Current Projects

Mentor: Nolan Hauck

Area: Functional Analysis 

Overview of the topic: Basic Functional Analysis forms the basis for many more advanced topics in math (like the PDE research I do), and is a natural next step after learning about metric spaces, general topology, and linear algebra. It allows interested students to start seeing connections between abstract linear algebra and analysis courses. The goal would be to comprehend the first few sections of Chapter 5 of Gerald Folland's Real Analysis, and hopefully present solutions to a few interesting exercises. 

Pre-requisites: Ideally comfortable with linear algebra (4140), metric spaces (4700), topology (4400 or 4700), and proofs (3000). If topology or metric spaces are shaky, there is an introduction in Folland. Having heard of Zorn's lemma and totally ordered sets in general would be nice, but if not we can start there. I think these things are usually taught in Topology 4400. 

Text: Real Analysis: Modern Techniques and their Applications by Gerald B. Folland. Available from Mizzou Library. I also have a pdf copy. 

Mentor: James Warta 

Area: Analysis 

Overview of the topic: To study two different (equivalent) constructions of the Lebesgue integral and explore how they lead to two different proofs of the monotone convergence theorem

Pre-requisites: Undergraduate analysis is a must  

Text: Two books - Wheeden & Zygmund, Folland 

Mentor: Nathan Bushman

Area: History of Mathematics

Overview of the topic: We will study Dunham's book and the proofs presented therein, and discuss the history of mathematics in general. Other sources may also be brought in where applicable. 

Pre-requisites: Student should have a strong grasp of college-level algebra, and have taken at least calc 1. They should be interested in math history and proof writing. Beyond that, no hard requirements; the rigor and depth of this project can be adjusted to fit any level of proficiency. 

Text: "Journey Through Genius," by William Dunham 

Mentor: Erik Amezquita 

Area: Applied Mathematics (Topological Data Analysis [TDA]) 

Overview of the topic: We will study the potential of Topological Data Analysis (TDA) for plant shape quantification. TDA is a combination of different mathematical and computational results based on algebraic topology that seeks to describe concisely and comprehensively the shape of data in a general setting. In very succinct terms, TDA consists of two basic ingredients. First we think the data as a collection of points. Second, we define a notion of distance between every pair of points. The points could be atoms, biomolecultes, cells' nuclei, image pixels, or an organism itself. Distances between points could be the Euclidean, geodesic, genetic, or correlation-based. Once we have data points and distances, we merge systematically the points, starting with those that are closer to each other. The key idea is to keep track of distinct blobs, loops, and voids that form and disappear as we merge several points. This versatile idea is not constrained to a particular dimension or set of landmarks, which makes it ideal to compare a vast array of possible different shapes.

In particular, 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)](https://doi.org/10.1137/19M1241519). 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. 

Pre-requisites: The most important prerequisite is to not be afraid of using Python. You do not have to be an expert: just be self-driven enough to use it. My end goal is to wrangle already existing TDA code in Python and adapt it for new plant cell data. 

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](https://doi.org/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](https://doi.org/10.1137/19M1241519) 

Mentor: Arun Suresh

Area: (Applied) Mathematics of Data Science

Overview of the topic: We will explore facets of unsupervised learning starting with clustering algorithms, and eventually develop a focus on linear algebraic techniques for dimensionality reduction like Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA) and Singular Value Decompositions (SVD). If time permits, we will possibly use these techniques in tandem with Random Forest Classifiers on real world highly unbalanced data.

Pre-requisites: A background in linear algebra is very helpful. Familiarity with Python is highly recommended.  

Text: Introduction to Statistical Learning Ch 6, 12 (with applications in Python) and Linear Algebra Done Right by Sheldon Axler (Ch 7E)

Mentor: Brent Koogler

Area: Optimization

Overview of the topic: Optimization algorithms, including theory and implementation. Many applications want to minimize a cost function. Doing so is subtle and hard. We start with simple concepts from optimization like gradient descent and Newton's method, and we will choose topics of interest to the student, e.g., equality and inequality constraints, real time control, large scale optimization. 

Pre-requisites: Preferably, background in linear algebra, calculus, and python. These topics will be reviewed in any case.  

Text: Numerical Optimization (Nocedal and Wright)