Exploring AD vs FD in Machine Learning (01) 

Mentor: Aniruddha Bora (Postdoctoral Research Associate, Applied Mathematics)

Students: Anand Advani, Jay Philbrick, Daniel Xiong

Project Description: The key idea of the project is to explores the advantages and limitations of Finite Difference (FD) methods and Automatic Differentiation (AD) within the context of deep learning. FD methods are widely used for numerical differentiation in physics-based models and partial differential equation solvers, while AD provides precise gradients crucial for training neural networks. The study will compare the computational efficiency, accuracy, scalability, and applicability of these techniques across various domains, including scientific machine learning, optimization, and hybrid physics-ML modeling. Key deliverables include benchmarks, best practices, and guidelines for integrating these methods into deep learning workflows.

Suggested Prerequisites: I can suggest a few pre-reqs: Math 0090/0100 (Single Variable Calculus), Math 0520 (Linear Algebra) and programming experience in Python.

Spectral Graph Methods (02)

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

Students: Shayaan Chaudhary, Sophia Huang, Grace Wang

Project Description: Certain data sets can be represented as graphs. For example: social networks or neurons in the brain. These graphs can then also be associated with matrices. Spectral Graph methods are techniques which use these matrices in clever ways to solve relevant problems you might be interested in with regards to the original data. 

One good example is the PageRank algorithm used in the Google search engine. The key idea in the algorithm is the representation of the internet as a directed graph where webpages are "vertices" and links to and from webpages are "directed edges". The algorithm then determines a ranking through an iterative process involving a matrix derived from the connectivity of the graph. In fact, there are many other spectral graph methods with wide applications in machine learning and data analysis! The goal of this project will be to explore these methods and applications, analyze them, and per student interest, implement them.

Sample Text: Lecture Notes on Spectral Graph Methods by Michael W. Mahoney.

Suggested Prerequisites: The only strict prerequisite is Linear Algebra (MATH 0520 or 0540). Knowledge of graph theory (e.g. APMA 1860) would also be useful.

Stochastic Thermodynamics (03)

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

Students: Emily Hamp, Shivam Kogar, Megan Tanuwidjaja

Project Description: Traditional thermodynamics continues to play a crucial role in everyday science and engineering. However, it is an inherently equilibrium theory. This DRP explores stochastic thermodynamics, a recent and exciting framework extending classical thermodynamic concepts—such as work and entropy production—to individual trajectories of non-equilibrium systems. Our focus will be the chemical master equation and Langevin equation which provide such a generalization and are ubiquitous in chemistry, statistical physics, biology, and—more recently—machine learning. The goal of this DRP is to understand the relationship between these equations with the physical concepts of free energy, Gibbs distributions, fluctuation theorems, and entropy production. Depending on the student’s interests, we may explore more model-specific applications such as in biology or chemistry or mathematical topics like the Fokker-Planck partial differential equation. This DRP offers a rich set of ideas for students with a background in probability and differential equations.

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

Geophysical Fluid Dynamics (04)

Mentor: Lulabel Ruiz Seitz (Third-year Graduate Student, Applied Mathematics)

Student: Raymond Zhong

Project Description: Geophysical fluid dynamics (GFD) is -- you guessed it -- the sub-field of fluid dynamics concerning naturally occurring flows. This DRP project may be more applied compared to other projects, but there are still many interesting theoretical aspects. GFD is built on partial differential equations, the most important of which are the Navier-Stokes equations. Since in most situations in which we want to use these equations, no analytical solution is known, there are alternative ways to analyze this system. One of the most important ways is to use asymptotic methods. Another is to consider interactions between waves, using equations derived from Navier-Stokes. The first goal of this project will to have a physical and mathematical understanding of the Boussinesq equations, a commonly used approximation of Navier-Stokes. The goal after that will depend on the interest of the student. Some of the more applied possibilities include: understanding common physical balances used as further approximations to Navier-Stokes, fluid simulations using spectral methods (we would use the Python package Dedalus), and an overview of current methods and challenges in climate modeling. Some of the more theoretical possibilities include: asymptotic methods in GFD, possibly with an end goal of going through the derivation of quasigeostrophy (which is a beautiful and useful result), a discussion of fast singular limits, Lagrangian fluid dynamics, and a discussion of the turbulence closure problem.

Suggested Prerequisites: MATH 0180, 0520 and APMA 0350, 0360 (or equivalent), and APMA 1330. APMA 1180 would be good if the student want to do simulations but is not necessary. No physics background is required, but a desire to do physically-motivated problems is necessary. 

Nonparametric Bayes (06)

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

Students: Avery Guo, Chai Harsha, Sadia Qazi

Project Description: You're thrown onto a large but deserted island for a year and tasked with classifying all animals you encountered. You want to distinguish and keep track of all the species while avoiding a cap on the number of species that you could see. How do you do this? This project aims at exploring nonparametric Bayesian methods. We will put a particular emphasis on Dirichlet processes and their application in density estimation and clustering. We begin with an introduction in parametric Bayesian statistics before moving onto its nonparametric cousin, discussing modeling and computational issues involved in both. The project aims at exposing students to important concepts and techniques in Bayesian statistics such as exchangeability, conjugacy, MCMC, and (if time permits) frequentist guarantees.

