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

Project Description: Drawing the line-of-best-fit is arguably the most fundamental problem in statistics. However, existing theory for linear regression has gone much beyond the canonical set up, and has informed us how to think about many modern statistical problems such as deep learning. This DRP, we will start by going through classical theory and interpretations of linear regression. Then, depending on the students interest, explore more "modern" aspects such as the double-descent phenomenon, implicit regularization, sketching algorithms. 

Sample text: "The Truth about Linear Regression" by Cosma Shalizi (https://www.stat.cmu.edu/~cshalizi/TALR/TALR.pdf). 

Suggested Prerequisites: Fluency in probability (e.g., APMA 1650, 1660, 1690) and linear algebra (MATH 0520) is recommended. Familiarity in analysis (e.g., MATH 1010, 1630, 1640) would be helpful, but not required.

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

Project Description: Single cell transcriptomics (experimental approaches that reveal the gene expression of individual cells) has given us unprecedented resolution into the dynamic nature of cell state. The mathematics of dynamical systems offers a rigorous way of approaching this problem, with increasing interest from biologists. 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. Some areas we can explore are stochastic dynamical systems, optimal transport, and generative modeling, although the project is open-ended. More concretely, we can 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 modeling cell state dynamics and trajectory inference from a mathematically rigorous biophysical approach.

Suggested Prerequisites: Familiarity with probability (APMA 1650/1690) and ordinary differential equations (APMA 0350). Some programming experience is important for the implementation of algorithms. Background in chemistry/physics is helpful.

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. Emphasis will be placed on rigorous proofs of error bounds and the manner in which these apply in practice. The culmination will be to code an adaptive integration algorithm.

Sample Text: An Introduction to Numerical Analysis by Suli and Mayers, linked here.

Suggested Prerequisites: Calculus I & II required. MATH 180, MATH 520, Python, and APMA 160 relevant but optional.

Mentor: Simone Tetu (Second-year Graduate Student, Applied Mathematics)

Project Description: Statistical mechanics is a branch of physics that aims to answer a critical question: in a world made up of trillions of invisible atoms and molecules, how can one predict macroscopic observations from the fundamental laws that govern microscopic particles? To this day, statistical mechanics offers a rich set of ideas and mathematical tools that can help us study many-body systems in fields ranging from astrophysics to glaciology.

In this DRP, we will go over the contents of a standard undergraduate course in statistical mechanics, adapted for mathematically inclined students. We will learn about the (thermodynamical) motivation and fundamental assumption of statistical mechanics, various concepts including entropy, Helmholtz free energy and phase transitions, and some topics in non-equilibrium statistical mechanics (diffusion, transport, fluctuation-dissipation theorem). Approaching the material from a physicist’s perspective, we will also develop some useful analytical skills for applied mathematicians. 

Depending on our pace and the students’ interests, we may additionally implement numerical simulations, delve into specific modern applications of statistical mechanics, or discuss related philosophical questions.

Suggested Prerequisites: Required: APMA 0350/55, APMA 0650, and multivariable calculus (APMA 0260 or MATH 0180/200). Preferred but **not required**: Some (not necessarily extensive) physics or chemistry background. For example, having taken a few college physics classes would be an asset. 

Mentor: Elham Kianiharchegani (Postdoctoral Research Associate, Applied Mathematics)

Project Description: Phase separation plays a central role in the design and performance of many modern materials, from polymers and alloys to battery components and biological systems. Accurately modeling and controlling these processes typically requires solving complex partial differential equations, which can be computationally expensive and challenging in high-dimensional or data-limited settings. In this project, students will explore physics-informed neural networks (PINNs) as a framework for modeling and controlling phase separation dynamics. PINNs integrate physical laws, expressed as PDEs, directly into neural network training, enabling data-efficient and physically consistent learning. Students will study representative phase-field models (such as the Cahn–Hilliard equation), implement PINNs to approximate their solutions, and investigate how these models can be used to inform or optimize material design.

