UGA VIGRE Research Groups, Spring 2009

Circle Packing Theorems - Sa'ar Hersonsky

http://www.math.uga.edu/~saarh/CirclePacking1.html

This group continues work done during 2007–08 exploring circle packing (in the plane and other surfaces) with prescribed combinatorial data. The subtle point is that these combinatorial data will actually prescribe which surface one is working on. This is a field which leads quickly to open questions and so far the techniques in the field have been a beautiful mix of combinatorics, complex analysis, differential geometry, and topology.

Tabulating Composite Links - Jason Cantarella

http://www.jasoncantarella.com/webpage/index.php?title=Geometry_VIGRE_Group

Tie a knot in a piece of string and close the ends to form a loop. The result is a topological knot. If you do the same with several strands, the result is a topological link. Deforming the loops (without passing any strand through another one) constructs a class of loops called a knot or link type. There is also an algebraic structure on the set of knot and link types: Splicing knots or links together forms a product called the connected sum. The identity element for this product, the unknot, is a simple round circle.

The set of knots and links which are prime (expressible as products only of themselves and the identity) are well-understood and classified up to a certain set of symmetries, such as taking the mirror image of a knot. Knowing these symmetries, the group has in previous years classified all composite knots up to 9 crossings. Our project for 2008–09 will be to use computer methods to understand and classify composite links. This depends on understanding a new kind of link symmetry where different loops are exchanged for each other. The project requires some knowledge of algebra (MATH 4010 would be helpful) and some participants will need very good computer skills (Perl preferred).

Modelling the Health of Ecosystems - Caner Kazanci

http://www.math.uga.edu/~caner/08vigre/index.html

A major reason biological and ecological systems are so complex is the network structure with many interactions among multiple identities. In a genetic network, these identities are genes, and the interactions among genes are up-regulation and down-regulation. In a cellular pathway, identities are molecular species, and interactions are biochemical reactions. In ecological systems, identities can range from accumulated organic matter to hundreds of species, interactions may represent flow of energy, biomass, or a specific element such as C, N or P. We will focus on ecological networks (weighted digraphs) and try to identify what properties of this weighted digraph represents a “healthy ecosystem.” Furthermore, we will investigate equations that govern how these “health indicators” change for an evolving system.Also, to read about the related REU that Dr. Kazanci ran last summer, visit http://www.math.uga.edu/~caner/08reu/index.html.

How to Cut Large Data Small - Ming-Jun Lai

http://www.math.uga.edu/~mjlai/vigre.html

Many data sets, e.g., images, contain a huge number of numerical values. It takes space to store them and much time to transmit them on the internet. Since most data sets do not consist of arbitrary random numbers, it makes sense to try to compress the large data sets into smaller ones. We look for the smallest number of nonzero coefficients needed to represent a given data set. The data compression problem has many applications such as in error correcting codes and HDTV.

Geometry, Combinatorics and Fourier Analysis - Neil Lyall

http://www.math.uga.edu/~lyall/VRG/

The following projects have a unifying theme of interaction of combinatorics and Fourier analysis:

(a) Szemeredi-Trotter incidence theorem
(b) Erdös distance problem
(c) Sum/product estimates
(d) Besicovitch/Kakeya sets
(e) Discrepancy and irregularity of distributions.

Prerequisites may vary, depending on how deeply a student chooses to delve into one of the projects. With the exception of Project (d), the students will be able to go quite far and discover interesting things without actually doing any “real” analysis.