Mentor: Geraldo Soto-Rosa (Fourth-year Graduate Student, Applied Mathematics)

Project Description: Many problems in science and engineering involve quantities that change over time, from the motion of physical systems to the spread of heat or waves. But how can we use a computer to simulate these changes, and how do we know whether the resulting approximation is reliable? We will explore numerical time-stepping methods, which allow us to approximate how a solution evolves by moving forward one time step at a time. We will compare explicit and implicit approaches and investigate how the choice of method and time-step size can affect the accuracy, stability, and computational cost of a simulation. Through mathematical analysis and computational experiments, we will see why some methods work well in certain situations while others can become inaccurate or unstable.

Sample text: 

[1] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd ed. Cambridge: Cambridge University Press, 2008.

Suggested Prerequisites: (APMA 0330, 0350 or MATH 1110) and (APMA 0340, 0360 or MATH 1120), some programming experience.

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

Project Description: As much as we'd like to think we live in a typical world where we know what to expect, at the end of the day, we all buy insurance in preparation for the worst. Companies do the same: before releasing the newest version of Claude, Anthropic has to make sure rare but problematic outputs will not be generated even after billions of queries. Biology also relies on rare events: conformational changes in biomolecules occur after long waits (on a molecular scale) for a rare thermal fluctuation or ligand-binding event to drive them over the energy barrier.

In this DRP, we will dive into some of the mathematical notions of rare events, introducing ideas in large deviations and extreme value theory. In addition to theory, we would also place an emphasis on how to compute rare events, which is notoriously difficult because, well, they don't occur often! Application areas can be tailored to participants' interests.

Sample Text:

For large deviations:

https://www.tandfonline.com/doi/abs/10.1080/10451120410001733845

For a more biochemically motivated text: 

https://epubs.siam.org/doi/abs/10.1137/070699500

For extreme value theory:

https://users.cms.caltech.edu/~adamw/papers/book-2020-01.pdf

Suggested Prerequisites: Two courses in probability and statistics (APMA 1650 or APMA 1200 or APMA 1690 or similar), experience in any computing environment.

Mentor: Ritvik Teegavarapu (First-year Graduate Student, Applied Mathematics)

Project Description: Models in applied mathematics often have input parameters that are uncertain, and observed data can be used to learn about these parameters and thereby improve predictive capability. The purpose of the DRP is to describe the mathematical and algorithmic principles of this area, with applications drawn from the physical, biological, and data sciences.

We will focus on the challenge of inversion, which is the process of extracting hidden information from noisy, indirect measurements. Unlike classical deterministic methods, we will adopt a Bayesian perspective where unknown parameters are treated as random variables. This approach allows us to not only find a best solution but also produce uncertainty quantification on the estimates.

This subject is expanding so quickly that it is impossible to give an account of all developments. However, the selection of topics and structure of the course aim to provide the foundational tools necessary to understand and contribute to research in these areas, as well as stimulate interest in the subject. 

Sample Text: 

Primary reference:  Andrew Stuart's first book (https://arxiv.org/abs/1810.06191), which is a theoretical introduction to the framework of Bayesian inverse problems and data assimilation.

Suggested Prerequisites: 

To gain the most from the DRP, fluency in probability (e.g., APMA 1650, 1660, 1690) and linear algebra (MATH 0520) is required. Some familiarity in analysis (e.g., MATH 1010, 1630, 1640) and basic coding (ideally Python) is recommended but not required.

Mentor: Jiuru Lyu (First-year Graduate Student, Applied Mathematics)

Project Description: Have you ever wondered how computers solve mathematical problems? How can a computer approximate the value of an integral, solve a system of equations, or simulate the motion of a planet? These questions lie at the heart of numerical analysis: the study of the behavior and performance of numerical algorithms. Although computers can carry out enormous numbers of calculations very quickly, many mathematical computations must ultimately be approximated using finite-precision arithmetic. Throughout the project, we aim to understand what makes these approximations reliable (or unreliable). 

We will begin with floating-point arithmetic and basic error analysis. Then, depending on the interests and backgrounds of the mentees, we will explore one or more topics such as solving linear systems, iterative methods, interpolation, numerical differentiation and integration, ordinary differential equations, root-finding, or optimization. Mentees will also implement several classical numerical algorithms in Python or MATLAB, using computational experiments to investigate their accuracy, efficiency, and stability. 

