Project 1: Algebraic Geometry & Combinatorics, directed by Dr. A. Gholampour
Combinatorial Algebraic Geometry
Combinatorics and Algebraic Geometry have classically enjoyed a fruitful interplay. There are many topics in algebraic geometry with deep combinatorial connections. These will include, but are not limited to, Hilbert schemes, moduli spaces, Okounkov bodies, Schubert varieties, toric varieties, and tropical geometry. Our focus will be mostly on toric varieties (and Hilbert schemes if time allows). Toric varieties provide an elementary way to see many examples and phenomena in Algebraic Geometry. Even though toric varieties are very special, nevertheless, they have provided a remarkably fertile testing ground for general theories. All the basic concepts on toric varieties correspond to simple combinatorial notions. This makes many things more computable and concrete than usual. There are applications and interesting relations with commutative algebra and the number of lattice points in polyhedra. Our aim will be a mild introduction to the combinatorial and computational aspects of toric varieties and the problems within. To be successful in this project students are expected to have a background in linear algebra and be familiar with computational programs such as Matlab, Mathematica, or Maple.
Project 2: Applied Harmonic Analysis directed by Drs. W. Czaja, A. Cloninger, and V. Rajapakse
Heterogeneous Data Integration and Fusion
The abundance of easily available information is one of the most important characteristics of our digital age. This information comes in various forms and formats. Its individual pieces may be noisy, unreliable, and provide only very selective view of the whole picture. Yet, integrative approaches prove over and over to provide us with better and deeper understanding of the available data, in such a way that no individual analysis can compare to. Therefore, the goal of our summer project will be to understand ways in which data integration and fusion can provide added value in specific applications, such as classification, machine learning, or data recovery. Our starting point will be an overview of a number of existing mathematical theories for data fusion, including fusion frames, joint manifold representations, and data-dependent operator eigendecompositions. We shall then look at examples of interesting problems arising in the analysis of social media, or land use data. The summer project will provide a solution to a specific selected problem by utilizing state-of-the-art mathematical theories. To be successful in this project students are expected to have a background in linear algebra and numerical analysis (be very familiar with computational programs such as Matlab, C or python).
Project 3: Mathematical Biology directed by Drs. D. Levy and S. Wilson
The quasi-steady-state assumption (QSSA) is a model reduction technique used to model a number of chemical reaction networks that involve short-lived, intermediate chemical species. The QSSA is used to remove the highly-reactive, low concentration species from the model and produce a reduced model valid on the slow time scale. The time-course of slow scale chemicals are of particular interest for identifying reduced mechanisms, estimating model parameters, and designing experiments. The resulting reduced model involves differential equations for each of the “slow” chemical species and algebraic equations for each of the “fast” chemical intermediaries.
In this project, we will explore when this approximation is appropriate and when there is a guaranteed solution to our reduced system. Students will investigate : solvability of an algebraic system of equations, numerical simulations of the reduced versus full model, as well as alternate methods to approximating the concentration profiles of a chemical reaction network (rescaling intermediates, computational singular perturbation, etc).
Students interested in this project should have some experience in differential equations (introductory course) and some computer programming (e.g. Matlab).
Project 4: Numerical PDEs directed by Dr. J. Bedrossian
Mixing in fluid mechanics
Stirring milk into a cup of coffee is not something most people consider a mysterious physical process. In fact, the mathematics is poorly understood and is connected to many other physical processes such as the dynamics of hurricanes. These problems are surprisingly hard to study numerically, and students will begin by learning about the various numerical methods available. Students interested in numerical analysis will code their own solvers and, if time permits, compare the results of several different methods. Students more interested in analysis may be able to find a suitable open-source code to perform simulations. Once students have obtained accurate and efficient solvers, the students will begin a systematic study of the dynamics for a variety of fluid flows. Possible directions relevant to applications include: determining how mixing and decay rates, determine the expected behaviors of "randomly" chosen fluid flows, and to study the long-time dynamics with random or deterministic external sources. If time permits, students could extend their codes to handle 2D incompressible fluid dynamics and compare how close the dynamics of these nonlinear equations are to the the simpler dynamics of passively mixing scalars in various settings.
Project 5: Scientific computing directed by Dr. M. Cameron
Construction of stochastic networks
The contemporary development of communications, information technologies and powerful computing resources has made networks a popular tool for data organization, representation and interpretation. Networks allow us to create mathematically tractable models that preserve important features of underlying systems and avoid problems associated with high dimensionality and complex geometry. In particular, networks have demonstrated a strong potential for modeling and analysis of complex physical systems such as the dynamics of clusters of interacting particles.
The project will be concerned with developing computational tools for building networks representing the dynamics of interacting particles. The figure depicts an example of a joint network representing the dynamics of 6 and 7 atoms interacting according to the Lennard-Jones pair potential. One atom is allowed to dissociate from the cluster of 7 and associate back. Optimization methods, methods for finding saddle points, Monte-Carlo methods, and some geometric/combinatorial techniques will be explored, and networks representing aggregation processes of interacting particles of different nature will be created.
The students are expected to know multivariable calculus, linear algebra, and to have programming skills, e.g. in Matlab.