UGA VIGRE Research Groups, Spring 2010

Introductory VIGRE Research Groups

String Theory - Robert Varley and Cal Burgoyne

String theory (the ideas and some of its mathematical foundations)  (View the official website here.)

What do the string theory models proposed in mathematical physics explain about particle physics, gravity, and the structure of the universe, and what are the mathematical ingredients of the theories? There are many popular accounts of the subject. In this undergraduate research group we will explore some of the mathematics built into string theory. The plan is to work together and share the results through student presentations, individually or in small groups. A list of some of the possible topics follows:

• basic ideas of string theory - Veneziano's model, the different types of string theories and the need for higher dimensions (10, 11, and 26 dimensions!)
• classical string theory (1-D wave equation & vibrational modes of a string)
• geometry of the world sheet as a string propagates in space-time (including an introduction to the differential geometry of surfaces)

Advanced VIGRE Research Groups

Toy Surfaces, Circle Packing, and Approximation - Sa’ar Hersonsky, Michael Ching

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

Start with a triangulated (closed topological disk) with four distinguished points on its boundary; we will construct a realization of it composed of Euclidean triangles (equilateral). The result is a surface carrying a piecewise Euclidean metric with conical singularities. We can "circle pack" this surface and obtain a "discrete circle mapping" onto an Euclidean rectangle packed with circles. Iterate the process by refining the initial triangulation. Does the sequence of maps you obtain converge? If so, what is the limit? What is the situation with other planar domains? All of this and more...
 

Structure and Dynamics of Ecological Networks - Caner Kazanci, Malcolm Adams

http://www.math.uga.edu/~caner/09vigre/index.html

Ecosystems are often modeled using weighted digraphs, representing flow of energy or nutrients among compartments. Depending on the model, what a compartment represents may range from dissolved organic matter in a lake, to hundreds of species living in a specific area. Flows among compartments may represent feeding, uptake, excretion, etc. Study of ecosystems pose new interesting mathematical questions, which are related to the relation between graph structure and dynamics. We know very little about this relation in the context of mathematical biology and ecology.

There are various methods to study ecosystem models. A common way to simulate these systems is to form a set of differential equations where the solution represents the state of each identity changing in time. Another way to analyze these systems is by formulating system-wide organizational properties, which provide insights as to how the environmental inputs are shared among identities, how much energy or matter cycling occurs within the system, or how strong are any two identities in the system related to each other. Most such properties are defined based on the digraph structure, flow quantities associated with each edge, and storage values associated with each compartment (vertex).

One of the fundamental challenges of analyzing ecosystems lies in deciding how best to represent them, known as "the modeling problem". Model complexity increases rapidly with an increasing number of model compartments, and parameterization of more complex models is exceptionally challenging. Decomposing a complicated ecosystem into sub systems for easier analysis is often tempting. However, ecosystems function as a system, therefore essential behavior may be lost by breaking connections. In this research group, we will investigate a new decomposition technique; where an ecosystem is divided into simpler sub-ecosystems without breaking connections. We will investigate how essential ecosystem properties are carried into these decomposed sub-systems, and study the uniqueness or orthogonality properties associated with this decomposition.

Cohomology of Finite Groups: Computations and Interactions - Brian Boe, Jon Carlson, Lenny Chastkofsky, Dan Nakano

The Algebra VIGRE Research Group will focus on the topic of the cohomology of finite groups. This is a broad area with many connections to homological algebra, derived categories and algebraic topology. We will introduce students and postdocs to the fundamentals of the subject in a one-month crash course at the beginning of the year. We will then turn our attention to looking at specific examples including cohomology for finite groups of Lie type, symmetric groups and perhaps some sporadic simple groups with an emphasis on computing low degree cohomology. One paper of interest involves Cline-Parshall-Scott's calculation of the first cohomology with coefficients in an irreducible module with minuscule highest weight for finite Chevalley groups. This paper was written in 1974 and was used in Wiles' proof of Fermat's Last Theorem. There are natural questions about extending these results to modules with fundamental highest weight.

For the calculations, we will introduce techniques from other related areas and employ computer packages such as MAGMA. If time permits, we might look at more advanced topics such as complexity and the variety theory for modules. Visit the Algebra Group's Research page at: http://www.math.uga.edu/~nakano/vigre/vigre.html.

Hilbert Polynomials, Regularity, and Multiplier Ideals - Elham Izadi, David Swinarski

This VRG will focus on elementary problems related to Hilbert polynomials, regularity and multiplier ideals. The goal will be for the students to compute the Hilbert polynomials and hence the regularity of subvarieties of projective space using software such as Macaulay, then to formulate conjectures and attempt to prove them.