### 2022 research project descriptions

- Louis Fan: Coalescent theory and the Ewen's sampling formula
**Faculty mentor:**Wai-Tong (Louis) Fan**Description:**Coalescent theory is the study of stochastic processes where particles may join each other to form clusters as time proceeds. They are powerful mathematical tools in the analysis of modern genetic and genomic data, and are rooted in long history of population genetics. The Kingman coalescent, for instance, arise as a universal object that describes the genetic ancestry of a sample of DNA sequences. Mathematically, the Kingman coalescent is the scaling limit of the genealogy of a sample under the Wright-Fisher model, the Moran model and many other population models without selection. Such scaling limit results are analogous to the central limit theorem, where the Kingman coalescent plays the robust role of the Gaussian distribution.

In these few weeks, we will investigate mathematical problems in coalescent theory. We will analyse and visualize coalescent processes (e.g. Kingman coalescent) and sampling distributions (e.g. Ewen's sampling formula). We may also address the rising demand to develop and analyse models that incorporate more realistic features than the Kingman coalescent.**Prerequisites:**Basic knowledge in probability and combinatorics, including the concepts of random variables, joint distribution, conditional probability and central limit theorem. Familiarity with Mathematica and/or computer programming would be a plus, but not required.

- Salman Siddiqi: Entropy estimation in physical systems
**Faculty mentor:**Salman Siddiqi**Description:**The entropy of a deterministic system or process is a quantitative measure of the degree of chaotic behaviour the system exhibits, and can be a useful metric in a variety of practical applications. Estimating entropy is unfortunately not entirely straightforward, and there are a variety of approaches and point estimators to choose from. The goal of this project will be to investigate various point estimators for entropy, and perform some calculations in a few interesting cases.

**Prerequisites:**A successful project would likely require some familiarity with probability and statistics (at the level of M365 or higher). Any background in programming, differential geometry or differential equations may be helpful, but none of these are required.

- Paolo Piersanti: Numerical validation of reduced models in thermoelasticity.
**Faculty mentor: Paolo Piersanti****Description:**(see attached)**Prerequisites:**(1) Knowledge of multi-variable calculus; (2) Knowledge of basic physics; (3) Some exposure to PDEs; (4) Proficiency in the Python programming language.

- Dylan Thurston: Graph energies and Sierpinski carpets
**Faculty mentor:**Dylan Thurston**Description:**Sierpinski carpets are obtained by taking a large disk and removing more and more non-touching disks of smaller and smaller sizes. They arise naturally in many contexts, notably in the study of complex dynamics.

A new technique for studying them, and in particular a natural quantity called the conformal dimension, is viewing them as limits of graphs, and studying energies of maps between these graphs. In this project, we will clarify this connection and use it to give concrete estimates of conformal dimensions.**Prerequisites:**Linear algebra, complex analysis, and familiarity with programming (especially optimization) are all helpful.

- Shouhong Wang: Center Manifold Reductions and Dynamic Transitions
**Faculty mentor:**Shouhong Wang**Description:**In dynamical systems theory, a center manifold function (CMF) links the fast variables to the slow variables, and it plays a crucial role in studying nonlinear dynamics for many problems in mathematical physics. The computation of CMFs is often a difficult task. This project focuses on the backward-forward approach for finding leading-order approximations of CMFs for some problems in statistical physics.

**Prerequisites:**`The prerequisites include undergraduate level Calculus 1-3, linear algebra and differential equations. The student can either focus on the theoretical side of the project, or explore computational aspect of the project. More computational inclined student may wish to further explore numerical codes for computing CMFs using the backward-forward approach. In this case, working knowledge of Matlab will be needed.`

- Kevin Zumbrun: Guts Poker and Optimization of Discrete and Continuous Games
**Faculty mentor:**Kevin Zumbrun**Description:**In a previous REU project, we have analyzed with participants L. Castronovo and Y. Chen the popular poker game “Guts,” along with a continuous analog more convenient for analysis, determining most of its main features. For the 2-player version, this includes an optimal, winning strategy. For the 3-player version, as for many multiplayer games, the situation is less clear, due to existence of winning strategies for 2-player coalitions. In this followup, we propose to investigate further, finding the optimum winning strategy for a coalition, and various strategies for play. At the same time, we will look at the general question of numerical optimization given a known payoff function, whether continuous or discrete, starting with the (discrete) Python-based package NashPy, and the development of simplified models for coalition phenomena.

**Prerequisites:**general knowledge of Calculus and basics of finite probability; prior knowledge of, or interest in learning, use of Python and Python packages.