Directed Reading Program

Past Projects

Winter 2024

Matrix Groups: An Introduction to Groups and Topology Through Linear Algebra

Graduate Mentor: Melanie Ferreri

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

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

Arithmetic Geometry

Mentor: Luke Askew

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

Spring 2023

Polynomials

Grant Molnar:

Description: Polynomials in a single variable are some of the first functions we study in math. They arise naturally throughout science, math, and engineering. In this directed reading program, we will explore the fundamental properties of polynomials and polynomial inequalities. We will start with the basics, including polynomial division and factorization, and the fundamental theorem of algebra. Based on student interest, we will then move on to studying the roots of polynomials (for instance via the Gauss-Lucas Theorem, Jensen's formula), polynomial inequalities, and other important results in the theory of polynomials. Along the way, we will see how the study of polynomials connects with other areas of mathematics, including number theory, geometry, and analysis. Suggested References: Polynomials and Polynomial Inequalities (Peter Borwein and Tamas Erdelyi, 1995)

Suggested References: Polynomials and Polynomial Inequalities (Peter Borwein and Tamas Erdelyi, 1995)

Prerequisites: Math 8 would be helpful

Fractional graph theory

Grant Molnar

Description: Are you curious about graph theory beyond the standard integer-valued concepts? Consider exploring the fascinating field of fractional graph theory! In this directed reading program, we will delve into the world of graphs with fractional weights, allowing for a more nuanced representation of relationships. We'll start by exploring basic concepts such as fractional chromatic number and independence number, and see how they apply to real-world problems such as resource allocation and network design. From there, based on student interest, we can delve deeper into more advanced topics such as fractional flows, fractional matchings, and fractional edge colorings. Along the way, we will encounter tools from linear algebra and optimization theory, and develop a deep understanding of the connections between them.

Suggested References: Fractional Graph Theory: A Rational Approach to the Theory of Graphs (Edward R. Scheinerman and Daniel H. Ullman, 1997)

Prerequisites: None

Analytic Combinatorics

Grant Molnar

Combinatorics is the study of counting: it asks questions like “How many ways can we arrange the integers from 1 to n?” and “How many ways can we partition a set into at most 3 pieces?” Although combinatorics often furnishes explicit answers to these questions, for many applications we care more about the approximate size of the answers than their exact numbers. By encoding combinatorial sequences in generating functions, we can use real and complex analysis to estimate the asymptotic size of the nth coefficient of a combinatorial sequence. For instance, Stirling’s approximation says that n! is approximately sqrt(2 pi n) (n/e)^n. In this reading course, we’ll study how to encode combinatorial sequences in generating functions, and use these generating functions to extract both exact and asymptotic information about the behavior of these sequences.

Suggested Text: Analytic Combinatorics by Philippe Flajolet, Robert Sedgewick

Prerequisites: Math 8 or equivalent. Math 43 would be helpful.

Mathematics of Gerrymandering and Redistricting

Mentor: Atticus McWhorter

How can we formulate Gerrymandering into a mathematical question? This DRP explores the intersection of mathematics, politics, and law through the lens of Markov Chain Monte Carlo (MCMC) techniques. Through hands-on projects with real-world data, we will gain an understanding of the mathematics of gerrymandering as well as practical experience using MCMC techniques to analyze and create redistricting plans.

Suggested References: Political Geometry (Doochin, Walch)

Prerequisites: Probability and Statistics: Math 40 or both Math 10 and Math 20

Algebraic Numbers and Their Applications

Mentor: Steve Fan

The fundamental theorem of arithmetic asserts that every composite number has a prime factorization that is unique up to rearrangement of factors. This nice property of natural numbers (or equivalently, of the ring of rational integers) is especially useful when it comes to applications such as multiplying and dividing numbers, finding the greatest common divisors, and solving Diophantine equations like x(x + 1) = 6 and x^2 + y^2 = z^2. As a matter of fact, the idea of unique prime factorizations also applies to some larger rings, and it turns out that even for rings in which elements don't necessarily have unique prime factorizations, their ideals may still be factored uniquely into prime ideals. With this idea in mind, we will study the basic theory of algebraic numbers and algebraic number fields, which allows us to do arithmetic beyond rational integers, and explore their interesting number-theoretic applications, concluding with Kummer's treatment of a special case of Fermat's equation x^n + y^n = z^n.

Suggested References: Algebraic Number Theory and Fermat's Last Theorem (Stewart and Tall), Number Fields (Marcus)

Prerequisites: Math 22/24 and 25. Some exposure to abstract algebra would be very helpful

Markov Chain Monte Carlo and Bayesian Computation

Jonathan Lindbloom

The use of Markov Chain Monte Carlo (MCMC) methods for solving problems arising in Bayesian computation is ubiquitous. Our guiding question for this DRP is: how can we characterize a high-dimensional probability distribution for which we possess only the unnormalized density function? We will begin with an in-depth study of Monte Carlo methods and the theoretical basis for the Metropolis-Hastings algorithm. We will then proceed with a survey of many specialized instances of MCMC algorithms, including: random walk MH, adaptive methods, Langevin algorithms, Gibbs sampling, Hamiltonian MCMC, the NUTS sampler, and ensemble methods such as parallel tempering and affine-invariant samplers. To supplement our theoretical understanding, we will implement some of these methods from scratch to sample the posteriors of some simple Bayesian models. This program would be a great jumping-off point for anyone interested in the intersection of applied mathematics, computational science, and statistics.

Suggested reference: Monte Carlo Statistical Methods (Robert and Casella)

Prerequisites: Math 13, Math 22/24, Math 20/40/60, and familiarity with a programming language.