Sample Text: 

• Bayesian Data Analysis by Gelman, Carlin, Stern, Duson, Vehtari, and Rubin.

• Lecture Notes on Bayesian Nonparametrics by Peter Orbanz.

Suggested Prerequisites: A first course in probability and statistics (e.g., APMA 1080, 1650, 1690), preferably prior exposure to Bayesian statistics and/or stochastic processes. 

Computable Structure Theory (07)

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

Student: Ari Wang

Project Description: Computable structure theory is a branch of mathematical logic that studies the relationships between computable functions and structures within formal systems. It seeks to understand which mathematical structures can be described or manipulated using computational methods, specifically focusing on the concept of "computability" in the context of structures such as graphs, groups, rings, and other algebraic or relational systems. The goal of this project is to expose students to the foundational concepts and techniques of computable structure theory and to tackle a specific problem within the field, helping students develop their mathematical reasoning, critical thinking, and problem-solving skills in the context of computability theory.

Sample Text: 

• Weber, Rebecca. Computability theory. American Mathematical Soc., 2012

• Soto-Rosa, G., Ocasio-González, V. A characterization of strongly computable finite factorization domains. Arch. Math. Logic (2024). 

Suggested Prerequisites: No theoretical knowledge is required, but the student should be comfortable writing and understanding proofs.

The Long Time Behavior of Fluids (08)

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

Students: Lloyd Sangwoo Ko, Lean Quimbo

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: After reviewing basic ODE theory, we will study selected sections from the excellent lecture notes by V. Šverák [link] and T. Elgindi ("Mixing, Enhanced Dissipation, and Stationary Euler States").

Suggested Prerequisites: 

• APMA 0350, APMA 0360, APMA 1330 (or equivalent)

• APMA 1360, APMA 2110, APMA 2120 (not necessarily)

Spatial Point Processes (09)

Mentor: Michel Davydov (Postdoctoral Research Associate, Applied Mathematics)

Students: Jennifer Shim, Leo Zhang (APMA DRP Best Poster Award)

Project Description: Point processes are a class of stochastic processes describing random configurations of points. Whether it be tree patterns in forests, neural spiking times in neuroscience modeling, crystallization patterns, communication networks or many others, phenomena of interest in applications can be modeled by various kinds of point processes on different spaces. The goal of this DRP project is to explore select aspects of point process theory. After an initial phase of reading relevant sections from S. Resnick's Adventures in Stochastic Processes to get an overview of the theory, the student will have an opportunity to deepen their study of one or more of the following, depending on their interests and knowledge: simulation of point processes using Poisson embeddings; stationary point processes and Palm theory; Voronoi tessellations; determinantal point processes, or other related topics from the field of stochastic geometry.

Sample Text: Adventures in Stochastic Processes (Resnick)

Suggested Prerequisites: Basic probability course such as APMA 1650 or equivalent. Any advanced probability course such APMA 1690 or 1740, or measure theory knowledge, is appreciated but not required 

Asymptotic and Perturbation Methods in Applied Mathematics (10)

Mentor: Ryan Creedon (Prager Assistant Professor of Applied Mathematics and NSF Mathematical Sciences Postdoctoral Fellow)

Students: Cassidy Charles, Calvin Eng, Jason Wu

Project Description: Hard problems are abundant in math. Take, for example, a quintic polynomial equation. Unless the coefficients are cherry picked in just the right way, a powerful theorem in abstract algebra tells us that there is no hope of obtaining closed-form solutions. Yet, if one of these coefficients is relatively small or large compared to the others, one can obtain approximate formulas for the solutions that are stunningly accurate and, most importantly, do not require numerical methods to obtain! The means of obtaining these scarily good formulas are called asymptotic and perturbation methods, the so-called "dark arts" of applied mathematics.

Asymptotic and perturbation methods find their usefulness not just in abstract math problems, but across a wide range of applications in science and engineering. Fluid dynamics, quantum theory, cell dynamics, airfoil design---all of these areas and more have hard mathematical problems with no closed-form solutions. However, in regimes where a small or large parameter exists, suddenly we can apply asymptotic and perturbation methods to obtain formulas that well-approximate solutions. The goal of this reading project is threefold: (1) to introduce students to asymptotic and perturbation methods,  (2) to apply these methods to a problem of students' interests, and (3) to broaden students' problem solving strategies using methods complementary to numerical methods. 

Sample Text: Preferred Texts: (1) Introduction to Perturbation Methods by M. H. Holmes, (2) Advanced Mathematical Methods for Scientists and Engineers by C. Bender & S. Orszag 

Suggested Prerequisites: MATH 180 (Required), MATH 520 (Required), APMA 0350 (Required), APMA 0360 (Preferred), APMA 1330 (Preferred)

Reaction-Diffusion Pattern Formation (11)

Topics in Combinatorial Optimization (01)

Mentor: Teressa Chambers (Sixth-year Graduate Student, Applied Mathematics)

Student: Michael Kratzer

Project Description: The field of combinatorial optimization, which focuses on finding the ideal solution to a problem from a finite set of discrete possibilities, is critical to many modern logistical problems. Whether the goal is to assign time slots for exams such that no student has multiple exams at the same time, allocate packages to delivery vehicles for optimal route times without overpacking the vehicles, or optimize the placement of a limited number of cell towers to maximize coverage, the algorithms and methodologies of combinatorial optimization provide reliable ways to find the best solution – or at least very nearly the best. Linear programming is perhaps the most widely-known form of combinatorial optimization, but it is far from the only approach. The central idea uniting all techniques is that they provide a more computationally sustainable way to seek an optimal solution from the finite, discrete set of possibilities than simply evaluating each option to see which yields the best results.

