Project descriptions for the VIGRE REU2001 Program
|Amelia Taylor||(Professor, Mathematics)||
|Avy Soffer||(Professor, Mathematics)||
|Chris Woodward||(Professor, Mathematics)||
|Richard Gundy||(Professor, Statistics/Mathematics)||
|Mario Szegedy||(Professor, Computer Science)||
The following topics have been suggested by REU research mentors, and offer a range of both theoretical and applied problems. The topics are of current interest to mathematicians, but can be understood in a short period of time by a well-trained undergraduate. Mentors may also be willing to work on other topics, depending on the student's and their own interests.
Graphs of Matrices (a project in Computational Commutative Algebra and Combinatorics) (Amelia Taylor) (Co-mentor: VIGRE graduate student Aaron Lauve)
A certain class of m+1×m matrices that arise when working with certain ideals in polynomial rings are in a one to one correspondence with trees from graph theory. We will use Macaulay2 and possibly Maple as well as pen and paper to explore the implications of this association. There are both algebraic and combinatorial questions that arise. In particular finding such relationship for a larger class of matrices and graphs. No prior knowledge of ring theory or graph theory is required, nor does the student need to have prior experience with Maple or Macaulay2. It would be helpful if interested students have taken an undergraduate abstract algebra class.
Partial Differential Equations (Avy Soffer) (Co-mentors: VIGRE Postdoc Carlo Lancellotti and VIGRE graduate student Pieter Blue)
Dynamical wave phenomena requires the understanding of the large time behavior of partial differential equations. Quantum Mechanics is one such example. Optical and laser systems are of similar nature. Such analysis poses a special challenge both to theorists and computational analysis. The study of various aspects of the nonlinear Schrödinger equation by analytic and numerical methods is proposed. The topic and its nature will be determined according to the interests and experience of the student. Students should know linear algebra and differential equations. Also very useful are: programming, graphics, quantum mechanics, and perhaps complex analysis, advanced calculus/real analysis.
Geometry and Representation Theory (Chris Woodward) (Co-mentor: Hill Professor Matt Leingang)
Heron's formula in Euclidean geometry gives the area squared of a triangle in terms of its edge lengths. The positivity of this quantity is equivalent to the triangle inequalities, describing numbers occur as edge lengths of triangles. The same triangle inequalities govern the decomposition of irreducible representations of SL(2). I'm interested in exploring generalizations of this fact, involving the spherical version of Heron's formula and related topics such as the construction of codes in the unitary group. Prerequisite for this project is a good knowledge of linear algebra and an interest in geometry and representation theory.
The Mathematics of Card Shuffling (Richard Gundy) (Co-mentors: VIGRE graduate students Peter Kay and Alex Zarechnak)
Geometric data structures (Mario Szegedy) (Co-mentor: VIGRE graduate student Nick Weininger)
The project is about developing geometric data structures for quick display of moving objects. The dynamic data structures would allow speedy update of matrices of spatial relations in between objects moving in front of a landscape. The landscape may be topologically non-trivial. The goal is to develop efficient algorithms and to obtain complexity lower bounds for these problems.
Click for the DIMACS REU2001 project listing.