaker: Haydee Herrera, Tufts University
Title of talk: TBA
Abstract: TBA

3:45 - Coffee, Juice, Cookies, Room 409

4:15-5:00
Speaker: Tom Ivey, College of Charleston
Title of talk: "Elastic rods and isoperiodic deformations of NLS potentials"
Abstract: Solutions of the vortex filament flow (VFF) for space curves correspond to solutions of the nonlinear Schrodinger equation (NLS) under the Hasimoto/Sym transformation.  In particular, stationary curves for variational problems involving VFF-conserved Lagrangians, including Kirchoff elastic rods, correspond to finite-gap NLS solutions, at fixed time. Moreover, one-parameter  families of closed elastic rods, explicitly obtained by Ivey and Singer, correspond to the isoperiodic deformation of periodic NLS potentials developed by Grinevich and Schmidt.

The relationship between geometrical/topological properties of closed space curves and the spectrum of periodic potentials is the focus of ongoing joint work with Annalisa Calini (College of Charleston). While it is difficult, in general, to distinguish the spectra of closed curves, we have developed a way of harnessing the isoperiodic deformations to produce closed curves of arbitrary complexity.  Using the deformation family as organizing principle,  we also have a conjectural picture of how geometrical/topological  properties of closed elastic rods correspond to special properties of the spectrum.


5:15-6:00
Speaker: Weiqing Gu, Harvey Mudd College
Title of talk: Examples of Cayley Manifolds
Abstract:  We present several families of so-called Cayley 4-dimensional manifolds in the real Euclidean 8-space. Such manifolds are of interest because Cayley 4-manifolds and Cayley 4-cycles in Calabi-Yau 4-folds and Spin(7) holonomy manifolds are supersymmetric cycles that are candidates for representations of fundamental particles in String Theory. Moreover, some of the examples of Cayley manifolds discovered in this paper may be modified to construct explicit examples in our current search for new holomorphic invariants for Calabi-Yau 4-folds and for the further development of mirror symmetry. We apply the classic results of Harvey and Lawson to find Cayley manifolds which are graphs of functions from the set of quaternions to itself and which are invariant under certain three dimensional subgroups of Spin(7).


Saturday, April 20, 2002

Coffee, Juice, Bagels
9:00, Room 409

9:30-10:15
Speaker: John Sullivan, Univ. of Illinois
Title of talk: Simulations of Tight Links
Abstract: We consider certain smooth knot energies which approximate ropelength, and appropriate discretizations of these for polygonal links. We describe numerical simulations with Brakke's Evolver, looking for
tight configurations of small knots and links.


10:30-11:15
Speaker: Elizabeth Denne, Univ. of Illinois
Title of talk: Quadrisecants of Knots
Abstract: A _quadrisecant _is a straight line that intersects a knot in four distinct points. It is interesting to compare the linear ordering of intersection points along the quadrisecant with the cyclic ordering of the intersection points along the knot. There are two such orderings, called _NSNS_ and _NNSS._ NSNS quadrisecants have implications for other geometric properties such as the curvature of the knot. In this talk, I will outline a proof that any non-trivial tame knot has a NSNS quadrisecant.

11:30-12:15
Speaker: Yuanan Diao, Univ. of North Carolina, Charlotte
Title of talk: The Linear Growth In The Lengths Of A Family Of Thick Knots, (Joint work with Claus Ernst and Morwen Thistlethwaite)
Abstract: For any given knot $K$, a thick realization $K_0$ of $K$ is a knot of unit thickness which is of the same knot type with $K$. In this talk, we show that there exist a family of prime knots $\{{ K}_n\}$ with the property that $Cr({K}_n)\to \infty$ (as $n\to \infty$) such that the arc-length of any thick realization of ${K}_n$ will
grow at least linearly with respect to $Cr({K}_n)$.

Afternoon Session
2:00-2:45
Speaker: Conrad Plaut, Univ. of Tennesee.
Title of talk: Geometry of the groups L^p([0,1],Z)
Abstract: The group G^p of integer-valued functions in L^p is a metric group  with the metric induced by the ususal L^p metric. On the one hand, this group  is a kind of generalized lattice, with a covering radius and "deep holes." On
the other hand, the group is globally contractible through homotheties, and  has many self-similarities. G^1 is an inner metric space, whose non-trivial  geodesics are nowhere differentiable as curves in L^1. For p>1 all non-trivial
curves in G^p have the property that their p-dimensional Hausdorff measure is  positive. The special case G^2 turns out to be a kind of universal lattice, in  which every finite dimensional lattice can be isometrically embedded.


3:00-3:45
Speaker: Kris Tapp, SUNY Stony Brook.
Title of talk: Conditions for nonnegative and positive curvature on bundles.
Abstract:  I will discuss tools for addressing the following two questions: (1) which vector bundles admit nonnegative sectional curvature, (2) which sphere bundles admit positive sectional curvature?  I will explain old and new relationships between these two questions, and I will discuss a classification (joint with Gromoll) of nonnegatively curved metrics on
$S^2 \times R^2$.

