The DRP is running in Spring Term 2025!
Stay tuned for updates!
Probability Theory for the Binomial Asset Pricing Model
Mentor: Jiayi Chen
We will explore how probability theory could be applied to financial markets together. We’ll start with probability essentials, then dive into discrete-time models like the binomial asset pricing model, and build intuition for risk-neutral valuation. This DRP aims to build a good starting point for whoever is curious about quantitative finance. No measure theory required—just a willingness to think probabilistically!
Suggested References: Stochastic Calculus for Finance I & II (primarily volume I)
Prerequisites: Math 20
Hyperbolic Geometry
Mentor:Ahmadreza Hajsaeedisadegh
In this topic, we study one of the most well-known non-Euclidean geometries. We initially learn about complex numbers and then move on to experiment with models of the hyperbolic plane, such as Poincaré's half-plane and disk models. We learn the correct notion of line, length, and distance and compare results in Euclidean geometry with those in hyperbolic ones. We discuss Möbius transformations and their conformity. If time allows, we will cover Gauss-Bonnet's theorem
Suggested References:Hyperbolic Geometry by James Anderson
Prerequisites:Math 31 or equivalent
Characteristic Classes
Mentor:Ahmadreza Hajsaeedisadegh
Our goal is to study invariants of vector bundles over manifolds such as Steifel-Whitney classes, Chern Classes, and Pontryagin classes. In this direct reading program, we aim to equip the students with the necessary tools to engage in future advanced courses in topology and geometry and to develop an understanding of research programs in these areas
Suggested References:Characteristic Classes by John Milnor
Prerequisites:Math 104 or equivalent
Applications of Evolutionary Game Theory
Mentor: Mark Lovett
Description: Our goal of this course is to explore the applications of evolutionary games in industry and other domains of science. This course is focused on the individual level, meaning we will be exploring topics that the students are interested in. For the students who have a more in depth knowledge of game theory or evolutionary game theory, they will be mostly reading papers within the domain that they are interested in; which we will discuss weekly. For other students we will be reviewing the text book and model papers from traditional/evolutionary game theory literature, to build or reinforce their mathematical foundations. Students will be expected to have a lot of self motivation and interest; in exchange for getting a foothold in applied mathematics research with evolutionary game theory. The ultimate goal is for the students starting a project within their interested domain.
Reference Textbook: Martin Nowak Evolutionary Dynamics and notes from Dan Cooney(free)
Prerequisites: required: MATH 8, MATH 13, COSC 1 or intro level coding in Python. Recommended: MATH 22, MATH 23 MATH 30, MATH 40, and COCS 74
Matrix Calculus
Mentor:Patrick Addona
Like in single and multivariable calculus, differentiation is crucial in optimization problems. But what happens when our variables are not just scalars, but vectors or matrices? Many standard techniques break down, and new tools are needed. Matrix calculus is a black sheep of applied math: often extremely helpful or even necessary, but spoken about in hushed tones. In this program, students will shed light on the process of differentiating expressions involving matrices! As a bonus, we will discuss the differential, an alternative to the more familiar derivative, which is essential for linearization
Suggested References:Matrix Differential Calculus with Applications in Statistics and Econometrics by Magnus and Neudecker
Prerequisites:Math 3, 8, 13, and 22 (or equivalent courses on single/multivariable calculus and linear algebra)
Commutative Algebra
Mentor: Sarah Frei
Description: Commutative algebra is the study of commutative rings with 1. It sounds simple enough, but there is a lot to dig into here! It is the foundation for algebraic geometry, in which we give geometric structure to sets of ideals in a ring (e.g. the collection of prime or maximal ideals). Here we will focus on building this bridge between algebra and geometry. We will study ideals, modules, Noetherian rings, finite extensions of rings, localizations, etc, with motivation for these algebraic structures coming from algebraic geometry.
Reference Textbook: Undergraduate Commutative Algebra (Reid), Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud), or Introduction to Commutative Algebra (Atiyah-MacDonald
Prerequisites: Math 31 or the equivalent
Mathematics of Card Shuflfing
Mentor: GaYee Park
Description: Card shuffling is a classic problem that can be studied using various mathematical tools. Among the many shuffling techniques—ranging from the naive method of moving one card at a time to the dexterous riffle shuffle—which are the most effective? How can this knowledge help us understand card games, magic tricks, or even traffic flow? While probability theory provides a natural framework for exploring these questions, shuffling also connects to other mathematical fields, including combinatorics, representation theory, and geometry. In this program, we will first study the application of linear algebra and probability to analyze shuffling, then study algebraic structures of shuffles and its connection to combinatorics and geometry (and more given enough time!).
Reference Textbook: The Mathematics of Shuffling Cards by Persi Diaconis and Jason Fulman
Prerequisites: Math 31 (or equivalent courses on abstract algebra)
Randomized numerical linear algebra (RandNLA)
Mentor: Jonathan Lindbloom
Description: We will explore Randomized Numerical Linear Algebra (RandNLA), a modern approach to solving large-scale linear algebra problems using randomness. We will survey key topics such as randomized low-rank approximation, sketching methods, and iterative solvers for large linear systems. Our goal is to understand how randomness can be leveraged to speed up and improve classical numerical linear algebra techniques. In addition to studying the theory, we will implement and run computations to see these methods in action. This program is ideal for students with background in linear algebra and an interest in computational mathematics.
Reference Textbook: Randomized Numerical Linear Algebra: Foundations & Algorithms (Martinsson & Tropp)
Prerequisites: Math 20 or 40 (probability), Math 22 (linear algebra), Math 56 (computational methods) or 76 (computational inverse problems)
The Math of Games
Brian Mintz:
Description: Do you want to learn strategies that can win games 100% of the time? The field of recreational mathematics is devoted to thinking about combinatorial problems like this, and is the perfect answer to people who think math can't be fun! In this project, we'll select a handful of games to analyze following a seminal 1000 page book on the subject.
Reference Textbook: Winning Ways for Your Mathematical Plays Vol. 1 by Berlekamp, Conway, and Guy.
Prerequisites: None