 ## The DRP is running in Winter Term 2024!

### Matrix Groups: An Introduciton to Groups and Topology Through Linear Algebra

After you learn calculus and linear algebra, what's next? Why not both? Matrix groups, or somewhat more generally Lie (pronounced "Lee") groups, are a collection of matrices that are also a topological space, and you can use calculus to understand them! They're a beautiful blend of calculus and linear algebra that opens the doors to topics like group theory, topology, manifolds, and so much more. Moreover, Lie groups and Lie algebras pop up all over math, in Number Theory, Combinatorics, Geometry, and Differential Equations. In fact, Lie groups were invented to mimic Galois Theory ( used to study roots of polynomial equations) to applications in the study of Differential Equations. The subject of Lie Groups is very deep, but Matrix Groups provide a way in without too much work. Along the way, we'll see interesting mathematics like Group Theory, Topology, and maybe some calculus on manifolds!

Suggested References: Naive Lie Theory (Stillwell), Matrix Groups for Undergraduates (Tapp)

Prerequisites: Math 11 or 13, Math 22. A proof based Math class would be useful, but isn't required

### Analytic Combinatorics - Symbolic Methods

Symbolic methods constitute a unified algebraic theory dedicated to setting up functional relations between counting generating functions. As it turns out, a collection of general (and simple) theorems provide a systematic translation mechanism between combinatorial constructions and operations on generating functions. This translation process is a purely formal one. In fact, with regard to basic counting, two parallel frameworks coexist — one for unlabelled structures and ordinary generating functions, the other for labelled structures and exponential generating functions. Furthermore, within the theory, parameters of combinatorial configurations can be easily taken into account by adding supplementary variables.

Suggested References: Analytic combinatorics (Flajolet and Sedgewick)

Prerequisites: Math 28 (introduction to combinatorics) or equivalence

### Mathematical Art

#### Mentor: Brian Mintz

As students advance in their mathematical career, they learn to solve more abstract problems by being more imaginative. However, few people get to see this creative side of mathematics, so most think math is just a series of increasingly complex calculations. In this project, we'll explore some of the many ways art has been infused with mathematical ideas, creating some of our own along the way!

Suggested References: To be determined based on particapant's interests

Prerequisites: None

### Computational Measure Transport

#### Mentor: Jonathan Lindbloom

What is the 'best' way to transform a reference probability measure into a target probability measure? This is the classic Monge problem: how can 'earth' be excavated from one region and transported to fill in another prescribed region using the least amount of effort, i.e., minimizing the 'earth mover's' distance? In this project, we will study theory and computational methods for the solution of the optimal transport problem for both discrete and continuous distributions. We will also study methods for building non-optimal 'transport maps' between target probability measures and simpler reference measures such as the Gaussian. An emphasis will be placed on writing code for numerical demonstrations.

Suggested References: Computational Optimal Transport: With Applications to Data Science (Peyre and Cuturi)

Prerequisites: Math 22/24 (linear algebra), Math 20 (probability)

### Gaussian Processes for Machine Learning

#### Mentor: Dylan Green

Bayesian methods are powerful tools for solving a variety of problems in science and engineering. Gaussian processes specifically can be used to provide deep insights into a problem from only a relatively small amount of data. In this project, we will dive into the world of Gaussian processes and their applications in machine learning and other contexts. We will build up the theory of how these mathematical models work and put them into use with some practical coding examples. No prerequisite coding experience is required.

Suggested References: Gaussian Processes for Machine Learning by Rasmussen and Williams

Prerequisites: Math 22 or 24, some probability

### The Algorithmic Beauty of Plants

#### Mentor: Matthew Ellison

Check out the images on this page: https://en.wikipedia.org/wiki/L-system. In this reading course you'll work to create your own fractals and organic-looking plants using L-systems. In meetings I'll appreciate your work and help you debug, or we can discuss interesting relevant mathematical ideas --- such as geometric operations in 2D and 3D, language theory, or fractal dimension.

Suggested References: The Algorithmic Beauty of Plants, by Przemyslaw Prusinkiewicz and Aristid Lindenmayer

Prerequisites: Beginning/Intermediate Python skills

### Arithmetic Geometry

Legend has it that the followers of Pythagoras drowned Hippasus of Metapontum for proving that the third side of a triangle with two rational length sidelengths could be irrational. Diophantine equations such as a^2 + b^2 = c^2 have been studied for millenia, and with modern algebraic geometry, we are finally coming to have a reasonable grasp on the complex valued solutions to them. In this project, we will follow in the footsteps of the Pythagorians and ask for when the answers are honest fractions. To do this there are plenty of paths forward depending on your mathematical interests. We can explore the algebraic structure of elliptic curves, solutions over p-adic numbers and finite fields, rational points on algebraic curves, quadratic forms and the Hasse-Minkowski theorem, and applications to cryptography.

Suggested References: One of the following, depending on interest. Rational Points on Elliptic Curves by Silverman and Tate, Algebraic Geometry in Coding Theory and Cryptography Harald Niederreiter and Chaoping Xing, Introduction to Arithmetic Geometry (Notes) by Bjorn Poonen

Prerequisites: Math 71 or Math 31 or Equivalent

### De Rham Cohomology

#### Mentor: Kameron McCombs

De Rham Cohomology is a beautiful subject that explores the connections between calculus and topology on manifolds. This theory connects the ideas behind the curl and the divergence; it also gives an explanation for why the divergence of a curl-free vector field is zero and what this means for higher dimensional spaces. For this project, we will review manifolds before exploring differential forms and exterior derivatives. Then we will learn the basics of de Rham cohomology on different manifolds.

Suggested References: Differential Forms in Algebraic Topology by Bott and Tu

Prerequisites: Math 72

### A Tour of the Finite Element Method

#### Mentor: Jonathan Lindbloom

Many partial differential equations (PDEs) that arise in practice cannot be solved in closed-form. In these cases, we must rely on numerical methods for approximating their solution. In this DRP, we will take a tour through the finite element method (FEM), the most powerful general-purpose technique for computing accurate solutions to PDEs. We will emphasize implementing the FEM numerically for several test problems. Time permitting, we will also explore the open-source FEniCSx computing platform.

Suggested References: Understanding and Implementing the Finite Element Method (Gockenbach)

Prerequisites: Math 53 (partial differential equations), some knowledge of a programming language.