Sample text: 

Introduction to Scientific Computing using Matlab by Ian Gladwell, James G. Nagy, and Warren E. Ferguson, Jr.

Suggested Prerequisites: 

Familiarity with single-variable calculus is expected (MATH 100 or equivalent). Linear algebra (MATH 520 or APMA 260), or concurrent enrollment, is recommended. Some programming experience (e.g., APMA 160 or equivalent) is helpful but not required.

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

Project Description: Single cell transcriptomics, the readout of the gene expression of cells, has given us unprecedented resolution of cellular dynamics. Many principles of dynamical systems have been applied in reasoning through this wealth of complex, high-dimensional 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, applied dynamical systems, and biophysical modeling.

Suggested Prerequisites: A strong knowledge of differential equations from APMA 0350/0355, multivariable calculus, and probability from APMA 1650/1655 are important.

Mentor: Erin Okey (Third-year Graduate Student, Applied Mathematics)

Project Description: From invasive species to habitat destruction to climate change, mathematics can be applied not only to understand the conservation problems we face but to study possible solutions. In this project, we will explore some of the ways mathematics is being applied to the field of conservation.

We will begin with a brief literature review of some interesting mathematical approaches in conservation. This may include topics like Lotka-Volterra population models, metacommunity models, climate models, and species distribution models. The reading list is subject to change, but the sample papers below should give a sense of the literature review. Following this, we will explore a population model, as determined by the interest of the students, and attempt to implement it in Python. If time permits, we will also explore an extension of the model designed to represent some type of human intervention, as determined by the students. This project will require some familiarity with ODEs (ideally APMA 0350 or equivalent) and some limited experience with Python. 

Sample text: 

Species Distribution Modeling (Miller 2010), Identifying robust strategies for assisted migration in a competitive stochastic metacommunity (Backus and Baskett 2021)

Suggested Prerequisites: APMA 0350 (or equivalent), some experience with Python

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

Project Description: Due to the ubiquity of Partial Differential Equations (PDEs) across science and engineering, it is important to computationally approximate the solutions of PDEs. Many such algorithms have a well-established theory concerning convergence and error analysis. Meanwhile, recent AI-based approaches such as Physics-Informed Neural Networks (PINNs), created at Brown in 2017, continue to develop.

This DRP will cover recent literature on theoretical developments in PINNs, starting from the review paper (De Ryck & Mishra 2024). Emphasis will be placed on learning to independently digest and evaluate mathematical literature. Options for specialized topics include the Universal Approximation Theorem, error estimates of PDEs, regularization approaches for convergence, and/or implementation details.

Sample Text: 

Numerical analysis of physics-informed neural networks and related models in

physics-informed machine learning (De Ryck & Mishra 2024). https://arxiv.org/abs/2402.10926

Suggested Prerequisites: APMA 360 required, MATH 1010 relevant but optional.

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

Project Description: 

Tipping events has been a large area of study in dynamical systems as it answers a fundamental question: as we vary a parameter will a model change states, and can that change be reversed? When put into context, such as a model for Arctic sea ice, this tipping from one state to another (e.g. full ice coverage to no ice) can quickly be identified as a necessary phenomena to explore. Introduced in 2012, the most recent type of tipping to be discovered is rate-induced tipping (r-tipping), which looks at how varying the speed at which a parameter changes forces a system to different limiting states. We will start this project by reading the foundational papers on r-tipping to build and explore the theoretical framework. We will then apply this theory, with the help of computational tools, to a system of your choice to see if we can identify r-tipping in the model. Possible models/applications span climate, neuroscience, finance, sociology, and more!

Suggested Prerequisites: APMA 0350, some experience coding in Matlab or Python

Mentor: Noe Reyes Rivas (Fourth year Graduate Student, Applied Mathematics)

Project Description: Optimization arises in a variety of natural phenomena. For certain applications, such as portfolio optimization and population dynamics, we require that our observed variables satisfy conditions more general than equality constraints, such as inequality constraints (negative population sizes typically aren't interesting to study). As such, the necessity for accurate numerical methods to approximate solutions to constrained optimization problems is imperative. In this project, we will explore several numerical algorithms for constrained optimization, such as proximal point, the penalty method, and the augmented Lagrangian method, through a mathematical and numerical lens.

Sample Text: 

https://doi.org/10.1561/2400000003, https://doi.org/10.1016/C2013-0-10366-2

Suggested Prerequisites: APMA 0260, APMA 1160, experience with Python or MATLAB