| Supervisor Name |
Research Project |
|---|---|
| Richard Anstee |
Extremal set problems consider
families F of subsets of {1,2,...,m} and given an additional property,
try to determine either the maximum number of sets |F|, or perhaps the structure of set
families with the property. The particular problems that will be
studied involve `Forbidden Configurations' or the closely related
problem of forbidden submatrices. There are grand conjectures and also
very specific problems to study. This is part of a long term project. In general the project does not require a lot of background (simple induction is a basic tool) but needs much cleverness. |
| Martin Barlow and Gordon Slade |
The project is two topics in
probability theory/statistical physics. Both will involve computer simulation of random processes. Some knowledge of both probability theory and structured programming languages is desirable. 1. Random media and exclusion process. Physicists working on solar cells have recently observed the odd phenomena that putting more electrons into the system increases the conductivity of the material. This project will examine one model for this, and will involve doing computer simulations of a random walk in a random media. No physics knowledge is needed. 2. Branching random walk. The project will study random networks arising from brancing random walks. We are interested in the electrical resistance of the network, and time it takes a random walk in the network to move a distance R from the origin. |
| Ailana Fraser |
Title: Minimal surfaces Minimization problems arise naturally in many branches of mathematics and science. For example, problems in navigation involve finding paths of least length (`geodesics') on the earth's surface. Minimal surfaces, which are two-dimensional analogs of geodesics, are minimizers (or simply critical points) of the area function, and arise naturally in material science; for example in fluid interface problems and elasticity problems. A simple physical example of a minimal surface is the soap film that forms after dipping a wire frame into a soap solution. By the laws of surface tension this soap film has the property that it is stable, that is it becomes larger under slight deformations. The theory of minimal surfaces (and submanifolds) has had striking applications, for example to general relativity and low dimensional topology. This project will involve studying existence of minimal surfaces, properties of minimal surfaces, and applications. |
| Joel Friedman |
Graph theory. |
| Masoud Kamgarpour |
Title: Multiplication in Hecke
Algebras.
Hecke algebras are
central objects in studying representations of groups. For the general
linear group, understanding the multiplication in Hecke algebra amounts
to clever tricks with matrices. The project I have in mind involves
implementing linear algebraic trick for understanding the convolution
product of Hecke algebras. The interested student be
very comfortable with linear algebra, and have had some
exposure to
abstract algebra. Some background in problem solving, for instance,
Putnam or Math Olympiads, may also be helpful.
|
| Mahta Khosravi and Malabika Pramanik |
Title: Spectra of Sol-manifolds. This is a group project suitable for a group of 2-3 students. |
| Young-Heon Kim |
Title: Study of optimal
transport phenomena. The theory of optimal transport is concerned with phenomena arising when one matches two mass distributions in a most economic way, minimizing certain transportation cost of moving mass from one location to another. We consider such optimal transport problems for discrete or continuous mass distributions in various situations. We plan to study how such "optimal maps" look like. This requires some solid background in mathematical analysis (for example, in such courses as both MATH 320 and MATH 321). |
| Anthony Peirce |
Title: Novel Approximation Schemes for Hydraulic Fractures We plan to investigate the numerical solution of an integral equation that governs the propagation of a fracture in an elastic medium under conditions of plane strain. We will consider a piecewise cubic Hermite approximation to the crack-opening-displacement which is used along with a collocation technique on a uniform mesh. We will also consider various approximation strategies for a propagating fracture, which does not coincide with the prescribed uniform mesh. The Hermite scheme will then be used to solve the dynamic model for a hydraulic fracture propagating in an elastic medium. It is proposed that a computer code will be developed in MATLAB and the numerical solutions will be checked against existing asymptotic solutions. |
| Hamid Usefi |
Title: McEliece Public-Key
Cryptosystems
Diffie and Hellman in 1976 introduced the concept of public-key cryptosystem (PKC). Since then many PKCs have been proposed based on integer factoring, discrete logarithm, inverting polynomial equations and so on. However, the unbroken (secure for now) PKCs constitutes only a small class of such systems that includes elliptic curve cryptosystems and the widely used RSA. The McEliece PKC, proposed in 1978, is one of few alternatives for PKCs based on IFP or DLP. It is based on the decoding problem of a large linear code. It is preferred that the student has taken an elementary coding theory/cryptography course. Strong background in algebra and computing programs such as Maple or Matlab is a plus. |
| Michael Ward |
Title: Mean First Passage Time
in Domains with Traps: Brownian Simulations There are many biological diffusion problems where a protein receptor must diffuse either within a cell or on the surface of a cell in the presence of many localized traps where it can be absorbed. Recently, there has been an asymptotic theory developed to analytically calculate the mean first passage time in the limit of small trap size. The goal of this project is to use different numerical algorithms, based on stochastic simulations such as the "Walk-on-Sphere" or Gillespie algorithm, to numerically predict the mean first passage time through averaging over many realizations, and then comparing these results with the recent analytical results and some other results in the biophysical literature. It would be helpful to have some exposure to differential equations, a course in elementary probability, and some experience in programming. The student will be exposed to stochastic simulations, asymptotic analysis, PDE, and some diffusion problems in biology. |