For this project, we will start with the textbook A First Course in Combinatorial Optimization by Dr. Jon Lee. The first section of the book deals with linear programming, which we will examine as an entry point for some of the ideas underlying combinatorial optimization. We will then move into an introduction to matroid techniques and graphical analyses, at which point the precise direction of the project can be tailored to the student’s interests. It is likely that this project will primarily be a reading project rather than a research project, with the goal being a full understanding of high-level concepts in the field and potentially some analysis of current applications.

Sample Text: A First Course in Combinatorial Optimization (Jon Lee). It can be accessed online through the Brown Library, but the student may want to purchase their own copy for notation and reference (Newman).

Suggested Prerequisites: MATH 0520, APMA 0340

Multi-armed Bandits (02)

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

Students: Mia Busuladzic-Begic, Jennifer Shim, Sabrina Shipley

Project Description: Among the numerous colorful slot machines at a casino, each with different probabilities of winning (or losing) money, which machines should you choose to maximize reward and minimize regret? With countless (actually useful) applications outside of gambling, Multi-armed Bandit problems have become the foundation of sequential decision-making (reinforcement learning) and online learning. This reading program will explore various bandit algorithms with a strong emphasis on the probabilistic tools to put the algorithms on a rigorous footing. By the end, students can expect to learn to balance exploration and exploitation, mathematically justify their algorithms, and gain exposure to probability and related fields such as game theory, statistics, and information theory.

Sample Text: Bandit Algorithms (Lattimore and Szepesvari)

Suggested Prerequisites: A course in probability and familiarity with analysis of algorithms, e.g., APMA 1650 + CSCI 1570, APMA 1690

Stochastic processes and their applications (03)

Mentor: Michel Davydov (Postdoctoral Research Associate, Applied Mathematics)

Students: Erica Brown, Yunxi (Harper) Liang, Qiushi Yu

Project Description: Stochastic processes represent a ubiquitous theoretical tool to model phenomena across different applicative fields that involve variability in time. The goal of this DRP is to explore this fundamental component of modern probability theory through example models. After an initial phase of selected readings from S. Resnick's Adventures in Stochastic Processes to get an overview of the theory and motivating examples, the student will have the opportunity to deepen their understanding of some of the tools of stochastic processes - such as renewal theory, point process theory, branching processes or random walks - and use them on models from their favorite domain among queuing theory, genetics, sociology, neuroscience,... 

Sample Text: Adventures in Stochastic Processes (Resnick)

Suggested Prerequisites: Basic probability course such as APMA 1650 or equivalent. Any advanced probability course such APMA 1690 or 1740, or measure theory knowledge, is appreciated but not required.

Topics in Random Matrix Theory (04)

Mentor: Jake Mundo (Fifth-year Graduate Student, Applied Mathematics)

Students: Shivam Kogar, Sofia Tazi (APMA DRP Best Poster Award)

Project Description: Random matrix theory, which concerns matrices whose entries are random variables, has numerous applications throughout mathematics, statistics, and physics. A typical question of random matrix theory may be something like the following: suppose the entries of a matrix X are independent random variables distributed according to the standard normal distribution; then how are the eigenvalues of X distributed? One can vary this question in countless ways: What if the matrix is required to be symmetric? What if the matrix is somehow evolving over time? What happens as the size of the matrix grows to infinity? This directed reading will explore questions like these; we will begin by building the foundations of the theory and discussing classical random matrix ensembles, and then explore further topics, including applications, according to student interest. Some familiarity with linear algebra and probability will be useful for this directed reading; beyond that, the content of this directed reading is very flexible according to student background. 

Sample Text: Introduction to Random Matrices (Livan, Novaes, and Vivo)

Suggested Prerequisites: MATH 520 (Linear Algebra) or equivalent, APMA 1650 (Probability) or equivalent

Randomized Numerical Linear Algebra (05)

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

Student: Jesus Rodriguez

Project Description: Numerical Linear Algebra (NLA) is the field of math regarding the application of linear algebra through computation. Considering how general that statement is, it is no surprise that the subject is of particular importance to a large number of other fields including but not limited to computer graphics, signal processing, scientific computing, computational finance, and bioinformatics. Hence, a large amount of research has gone into the development and improvement of NLA algorithms. More recently, in the last 15 years, a promising set of techniques now referred to as Randomized Numerical Linear Algebra (RNLA) has risen in popularity. The key idea is that by adding a certain amount of noise, and allowing for a slight loss of precision, fundamental NLA problems such as matrix multiplication, least-squares approximation, and low-rank matrix approximation, etc. have seen significant improvements in efficiency.

The aim of this project will primarily be to read and understand the review articles listed below. Based on student interest, we can either focus more on the theory and proofs for the important results or the implementation of the techniques.

Sample Text: 

• Lectures on Randomized Numerical Linear Algebra (Drineas and Mahoney)

• Randomized Numerical Linear Algebra: A Perspective on the Field With an Eye to Software (Murray et al.)

