Project 1: Applied Harmonic Analysis directed by Drs. R. Balan and K. Okoudjou
This project involves two different topics. Please specify which topic you are interested in on your statement of interest.
1-1 (This project will be directed by Dr. Balan) Standard Inverse Model problems and Signal Processing techniques use linear decompositions of various classes of functions. However many of real world problems have intrinsic nonlinear models at core. A special case of nonlinearity is presented by taking the absolute value of coefficients of a redundant linear representation. The X-Ray Crystallography presents such an application where the underlying linear transformation is a 3D Fourier transform of the electron density. However the issue of reconstruction from magnitudes of frame coefficients is related to a significant number of problems in engineering and science: noise reduction and source separation in signal processing; optical data processing; quantum computing; low rank matrix completion problem; deep scattering networks.
This summer REU program will focus on analytic results and inversion algorithms for the problem
where is a fixed set of spanning vectors in .
1-2 (This project will be directed by Dr. Okoudjou) In this project students will study frames for finite dimensional Euclidean spaces. In particular, they will explore construction methods for tight frames by exploiting (a) the fact that these frames minimize certain potential functions, and (b) the association of this class of frames to certain finite graphs. To gain intuition, the students will start by investigating some of these questions for small dimensions, e.g., dimension 2 or 3. The project will involve theory as well as computer simulations and will build upon results obtained by last year REU students.
Students interested in either of these two topics are expected to have some familiarity with Linear Algebra, and some computer programing, e.g., Matlab
Project 2: Chaotic Dynamics directed by Dr. J. Yorke
Chaos is a highly geometric field. There is a wide variety of behaviors that are possible in simple dynamical systems, and student projects can take advantage of these phenomena. One phenomenon that the students will investigate in detail is horseshoe maps. Even easier are closely related examples based on so-called snap-back repellers. Students will investigate dynamical systems both using paper and pencil examples and computer examples. The latter will vary according to the students computational skills. While students will be encouraged to write some of their own software, free software packages are available that allow students to explore more complicated behaviors.
Students will be given projects that allow them to tease out the complex behaviors of trajectories that occur in maps. For a map F with transverse homoclinic points, one usually points out that there is a horseshoe map for sufficiently large iterate F^n of the map, but that does not reveal all of the complex dynamics that must occur because the different horseshoe maps for different n interact with each other. Students can discover the true nature of such phenomena on their own -- with some subtle hints. To avoid too much complexity, one-dimensional maps are an excellent initiation into dynamical systems. Some of the most interesting behaviors students will explore are the sudden changes in the dynamics that can occur as parameters are varied. These have names like crises (sudden changes in attractors), metamorphoses (of basin boundaries), and period-doubling cascades. This is a rich environment for student discovery.
Project 3: Geometry directed by Dr. K. Melnick
"Local isometries of 3D spacetimes"
In this project students will study symmetries of three-dimensional spacetimes. We are looking for Lorentzian metrics with unusual local properties. Spaces that look the same around every point, such as Euclidean space or the sphere, are called locally homogeneous. We are looking for spacetimes that are locally homogeneous in an open set, but not on a lower-dimensional closed subset. It is not known whether such a spacetime exists, but it is known that if it does, it must be a so-called Kundt spacetime. We will work from a classification of Kundt metrics and investigate their symmetries. Our calculations will be in terms of explicit coordinate expressions of Lorentzian metrics on R^3; students will perform them using special packages in Mathematica and Maple. Prerequisites are a solid foundation in multivariable calculus and linear algebra. An introduction to differential geometry or familiarity with either of these software packages are also helpful.
Project 4: Mathematical Physics directed by Drs. D. Margetis and M. Grillakis
Two recent major advances in atomic physics were the Bose-Einstein condensation of atomic gases and quantum computing. The goal of this REU project is to study the connection of mathematical concepts that underlie these advances. This study will focus on the use of Differential Equations, especially Schroedinger equations, for the description of quantum phenomena. The objective is to demonstrate how quantum mechanics can predict interesting phenomena at small as well as large scales, from atoms to manmade devices.
Prerequisite: Some background (introductory course) in ordinary or partial differential equations.
Project 5: Probability directed by Dmitry Dolgopyat and Leonid Koralov.
`"Asymptotic problems for random processes"
During this project the students will work on several asymptotic problems for random processes. Here 'asymptotic' refers to the behavior of the process at large time and/or spatial scales. In particular, branching processes will be studied. These are used to describe the evolution of various populations such as bacteria, cancer cells, carriers of a particular gene, etc., where each member of the population may die or produce offspring independently of the rest.
Possible problems include describing the long-time behavior of the population in different regions of space when, in addition to branching, the members of the population move in space and the branching mechanism depends on the location.
At the first stage of the project, students will be taught the main facts about random walks and branching processes. We will also introduce students to the Brownian motion and diffusion processes, which are the continuous-time analogues of the random walks. A relatively accessible problem is the description of front propagation and the behavior near the front for branching diffusions with a slowly changing branching rate. Other open problems concern the distribution of particles in the critical case of branching diffusions and the behavior of branching diffusions with periodic branching potential.
Prior exposure to proof-based mathematics (e.g., advanced calculus or mathematical analysis) is required.