Suggested Prerequisites: Programming experience in Python. Prior experience with deep learning frameworks such as PyTorch or TensorFlow. Familiarity with partial differential equations (e.g., APMA 0350 or APMA 0360), Multivariable calculus (e.g., APMA 260 or MATH 0180/200). Background in numerical methods, scientific computing, or probability/statistics (e.g., APMA 1650/1660) is helpful but not required.

Mentor: Additi Pandey (Third-year Graduate Student, Applied Mathematics)

Project Description: Turbulence is a phenomenon that we have all experienced, especially in airplanes. It remains one of the most challenging problems in fluid mechanics. Instabilities lie at the heart of turbulence leading to the transition to turbulence. In this DRP, we will cover linear instability theory as a first step towards investigating turbulence. We will study Kelvin–Helmholtz instability, Tollmien–Schlichting instability, Rayleigh stability criterion and transition to turbulence in a pipe flow. We will also study characteristics, energy cascades and statistical properties of turbulence. Time permitting, we will learn about closure models as well as the experimental and numerical ways to model turbulence.

Sample text: While I believe that consulting multiple sources often provides a wider perspective and deepens the understanding of a topic, I recommend the following:

1. Introduction to Theory and Applications of Turbulent Flows by  Frans T.M. Nieuwstadt, Bendiks J. Boersma and Jerry Westerweel

2. A First Course in Turbulence by H. Tennekes and J. L. Lumley

Suggested Prerequisites: APMA 0360 or APMA 0365 is needed, ENGN 0810 or some equivalent coursework on fluid mechanics is desirable but optional. 

Mentor: Ezra Seidel (Fourth-year Graduate Student, Applied Mathematics)

Project Description: An oscillator is a process which exhibits repeating periodic motion, such as the pendulum of a clock. If we have a collection of oscillators which interact with each other, this is called a system of coupled oscillators. Many phenomena in biology can be represented as systems of coupled oscillators, from neurons and heart cells to swarms of communicating fireflies. A surprising and counterintuitive feature of such systems is that they can exhibit spontaneous synchronization: beginning from a random state, they can, without any external guidance, end up in a state where all the oscillators are moving in unison. In this DRP, we will study how the phenomenon of synchronization can be analyzed mathematically. We will begin by learning the theory surrounding the classic Kuramoto model, a simple mathematical model in which synchronization can be proven to occur; from there we will move to more elaborate (and perhaps more interesting) cases such as complex networks of oscillators. We will touch on applications to biology to a greater or lesser degree depending on the interest of the student(s). We may also do a coding/simulation project depending on the interest and background of the student(s).

Suggested Prerequisites: APMA 0350, minor coding experience helpful but not necessary

Mentor: Kiran Chandrasekher (First year Graduate Student, Applied Mathematics)

Project Description: We’ve observed dye spread through water without any visible stirring or tiny particles dancing unpredictably under a microscope. These seemingly random motions are all examples of Brownian motion, a fundamental phenomenon arising from microscopic interactions between particles and their surrounding environment. We will start by studying Brownian motion simulations, focusing on how stochastic models capture random particle dynamics. (Sample text on Brownian motion: https://empslocal.ex.ac.uk/people/staff/NPByott/teaching/FinMaths/2006/Brownian.pdf) 

We will then apply these ideas to problems in biology and chemistry, where Brownian motion plays a central role in how molecules move and interact. In biology, it describes how proteins and other molecules wander through the crowded interior of a cell, how enzymes find their targets, and how substances spread within and between cells. In chemistry, Brownian motion helps explain how molecules mix in solution, how chemical reactions occur when reactants randomly encounter each other, and how small particles assemble into larger structures. We will examine examples from the literature, beginning with a recent review (https://www.sciencedirect.com/science/article/pii/S0304416524001831) and the works it cites, to understand how simulations are used to study these systems in practice.

Suggested Prerequisites: APMA 1650