Suggested Prerequisites: Knowledge of MATH 0520 (or 0540) and APMA 1650 are necessary. Knowledge of APMA 1170 is (highly) recommended, and some knowledge of APMA 1690 could be useful.

Opinion Formation on Networks (06)

Probabilistic Programming (01)

Mentor: Aaron Kirtland (Second-year Graduate Student, Applied Mathematics)

Students: Jack Cheng, Lucas Gelfond

Project description: This DRP will focus on probabilistic programming, a topic that promises to bridge artificial intelligence, machine learning, cognitive science, and programming languages. We will begin with an introduction to the topic in van de Meent et al., then decide if we want to continue following that textbook and gradually build up to a differentiable language capable of supporting efficient deep generative models. Alternatively, we could read a wider range of papers about implemented probabilistic programming languages (PPLs), their applications, and the philosophy behind why to use them. We can also implement models in PPLs like Church, WebPPL, and Gen, or work towards implementing our own PPL.

Main text/reading list:

• An Introduction to Probabilistic Programming (van de Meent et al.)

• MIT probabilistic computing project reading list

Other texts/reading lists:

• Probabilistic models of cognition, a text introducing PPLs that ties them to cognitive science

• An Introduction to Bayesian Inference, Methods and Computation (2021) by Nick Heard, another book on theory

• Probabilistic Programming and Bayesian Methods for Hackers, a book emphasizing implementational details

• The Bayes Way, a reading list

Theory and Application of Networks (02)

Mentor: Teressa Chambers (Fifth-year Graduate Student, Applied Mathematics)

Student: Naomi LeDell 

Project description: Networks are a highly versatile and widely applicable mathematical structure for modeling complex systems from food webs to shipping lanes to online social connections. This project is intended to introduce the theoretical foundations of network science, beginning with the graph theory underlying our understanding of networks and moving on to the practical question of how to construct and analyze network models. The initial phase will be facilitated through selected readings from “Networks: An Introduction” by M. E. J. Newman, which provides consistent practical context alongside the theoretical foundations. Once these tools have been established, the student will have the opportunity to construct a network model for a system of interest to them, and we will work together to ask and answer interesting questions about their network.

A solid background in linear algebra will be necessary for this project, but otherwise there is very little prerequisite knowledge. The reading for this project will involve both textbook selections and papers in network science, so students are advised to have some experience with mathematical writing as well as proof structures. Basic familiarity with MATLAB is also strongly recommended.

Sample text: Networks: An Introduction (M. E. J. Newman)

Prerequisites: MATH0520 Linear Algebra and ideally an exposure to an upper level APMA or MATH class for example: APMA1210 Operations Research, MATH1010 Real Analysis or MATH1530 Abstract Algebra.

Introduction to Fractional Calculus (03)

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

Student: Austine Zhang

Project description: When learning calculus, one learns to take derivatives and integrals and explores the effect of these operations on various types of functions. However, one interesting question one might consider is "Can I take half a derivative of a function?". What about pi'th derivatives? Imaginary order derivatives? From these questions the theory of fractional calculus was developed. At the time of development, very few applications were found for the theory. However, in more recent years fractional order models of physical phenomena have become more widespread. Applications have been found in physics, chemistry, biology, and economics for instance.

In this project, the student will learn the definitions of fractional derivatives and fractional integrals and how to compute them. Note that these operators are referred to as "fractional", but have actually been extended to irrational and complex orders as well. From these definitions, applications will be explored as per student interest.

Suggested Prerequisites: A very solid understanding of calculus and having taken APMA 0350 and/or 0360 would be helpful but not necessary.

Sample text: The Fractional Calculus (Oldham, Spanier)

The Probabilistic Method: a non-constructive approach (04)

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

Students: Anand Advani, Caroline Zhang

Project Description: How do we show that a mathematical object with certain properties exists without constructing an explicit example? For example, loosely speaking, without picking, can we know if it is possible to pick out a very small group of people in a party so that everyone else in the party knows at least one person in the group? With the "constructive approach", we either explicitly construct an example or devise a method to do so. The probabilistic method, on the other hand, is a "non-constructive approach". With this method, one shows that if we randomly choose objects from a specified class, we can choose the desired object with (strictly) positive probability, thus there must be an instance where this desired object appears, and so it exists.

Though the idea is simple, the method is actually quite powerful. Developed by Paul Erdos and initially used in combinatorics and graph theory, the method has since yielded non-trivial and rich applications in computer science, real analysis, number theory, and even in fields as far removed from probability as algebraic number theory.

We will begin by reading the first 5 chapters of "The Probabilistic Method" by Noga Alon and Joel H. Spencer, introducing motivations and examples of this method, and a few basic techniques. Afterwards, depending on student's interests, we can either learn other techniques beyond the first 5 chapters or explore a few specific applications of this method. Students are assumed to be comfortable with calculus, probability, and enumeration (i.e. counting permutations, combinations, partitions, etc). Familiarity with graph theory, number theory, and/or algorithms is not required, but would be useful.  

Prerequisites: APMA1650 Statistical Inference I or MATH1610 Probability or equivalent

Textbook: The Probabilistic Method (Alon and Spencer)

Constructive Approximation of Functions (05)

Mentor: Wenjun Zhao (LFZ Assistant Professor, Applied Mathematics)

Student: Benjamin Shih

Project description: Approximation theory is an established field that is concerned with how functions like exp(x), sin(x) can be approximated simpler functions such as polynomial or rational (ratio of polynomial) functions. The objective is to characterize the errors quantitatively, and make the approximation as close as possible to the actual function. Through the program, we will learn about the classical ideas in this field, illustrated by corresponding numerical examples.