Coffee, Juice, Cookies
3:45, Room 409

4:15-5:00
Speaker: Chaim Goodman-Strauss, Univ. of Arkansas.
Title of talk: "Triangles"
Abstract: We introduce the use of "regular substitution systems"--- a certain generalization of symbolic substitution systems--- as a tool for analyzing tilings and more arbitrary complexes defined by combinatorial rules, though
in this talk we are particularly interested in tilings of the hyperbolic plane.

As an application, we conjecture necessary and sufficient conditions under which we may tile the sphere, hyperbolic or Euclidean plane by copies of a given triangle, and prove the conjecture on all but a measure-zero set in the space of all triangles. We give a new proof of Poincar\'e's Triangle theorem as an aside.

We also show most triangles that do tile are  "weakly aperiodic"; that is, they admit tilings, and admit tilings that are invariant under some infinite cyclic symmetry, but do not admit tilings with a compact fundamental domain.


5:15-6:00
Speaker: Jeanne Clelland, University of Colorado, Boulder
Title of talk: Backlund transformations of hyperbolic Monge-Ampere equations
Abstract: Backlund transformations provide a method for contructing  new solutions of a partial differential equation from a known  solution.  The new solutions are constructed by solving ordinary  differential equations.  These transformations are known to exist for  certain special PDEs - in particular, for most integrable systems -  but it is not known what conditions a PDE must satisfy in order to  have a Backlund transformation.  In this talk I will describe some  classical Backlund transformations of hyperbolic Monge-Ampere  equations in terms of exterior differential systems.  Using Cartan's  method of equivalence we can classify the homogenous examples, i.e.,  those transformations having maximal symmetry.  In the process, we  discover a family of previously unknown Backlund transformations  between timelike surfaces of constant mean curvature in 3-dimensional
Lorentzian space forms.

Sunday, April 21, 2002

Coffee, Juice, Bagels
9:00, Room 409

9:30-10:15
Speaker: Casim Abbas, Michigan State University.
Title of talk: The Chord Problem in Contact Geometry and Fillings by Pseudoholomorphic Curves
Abstract: On a contact manifold there is a distinguished vector field, the Reeb vector field, and its dynamics can be used to define invariants for contact manifolds. The interesting objects are periodic orbits of the Reeb vector field and characteristic chords for Legendrian knots. In my talk I will present a global existence result for characteristic chords using pseudoholomorphic curve techniques.


10:30-11:15
Speaker: Sungwok Lee, Univ. of Southern Mississippi.
Title of talk: Space-Like Surfaces of Constant Mean Curvature in the De Sitter 3-Space and the Gauss Map
Abstract:  A Bryant type representation formula for space-like surfaces of constant mean curvature 1 (abbreviated as CMC 1) in the de Sitter 3-space ${\mathbb S}^3_1$ is obtained. The formula is used to investigate a 1:1 correspondence between CMC 1 space-like surfaces in ${\mathbb S}^3_1$ and maximal space-like surfaces in the Minkowski 3-space ${\mathbb L}^3$. Three types of Gauss maps (the secondary, hyperbolic, and generalized Gauss map are discussed, and their relationships to each other are investigated. A duality property of CMC 1 space-like surfaces in ${\mathbb S}^3_1$ is also studied. Some examples of CMC 1 space-like surfaces in ${\mathbb S}^3_1$ are presented.


11:30-12:15
Speaker: Chris Mosely, Agnes Scott College
Title of talk: Geodesics of Sub-Riemannian Engel Manifolds
Abstract:  An Engel system is a smooth 2-plane field $D$ in a four-manifold M with the property that $D + [D,D]$ has rank 3 everywhere and $D + [D,D] + [D, [D,D]] = TM$.  The equations of for both regular and non-regular sub-Riemannian geodesics in $M$ are derived from canonical local structure equations on $M$, a result of a covering space theorem for Engel manifolds.  Examples of sub-Riemannian geodesics on Lie groups are explicitly solved.
Accomodations

A block of rooms has been set aside for conference participants at the Days Inn, Athens
230 N. Finley Street, Athens, GA  706-543-6511.

Travel Information

Direct flights to Athens are available through USAir. However it is likely to prove more convenient and cost-efficient to fly into Hartsfield International Airport in Atlanta. The shuttle from Hartsfield to Athens goes directly to the Georgia Center.

Dining On Campus

UGA Creamery

Georgia Center for Continuing Education

Tate Student Center (The Bulldog Room) 7:30 am - 3:00 p.m., Monday-Friday.


UGA Visitor's Information
Athens Convention and Visitors Bureau
Athens Weather

Page last updated April 9, 2002