Mentor: Hyunwoo(Will) Kwon (Fourth-year Graduate Student, Applied Mathematics)

Project Description: We observe water flowing every day, watch clouds drifting across the sky, and see swirling cream in the coffee we make each morning. However, despite witnessing these phenomena daily, our mathematical understanding of them remains limited. 

As an attempt to understand these phenomenon, we will explore the long-term behavior of fluids, focusing on the incompressible Euler equations. We will study several example of solutions to Euler equations and the modern theoretical framework to study those problems and study numerical evidence as well. 

Sample Text: https://www.dropbox.com/scl/fi/f9z707orxaetgu9yo6jy4/Notes.pdf?rlkey=7iqou1w0vtlpj2x0ra7r3295a&dl=0

Suggested Prerequisites: APMA 0350, APMA 0360, APMA 1330 (or equivalent), APMA 1360, APMA 2110, APMA 2120 (not necessarily)

Mentor: Ethan Brady (Third-year Graduate Student, Applied Mathematics)

Project Description: Many integrals that arise in practice cannot be computed exactly, so the field of Numerical Analysis seeks to reliably and efficiently approximate them. The tools for this task come from introductory calculus, which ironically focuses much more on exact solutions than approximation. A key trick to integrate any function is to approximate that function by a polynomial, then to integrate the polynomial exactly using antiderivative formulas. In effect, we can use calculus to approximate integrals even when calculus cannot exactly compute those integrals.

      This DRP will analyze the common methods for computationally approximating integrals, including midpoint, trapezoidal, Simpson, and Gauss rules. Depending on programming experience, the culmination will be to code an adaptive integration algorithm. This detailed study of integration will complement APMA 160 and preview APMA 1160, 1170, 1180, and 1690, which cover various aspects of Numerical Analysis.

Sample Text: Sample Text: An Introduction to Numerical Analysis by Suli & Mayers.

Suggested Prerequisites: AP Calculus BC

Mentor: Stanley Nicholson (Third-year Graduate Student, Applied Mathematics)

Project Description: Single cell transcriptomics  (sets of methods that measure gene expression (RNA levels) in individual cells) has given us unprecedented resolution of the dynamics of cells. Many principles of dynamical systems have been applied in reasoning through this wealth of data. In this DRP, we will explore methods rooted in mathematics, statistics, and biophysical models to answer—from data— questions of fate determination, cell state transitions, and gene-knockout. We will explore tools from stochastic dynamical systems, optimal transport, and generative modeling. We will explore some theoretical ideas in the paper (https://arxiv.org/abs/2102.09204) and applications to biological data (https://pubmed.ncbi.nlm.nih.gov/35108499/). The goal of the DRP is to explore the research landscape of trajectory inference and learn about relevant biophysical approaches.

Suggested Prerequisites: Familiarity with probability (APMA 1650/1690) and ordinary differential equations (APMA 0350). Background in chemistry/physics is helpful.

Mentor: Malindi Whyte (Third-year Graduate Student, Applied Mathematics)

Project Description: In this project, we will begin to cover the material presented in APMA 2190 (Graduate Dynamics 1). We will be rigorous and learn the analysis needed as we go but we will not assume a background in analysis. We will dive into topics that may have been covered in APMA 1360 but with full rigor and generality in addition to new topics. Topic selection will be very much guided by the interest of the undergrad(s) involved. We will spend half the semester on building up theory and then the second half on a more applied topic that the undergrad(s) will choose. Such applications could include economics, biology, sociology, neuroscience, physics and more!

Suggested Prerequisites: APMA 1360

Mentor: Daniel Chen (Third-year Graduate Student, Applied Mathematics)

Project Description: In competitions---biological, social, political---how often do we see rock-paper-scissors-type "cyclic" dynamics? In theory, they are generic---for example, a quarter of the triangles formed in a graph constructed by picking each directed edge independently forms a cycle. On the other hand, competitive systems in real-life are rarely cyclic; voters rarely prefer one candidate over another while lacking a dominant preference; one clear winner emerges from the many variants of Deepmind's AlphaGo agents; though contain cycles, food chains in the ecosystem are generally "chains" and not cycles; the most recent NBA championship was won by an overwhelming favorite and less by favorable seeding that promoted certain match-ups. In this DRP, we will investigate how hierarchies emerge from competitions using tools from probability, graph theory, and algebraic topology.

Suggested Prerequisites: Fluency in probability (ideally two courses in probability), fluency in linear algebra, basic familiarity in graph theory.

Sample text: https://epubs.siam.org/doi/abs/10.1137/20M1321012

Mentor: Kamaljyoti Nath (Postdoctoral Research Associate, Applied Mathematics)

Project Description: This project aims to systematically investigate how a neural network approximates data with varying frequency content. A systematic study will be carried out by considering synthetic datasets that contain single- and multiple-frequencies in data spanning a broad range of frequencies. We will compare different network architectures (e.g., MLP, Kolmogorov–Arnold Networks), activation functions, etc, to determine their sensitivity, generalization, and degradation patterns across the spectrum. Time-domain and frequency-domain representations will be tested to assess how neural networks approximate different frequencies. The results will provide a systematic study to understand the frequency-dependent behavior of neural networks.

Suggested Prerequisites: Basic knowledge of Python programming, deep learning library (e.g. TensorFlow, Pytorch)

Mentor: Pratyush Potu (Fourth-year Graduate Student, Applied Mathematics)

Project Description: Splines (or piecewise polynomials) are powerful mathematical tools with wide-ranging applications in computer graphics (animation, font design), data science (spline regression), machine learning (e.g. KANs), and engineering design (CAD models). This project will explore the theory splines from a mathematical perspective, with a focus on interpolation and approximation. Some topics will include the unique properties and construction of cubic splines, B-splines, and higher dimensional splines (if time permits). Based on student interest, we may also look into implementation of spline based methods in applications.

Sample Text: An Introduction to Spline Theory by Michael S. Floater

Suggested Prerequisites: Knowledge of Calculus at the level of Math 0100 and Linear Algebra (e.g. Math 520/540) required.

Mentor: Phuc Lam (Fourth-year Graduate Student, Applied Mathematics)

Project Description: Games and competitions has been such an important aspect of all cultures that when humankind examines themselves, a theory for games is fated to arise at some point. Economic game theory, which is probably synonymous with the term “game theory” in our daily use, was established as its own field with John von Neumann’s works in the early 19th century. However, there is a plethora of different games - to name a few, evolutionary game theory, mean field game theory, etc.

In this project, we will explore classical game theory in tandem with combinatorial games, which are typically two-player sequential games with perfect information available to all players (unlike, say, poker) and no chance moves (without, say, a dice). Chess, checkers, and Go are typical examples of such games. We will start with the first 4 chapters of “Game Theory - A Playful Introduction” by DeVos and Kent. Afterwards, depending on students’ interests, we may explore classical game theory in subsequent chapters or make a deeper dive into combinatorial game theory.

Sample Text: “Game Theory - A Playful Introduction” (DeVos, Kent) (link: https://bookstore.ams.org/stml-80)

Suggested Prerequisites: Some background in calculus/ linear algebra is helpful but not necessary.