Prerequisites: single-variable calculus. The beginning of this program may overlap a bit with APMA 0160. Programming experience with MATLAB will be also helpful, as lots of illustrations were done with MATLAB. 

Textbook: Approximation Theory and Approximation Practice (Lloyd N. Trefethen). 

Observing the Universe at the Quantum Scale (06)

Mentor: Victoria Antonetti (Third-year Graduate Student, Applied Mathematics)

Student: Cerulean Ozarow

Project description: Black body radiation confounded classical physicists. A black body is an object that absorbs all energy that is transferred to it. In the 1850s, Gustav Kirchhoff challenged physicists of his time to find the form of the energy of such an object knowing that its form depended only on temperature and frequency. Solutions to the problem worked in certain frequency (energy) regimes but not on the whole until the breakthrough of Max Plank, in which he assumed that energy could be thought of as a discrete variable made up of what he then labeled quanta. Here begins the story of quantum mechanics. The notion that science appears to behave differently at different scales is a profound one throughout the different fields of science from pattern scaling and formation in living systems to the mechanics of matter. In this reading project, we will focus on developing some of the mathematical tools that allowed quantum physicists in the second wave of quantum mechanics, the quantum we learn today, to address the scaling problem and understand phenomena that the formalism of classical mechanics failed to capture. As we develop some of the mathematics of quantum mechanics, which ranges from functional analysis to linear algebra to differential equations, we will consider some of the foundational triumphs of the theory like the Heisenberg uncertainty principle and solutions to the Schrödinger equation. 

Sample Text: Quantum Mechanics, Concepts and Applications, 2nd edition (Nouredine Zettili)

Suggested Prerequisites: Classical mechanics + linear algebra 

Evolutionary Game Theory (07)

Theory and Applications of ODEs (01)

Mentor: Katie Slyman (Postdoctoral Research Associate, Applied Mathematics)

Student: Alessandra Darcy

Project description: A dynamical system, loosely, is a system of physical quantities that evolves over time. Many dynamical models of complex phenomena in biology, chemistry, and physics are nonlinear. This subject is a natural follow-up to APMA 0350 or MATH 1110, where we now study the behavior of solutions to non-linear ordinary differential equations, often in a highly qualitative manner. The goal of this project is to learn the basics of dynamical systems from a qualitative and applied perspective, building up to an application at the end. We will read various parts of Strogatz’s text, primarily portions of chapters discussing 1D and 2D flows (with a focus on stability analysis, phase portraits, and bifurcation theory), along with different or additional topics depending on the interest level. Since modern applied mathematics is intrinsically interdisciplinary, we will apply the learned techniques to a model of the oceanic carbonate cycle, a model of arctic sea ice, or an infectious disease model. The preferred prerequisites are some knowledge of eigenvalues and eigenvectors, and some programming experience.

Prerequisites: MATH0180 and APMA0350 or equivalents and ideally MATH0520 or equivalent.

Optimal Transport: Theory and Algorithms (02)

Mentor: Sicheng Liu (Third-year Graduate Student, Applied Mathematics)

Student: Yuechuan Yang

Project description: Imagine a worker on a construction site holding a shovel, tasked with moving a large pile of sand. The worker's goal is to shape that sand into a specific form, like a giant sandcastle. The worker wished to minimize the total effort expended, which can be measured by factors such as the overall distance traveled or the time spent carrying shovelfuls of sand. The above scenario is an informal illustration of the optimal transport (OT) problem. Mathematicians are interested in the properties of the least costly transport, as well as in its efficient computation. The least costly transport not only gives a way to compare two probability distributions - two different piles of sand of the same volume, but also entails a rich geometric structure on the space of probability distributions, which gives rise to many active research questions.

In this DRP we will focus on learning the mathematical foundation of OT under discrete measure settings, as well as the efficient practical algorithm for solving OT. We will mainly follow the book “Computational Optimal Transport” by Gabriel Peyré and Marco Cuturi. Time permitting, we can also explore the modern application of OT to data science or go deeper towards the theory of OT, depending on the student's interest.

Prerequisites: APMA1650 Statistical Inference I or equivalent, and ideally exposure to another upper level probability course for example: APMA1720, APMA1740 or APMA1200.

The Probabilistic Method (03)

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

Student: Luis Gomez

Project description: The probabilistic method is a non-constructive technique for proving the existence of a structure with given properties. One defines an appropriate probability space, then shows that the desired structure exists with positive probability, thus there must be an instance where this structure appears. Though the idea is simple, the method is actually quite powerful. Developed by Paul Erdos and initially used in combinatorics and graph theory, the method has since yielded non-trivial and rich applications in computer science, real analysis, number theory, and even in fields as far removed from probability as algebraic number theory. We will begin by reading the first 5 chapters of "The Probabilistic Method" by Noga Alon and Joel H. Spencer, introducing motivations and examples of this method, and a few basic techniques. Afterwards, depending on the student's interests, we can either learn other techniques beyond the first 5 chapters or explore a few specific applications of this method. Students are assumed to be comfortable with calculus, probability, and enumeration (i.e. counting permutations, combinations, partitions, etc). Familiarity with combinatorics, graph theory, and/or algorithms is not required, but would be useful. 

