Project 1: Zeta functions of graphs, directed by Niranjan Ramachandran
This project aims to understand finite and infinite graphs by means of their zeta functions. Students will start with the definitions of a graph and the associated zeta function; they will compute examples for various families of finite graphs. The main goal will be to identify invariants of the graph from the zeta function such as Euler number etc. If time permits, these investigations will be extended to infinite graphs. Background and context for the project can be obtained from the book "Zeta functions of graphs: A stroll through the garden" by Audrey Terras see also a very useful review in the Bulletin of the AMS. Graphs are used in all fields of science and engineering (neurons, networks, telecommunications, information technology).
Project 2: Finite frames, graphs and optimal configurations, directed by John J. Benedetto and Kasso Okoudjou
In this project students will study frames for finite dimensional Euclidean spaces. Applications of finite frames abound. 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. Applications of finite frames abound. For example, certain class of generator (or window) functions for finite Gabor frames lead to sparse solutions in deterministic compressive sensing. These generators are number theoretic sequences constructed to obtain optimal ambiguity function behavior and are used in radar and communications.
Project 3: Visualization of geometric structures in 2 and 3 dimensions, directed by William Goldman
In this project the students will work in the Experimental Geometry Lab and explore the relationship between geometric structures and discrete groups is various (interrelated geometries). This subject began in the nineteenth century with the application of group theory to crystallography. The symmetries of crystals led theoretically to discrete groups, which have now become called crystallographic groups. Recently the theory of ''affine crystals" (where the symmetry groups preserve the Euclidean notion of parallelism, but perhaps not the finer notions of distance and angle, have led to unexpected relations which hyperbolic onEuclidean crystallography. In particular regular tilings of the hyperbolic plane have led to Lorentzianisometric tilings of Minkowski 3space (the geometry of flat spacetime with 2 space dimensions) by polyhedra bounded by ''crooked planes,'' geometric substitutes of planes adapted to Minkowski geometry invented by Drumm. Students will work on visualization projects relating 2d hyperbolic nonEuclidean geometry with 3d Lorentzian ''crooked geometry.''
Project 4: Statistical Estimation via Calibration, directed by Eric V. Slud
This project will introduce students to estimation of means in surveys and other statistical data settings in which some planned observations are missing. The estimation methods studied are a class of methods called Weighting and Calibration. The idea is initially to weight observational units inversely by their probability of providing data, and then to modify the weights to reflect known mean values of observed data components as constraints. This is a statistical topic arising in biomedical studies, sample surveys, and observational studies in social science. The research topic of primary interest in this project is the interplay between the number of constraints imposed and the variability of the resulting survey estimates. The mathematical elements of this project are linear regression, some basic survey sampling theory, linear algebra, and quadratic programming. After being introduced to each of these topics through readings in books and journal articles, students will explore the properties of calibration estimators by designing constraintselection schemes and analyzing their performance through simulation experiments.
Project 5: 3manifolds, triangulations, and volume computations, directed by Christian Zickert
A basic fact of topology is that every surface can be obtained by gluing together a collection of triangles. This is for example used in computer animations, where a complicated surface is represented by a polyhedron. Similarly, a 3manifold (curved space which is locally 3-dimensional) can be represented by gluing together tetrahedra. This project deals with the combinatorics of triangulations and methods for computing invariants. Given a triangulation, one can set of a system of polynomial equations whose solutions are invariants of the manifold. To each solution one can compute a number called "volume", and the goal of the project is to explore a recent conjecture stating that all "volumes" are linear combinations of real volumes of hyperbolic manifolds. Knowledge of Python programming is required, and students will be expected to write code. We will use the software SnapPy for dealing with hyperbolic manifolds (available at http://www.math.uic.edu/t3m/SnapPy/). Some knowledge of basic topology and algebra is required. Knowledge of Groebner bases is helpful, but not required.