Mentor: Brent Koogler 

Area: Applied Math

Overview: Develop an understanding of the theory and tooling around nonlinear optimization. If time permits, we will analyze and implement a real-time model-predictive-control (MPC) algorithm, which can be used in robotics applications.  

Pre-requisites: Calculus II and some programming. We can get more/less sophisticated depending on background. 

Text: N/A 

Mentor: Nathan Bushman

Area: Number Theory, proof writing, algebra

Overview: We will introduce the basic concepts of number theory (primes, divisibility, congruences), with a goal of developing both proof writing skills and an early understanding of key concepts in abstract algebra.   

Pre-requisites: Student may be currently enrolled in math 3000, but should not have already completed it. Should not already be in/have taken advanced linear algebra or introductory abstract algebra. 

Text: "A Friendly Introduction to Number Theory," Joseph H. Silverman  

Mentor: Luis Flores

Area: Representation theory (of groups)

Overview: In algebra, only linear algebra is completely developed. However, abstract algebra is not just limited to linear algebra. In order to deal with various algebraic structures such as rings, modules, quivers, and groups, one can represent them as linear transformations with which one can then do linear algebra. That is, representation theory is a way to do linear algebra in contexts where doing so is normally not possible. We will begin our journey through representation theory through representations of groups. 

Pre-requisites: Linear Algebra and Abstract Algebra. 

Text: A Journey Through Representation Theory by Gruson and Serganova. 

Mentor: Arun Suresh

Area: Applied mathematics

Overview: After establishing some theory on probability and generating functions, we will spend most of our time building and analyzing Stochastic processes in the context of discrete time Markov chains as they apply to biological and epidemiological processes (i.e, spread of disease - an epidemic). Based on student interest, we could introduce a programming component. If time permits, we will continue to explore these concepts in the continuous setting and introduce the notion of a stochastic differential equation.

Pre-requisites: Linear algebra, Calculus 2. Some programming knowledge is useful, not necessary. 

Text: An introduction to stochastic processes with applications to biology - Linda J.S. Allen.

Mentor: Erik Amezquita Morataya

Area: Topological Data Analysis (TDA)

Overview: I am interested in understanding applications of TDA to measure spatial patterns to the apply them to plant cell data. This will require reading relevant work from the Porter Lab, mainly Feng and Porter (2021). In that paper, they provide ways to analyze the intrinsic shape of geospatial patterns of voting districts despite the heterogeneity of district sizes and shapes. Being able to parse through such size and shape heterogeneity will be key to study the shape of cellular patterns. More details available at https://ejamezquita.github.io/docs/undergrad_tda_spatial.pdf 

Pre-requisites: The student should feel comfortable with some linear algebra and calculus at the bare minimum. Some knowledge on the basics of group theory will be useful to get a quick grasp of the TDA fundamentals. Basic coding skills (e.g. feel comfortable using `numpy` in python) would be very helpful but not required. The exact path of the research project will vary depending on the student's strengths.The student should be open to discuss their knowledge with biologists and be receptive of biology input when it comes to data analysis and interpretation.

Text: 

• Feng, M., Porter, M.A. (2020) Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence. Phys. Rev. Res. 2: 033426. DOI: 10.1103/PhysRevResearch.2.033426

• Feng, M., Porter, M.A. (2021) Persistent Homology of Geospatial Data: A Case Study with Voting. SIAM Review 63(1): 67--99. DOI: 10.1137/19M1241519

• Feng, M., Hickok, A., Porter, M.A. (2022) Topological Data Analysis of Spatial Systems. In Higher-Order Systems. Springer: Cham. p. 389-399. DOI: 10.1007/978-3-030-91374-8_16

• Hickok, A., Needell, D. Porter, M.A. (2022) Analysis of Spatial and Spatiotemporal Anomalies Using Persistent Homology: Case Studies with COVID-19 Data. SIAM Journal on Mathematics of Data Science 4(3): 1116--1144. DOI: 10.1137/21M1435033