Sample text: The Probabilistic Method (Alon, Spencer) 

Suggested Prerequisites: APMA1650 Statistical Inference I or MATH1610 Probability or equivalent.

Introduction to Mathematical Fluid Mechanics (04)

Mentor: Qian Zhang (Third-year Graduate Student, Applied Mathematics)

Student: Alexander Mark

Project description: Fluid mechanics is important in engineering applications and has great value in physical science and mathematics. It pioneers the development of applied math and remains a keystone of modern applied mathematics. This project aims to demonstrate how to use mathematical tools to model engineering problems related to fluid mechanics, and how to analyze the physical phenomena that arise from these problems by deduction and computation. For the fluid part, we will try to cover the governing equations of fluid mechanics, vortex, potential flow, boundary layer, and shocks. For the mathematical part, the student will learn to deal with a considerable amount of mathematical deduction, and obtain some taste for applied PDEs and continuum mechanics.

The focus will be flexible depending on the student's background and interests. Prerequisites would be limited to multivariable calculus, linear algebra, and basic Newtonian mechanics. Basic complex analysis is strongly encouraged but not required.

Sample text: We will use A Mathematical Introduction to Fluid Mechanics (Chorin and Marsden), but not the full book: we will cover Chapter 1, half of Chapter 2 and half of Chapter 3. If time permits, we may have one more advanced topic in either compressible or incompressible flow.

Suggested Prerequisites: APMA0360 Applied PDEs, MATH0200/0350 Multivariable Calculus, MATH0520/0540 Linear Algebra and ideally PHYS0030/0050 Basic Physics A 

The Dynamics of Language: Competition and Spread (05)

Mentor: Erik Bergland (Fifth-year Graduate Student, Applied Mathematics)

Student: Laura Romig

Project description: Language is, without question, one of humanity’s most crucial tools. It has been with us since the beginning, always changing and evolving to fill the needs of our communities. Today, humanity boasts over 6,500 spoken languages. With so many different ways to communicate in an increasingly interconnected world, a natural question arises: what factors determine a region’s dominant language?

Unsurprisingly, this question has been taken up by mathematicians in recent years. Drawing on insights from mathematical epidemiology, dynamical systems and partial differential equations have allowed researchers to make substantial progress. This project will roughly fall into 3 stages. First, we will review literature on past models. Then, we will select a subset of these models and perform numerical simulations to gain understanding of the predictions they make. Finally, we will consider the assumptions these models make and attempt to propose modifications, evaluating the choices numerically. Students are encouraged to analyze both theirs and the original assumptions of the models through the lens of social justice or fairness. Familiarity with both ordinary and partial differential equations is assumed. Experience with MATLAB or other mathematical computing software is useful but not required.

Sample Text: Modeling the dynamics of language death (Abrams & Strogatz), A reaction-diffusion model for competing languages (Walters)

Suggested Prerequisites: APMA0350 Applied ODEs and APMA0360 Applied PDEs

Theory and Application of Networks (06)

Mentor: Teressa Chambers (Fifth-year Graduate Student, Applied Mathematics)

Student: Marwan Ali

Project description: Networks are a highly versatile and widely applicable mathematical structure for modeling complex systems from food webs to shipping lanes to online social connections. This project is intended to introduce the theoretical foundations of network science, beginning with the graph theory underlying our understanding of networks and moving on to the practical question of how to construct and analyze network models. The initial phase will be facilitated through selected readings from “Networks: An Introduction” by M. E. J. Newman, which provides consistent practical context alongside the theoretical foundations. Once these tools have been established, the student will have the opportunity to construct a network model for a system of interest to them, and we will work together to ask and answer interesting questions about their network.

A solid background in linear algebra will be necessary for this project, but otherwise there is very little prerequisite knowledge. The reading for this project will involve both textbook selections and papers in network science, so students are advised to have some experience with mathematical writing as well as proof structures. Basic familiarity with MATLAB is also strongly recommended.

Prerequisites: MATH0520 Linear Algebra and ideally an exposure to an upper level APMA or MATH class for example: APMA1210 Operations Research, MATH1010 Real Analysis or MATH1530 Abstract Algebra.

An Exploration of the Calculus of Variations (07)

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

Project description: What is the shortest path between two points on Earth? What about the shape of bubbles inside different shapes? The Calculus of Variations is a particular extension of calculus that allows us to answer these questions and many more. In particular, all equations of motion that you learn in a Mechanics course can be explained using the calculus of variations and we can even get more insight into the physics of various physical phenomena. Also, the principles of dynamic programming and optimal control are intimately related to the calculus of variations as well. Finally, the calculus of variations was actually the inspiration for the sci-fi movie Arrival!

This project will begin with an introduction to the calculus of variations assuming only knowledge of calculus. Then, based on the student(s)' interest, we can explore more into particular interesting applications.

Prerequisites: APMA0350 Applied ODEs and a strong background in calculus.

Mathematical Models of Disease Spreading (08)

The Dynamics of Language: Competition and Spread (01)

Mentor: Erik Bergland (Fourth-year Graduate Student, Applied Mathematics)

Students: Emma Berman, Jiayuan Sheng

Project description: Language is, without question, one of humanity’s most crucial tools. It has been with us since the beginning, always changing and evolving to fill the needs of our communities. Today, humanity boasts over 6,500 spoken languages. With so many different ways to communicate in an increasingly interconnected world, a natural question arises: what factors determine a region’s dominant language?

Unsurprisingly, this question has been taken up by mathematicians in recent years. Drawing on insights from mathematical epidemiology, dynamical systems and partial differential equations have allowed researchers to make substantial progress. This project will roughly fall into 3 stages. First, we will review literature on past models. Then, we will select a subset of these models and perform numerical simulations to gain understanding of the predictions they make. Finally, we will consider the assumptions these models make and attempt to propose modifications, evaluating the choices numerically. Students are encouraged to analyze both theirs and the original assumptions of the models through the lens of social justice or fairness. Familiarity with both ordinary and partial differential equations is assumed. Experience with MATLAB or other mathematical computing software is useful but not required.

Sample Text: Modeling the dynamics of language death (Abrams & Strogatz),  A reaction-diffusion model for competing languages (Walters)

Suggested Prerequisites: APMA 350, APMA 360

Mathematics of Chemical Reaction Networks (02)

Mentor: Nidhi Kaihnsa (Prager Assistant Professor, Applied Mathematics)

Student: Greta Brablec

Project description: In the past half century Chemical Reaction Network Theory (CRNT) has seen tremendous growth. This theory focuses on developing methods and mathematical models to study the behaviour of (bio)chemical networks. Primarily, it is concerned with the evolution of the quantity of reaction species over time. Under certain ``nice"(mass-action kinetics) conditions, this evolution for a deterministic system can be modelled using polynomial dynamical systems. This often allows us to develop and use sophisticated techniques from algebra (real algebraic geometry) to study their behaviour such as stability and oscillations. 

This project will first provide a gentle introduction to the framework of reaction networks. After getting familiar with the basic setup, we will next look at the dynamics of continuous time and state deterministic models, particularly from an algebraic point of view.

Sample Text: Reaction Kinetics: Exercises, Programs and Theorems (János Tóth, Attila László Nagy, Dávid Papp)

Prerequisites: APMA 350

Mean Field Models for Spin Glasses (03)

Mentor: I-Hsun Chen (Second-year Graduate Student, Applied Mathematics)

Students: David Heffren, Catherina Niu

Project description: Models arising from Statistical Physics/Statistical Mechanics have a growing importance in modern probability theory. They provide a framework for relating the random microscopic fluctuation of individual atoms to the macroscopic properties that can be observed in everyday life. One of the most well-known models is the Ising model, which successfully explained the occurrence of ferromagnetic phase transition in iron using the interaction of individual ”spins”.

These complex systems have been studied by physicists for a couple of centuries, but can be difficult to describe mathematically. In around 1980, Giorgio Parisi discovered hidden patterns in disordered complex materials, he also predicted the famous Parisi formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model. The formula was verified in the seminal work of Michel Talagrand in 2006. In 2021, Giorgio Parisi was awarded the Nobel Prize in Physics ”for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales”.

In this program, we will focus on the ”original” spin glass model called the Sherrington-Kirkpatrick model, and try to get a taste of some basic phenomena that turn out to be somehow universal in statistical physics. Please be aware that this is a pure theoretic topic that is only suitable for those who are interested in the connection between physics and probability theory. Also, please note that we will not have time to cover the Parisi formula.

Prerequisites: APMA 1650 or 1655. No knowledge of physics is required.

Recommended: A heart that’s not afraid of complicated but straightforward computations.

Reference: Chapter 1 of Mean Field Models for Spin Glasses by Michel Talagrand.

The Geometry of the Poincaré Upper Half-Plane (04)

Mentor: Jake Mundo (Third-year Graduate Student, Applied Mathematics)

Students: Calvin Eng, Nathan Haronian

Project description: The Poincaré upper half-plane is a model of hyperbolic geometry, a type of non-Euclidean geometry, with a very rich structure. The purpose of this project is to use the Poincaré upper half-plane as a guiding example from which to introduce and explore several core areas of mathematics which emerge naturally in the study of this space. Depending on interest and background, these topics may include an introduction to Riemannian geometry, the calculus of variations, complex analysis, and/or the theory of matrix groups. No familiarity with any of these topics will be assumed.

The focus of the project will be very flexible according to student background and interest. There are no strict prerequisites, though some familiarity with any or all of ordinary differential equations, multivariate calculus, and/or linear algebra would be useful. In contrast, familiarity with hyperbolic geometry and the Poincaré upper half-plane itself will not be assumed. Readings will be pulled from a variety of texts to explore different perspectives.

Suggested prerequisites: MATH 0520/0540, APMA 0350 (or equivalent).

Sample text: Hyperbolic Geometry (James W. Anderson)

The Discrete Fourier Transform and Applications (05)

Language Processing (01)

Mentor: Erik Bergland (Third-year Graduate Student, Applied Mathematics)

Student: Megan O'Connor

Project description: Language is, without question, one of humanity’s most crucial tools. It has been with us since the beginning, always changing and evolving to fill the needs of our communities. Today, humanity boasts over 6,500 spoken languages. With so many different ways to communicate in an increasingly interconnected world, a natural question arises: what factors determine a region’s dominant language?

Unsurprisingly, this question has been taken up by mathematicians in recent years. Drawing on insights from mathematical epidemiology, dynamical systems and partial differential equations have allowed researchers to make substantial progress. This project will roughly fall into 3 stages. First, we will review literature on past models. Then, we will select a subset of these models and perform numerical simulations to gain understanding of the predictions they make. Finally, we will consider the assumptions these models make and attempt to propose modifications, evaluating the choices numerically. Students are encouraged to analyze both theirs and the original assumptions of the models through the lens of social justice or fairness. Familiarity with both ordinary and partial differential equations is assumed. Experience with MATLAB or other mathematical computing software is useful but not required.

Sample Text: Modeling the dynamics of language death (Abrams & Strogatz),  A reaction-diffusion model for competing languages (Walters)

Suggested Prerequisites: APMA 350, APMA 360

Stochastic Processes, the Heat Equation, and Applications (02)

Mentor: Kevin Hu (Third-year Graduate Student, Applied Mathematics)

Student: Lucas Kuan

Project description: Probability and PDE theory are the two cornerstones of applied mathematics. In this project, we will connect the two by studying the relationship between three fundamental objects: the random walk, Brownian motion, and the heat equation. 

This course will provide a gentle introduction to the theory of stochastic processes in both discrete and continuous time. We will closely follow the first two chapters of "Random Walk and the Heat Equation" by Greg Lawler. Depending on the interest of the student, we may continue with the latter chapters or dive deeper into stochastic calculus. If time permits, we will cover applications to mathematical finance and physics. 

Sample Text: Random Walk and the Heat Equation (Greg Lawler)

Prerequisites: APMA 1650, APMA 360

An Excursion into Mathematical Billiards (03)

Mathematical basics for Neural Network-based Scientific Computing (01)

Mentor: Aniruddha Bora (Postdoctoral Research Associate, Applied Mathematics)

Student: Jennifer Wang

Project description: The first idea of how a neuron works was given by Warren McCulloch, a neurophysiologist, and Walter Pitts, a mathematician in 1943 where they modeled a simple neural network using electrical circuits. Neural Networks have come a long way since then. In today’s world of scientific computing, neural networks are being used in many applications like: solving partial differential equations, image analysis, navigation systems, surgery, medicine, self-driving cars, chatbots (ChatGPT)etc. Although these methods seem like a black-box (like magic), they are based on proper mathematical and statistical foundation.

The focus of the project will be very flexible according to student background and interest. No strict prerequisites (some familiarity with ordinary differential equations and linear algebra would be useful). The main idea of the project will be to learn the fundamentals behind neural networks, deep-learning and learning to apply it to a practical application of the student’s area of interest.

Suggested Prerequisites: Math 0090/0100 (Single Variable Calculus), Math 0520 (Linear Algebra)

The Dynamics of Language: Competition and Spread (02)

Mentor: Erik Bergland (Fourth-year Graduate Student, Applied Mathematics)

Students: Marc Fernandez, Carlos Ramos

Project description: Language is, without question, one of humanity’s most crucial tools. It has been with us since the beginning, always changing and evolving to fill the needs of our communities. Today, humanity boasts over 6,500 spoken languages. With so many different ways to communicate in an increasingly interconnected world, a natural question arises: what factors determine a region’s dominant language?

Unsurprisingly, this question has been taken up by mathematicians in recent years. Drawing on insights from mathematical epidemiology, dynamical systems and partial differential equations have allowed researchers to make substantial progress. This project will roughly fall into 3 stages. First, we will review literature on past models. Then, we will select a subset of these models and perform numerical simulations to gain understanding of the predictions they make. Finally, we will consider the assumptions these models make and attempt to propose modifications, evaluating the choices numerically. Students are encouraged to analyze both theirs and the original assumptions of the models through the lens of social justice or fairness. Familiarity with both ordinary and partial differential equations is assumed. Experience with MATLAB or other mathematical computing software is useful but not required.

Sample Text: Modeling the dynamics of language death (Abrams & Strogatz),  A reaction-diffusion model for competing languages (Walters)

Suggested Prerequisites: APMA 350, APMA 360

The Geometry of the Poincaré Upper Half-Plane (03)

Mentor: Jake Mundo (Third-year Graduate Student, Applied Mathematics)

Project description: The Poincaré upper half-plane is a model of hyperbolic geometry, a type of non-Euclidean geometry, with a very rich structure. The purpose of this project is to use the Poincaré upper half-plane as a guiding example from which to introduce and explore several core areas of mathematics which emerge naturally in the study of this space. Depending on interest and background, these topics may include an introduction to Riemannian geometry, the calculus of variations, complex analysis, and/or the theory of matrix groups. No familiarity with any of these topics will be assumed.

The focus of the project will be very flexible according to student background and interest. There are no strict prerequisites, though some familiarity with any or all of ordinary differential equations, multivariate calculus, and/or linear algebra would be useful. In contrast, familiarity with hyperbolic geometry and the Poincaré upper half-plane itself will not be assumed. Readings will be pulled from a variety of texts to explore different perspectives.

Suggested prerequisites: MATH 0520/0540, APMA 0350 (or equivalent).

Sample text: Hyperbolic Geometry (James W. Anderson)

Applying Differential Equations to Biology (04)