The Symmetric Group

Mentor: Melanie Ferreri

The symmetric group S_n is the group of permutations of n elements, which you may have worked with in a class on combinatorics or group theory. Representation theory allows us to view group elements as linear transformations from a vector space to itself. For this topic, we would look at representations of the symmetric group from a combinatorial perspective, using Young tableaux, generating functions, and symmetric functions.

Suggested References: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions (Sagan)

Prerequisites: Math 28 and 31 or the equivalent

Algebraic Geometry

Mentor: Richard Haburcak

You've done linear algebra, so what about solving equations that aren't linear? Algebraic geometry is the study of geometric objects coming from algebraic stuctures and using both geometry and algebra to understand the other better. Much like vector spaces are solutions to linear equations, we'll learn about how solutions to polynomial equations can be viewed geometrically and begin an exploration of affine varieties. We'll begin by getting a glimpse into what algebraic geometry is all about, and quickly realize that there's a lot of cool algebra that geometry elucidates. We'll encounter a little commutative algebra, some topology, and even some category theory! Time permitting, we may explore more advanced topics such as an introduction to curves.

Suggested References: Elementary Algebraic Geometry (Hulek, 2003), Algebraic Geometry (Perrin), Algebraic Curves (Fulton)

Prerequisites: Math 31 or the equivalent

Matrix Groups: An Introduction to Groups and Topology Through Linear Algebra

Mentor: Melanie Ferreri

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

Modular Forms

Mentor: Eran Assaf

In how many ways can you represent a number as a sum of four squares? Can you hear the shape of a drum? What are all the integers x, y, z satisfying x^n + y^n = z^n for n a positive integer? Is the number 1 + 1/2^3 + 1/3^3 + 1/4^3 + …. rational? All these questions (and more) are answered by the beautiful theory of modular forms, which will be explored in this reading course. We will learn about elliptic curves, modular curves and classical modular forms, with a view towards the modularity theorem - every rational elliptic curve arises from a modular form. This program would be a good preparation for those interested in pursuing research in number theory in the future.

Suggested References: A First Course in Modular Forms (Diamond and Shurman), Modular Forms, a Computational Approach (Stein)

Prerequisites: Math 25, Math 31, Math 35, Math 43

Representation Theory

Mentors: Eran Assaf

Symmetry is an important tool used consistently throughout mathematics in various ways, and formalized through the notion of a group action. Groups can act on many different objects, such as sets, topological spaces or vector spaces. An action of the group on a vector space gives rise to a representation of the group, and representation theory studies these representations. These have interesting applications in both physics and quantum chemistry. It also turns out one can infer quite a lot about the group from its representations. In this program, we embark on a journey, which begins by learning the theory of finite-dimensional representations of finite groups, and time permitting, goes on to the study of finite-dimensional representations of Lie groups and Lie algebras.

Suggested References: Representation Theory: A First Course (Fulton and Harris), Linear Representations of Finite Groups (Serre), Representations and Characters of Groups (James and Liebeck)

Prerequisites: Math 31

p-adic Numbers

Mentor: Eran Assaf

Did you know that it is impossible to dissect a square into an odd number of triangles of equal area? And that in some (p-adic) places, every triangle is isosceles? In this reading course we dare to go on an adventure beyond the standard realms of undergraduate mathematics, and visit the exotic p-adic universe. While we are having fun, we will learn more about the importance of p-adic numbers in modern number theory, featuring key ideas such as the local-global principle. This program would be a great place to anyone interested to start exploring the realms of number theory, in a meeting point between algebra and analysis.

Suggested References: p-adic Numbers: an Introduction (Gouvêa), p-adic Numbers, p-adic Analysis and Zeta-Functions (Koblitz)

Prerequisites: Math 3

Category Theory

Mentor: Richard Haburack

Category theory takes a bird's eye view of math by considering collections of objects all at once (e.g. vector spaces, groups, topological spaces...) and functions between them (linear transformations, group homomorphisms, continuous maps...). Furthermore there are relationships between different collections of objects, for example associated to every topological space is a group of paths on the space called the fundamental group, and continuous maps between topological spaces induce homomorphisms between the associated fundamental groups. Abstracting ideas like this and making them precise are at the heart of category theory. Many subjects in math require a basic understanding of the notions of category theory and for some such as algebraic geometry it is essential. Whilst category theory might initially sound wilfully abstract, an introduction is full of hands-on examples and exercises.

Suggested References: Category theory in context (Riehl), Basic Category theory (Leinster)

Prerequisites: Math 31

A Mathematical Foundation for Synthetic Aperture Radar

Mentor: Dylan Green

Synthetic aperture radar (SAR) imaging is a day and night, all-weather method of forming high-resolution images of large swaths of land and sea. SAR has many applications and is widely used by the military, academia, and private industry. The problem of SAR image formation has been an ongoing area of research in both engineering and applied mathematics for decades now, and advances are still being made to this day. In this reading course, we will develop a mathematical framework for the basic principles in SAR imaging and work our way up to forming an image using real SAR data.

Suggested Reference: Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Jakowatz, Wahl, Eichel, Ghiglia, and Thompson)

Prerequisites: Math 11 or 13, Math 22 or 24, some coding. Previous exposure to Fourier transforms is preferred, but not required.

Mean Field Games

Mentor: Bohan Zhou

Mean field games theory was created in 2006 by Jean-Michel Lasry and Pierre-Louis Lions. It is a branch of games theory which a particle physicist, a (micro- or macro) economist, a sociologist or an urban planner is interested in, much more than the community of mathematicians. Have you heard about the "prisoner's dilemma"? Then how about Nash equilibria? Let's explore more to a generalized case for the dynamics and strategies of n-player games, or even push to infinity cases!

Suggested References: Mean Field Games (Achdou, Cardaliaguet, Delarue, Porretta, and Santambrogio), Mean Field Games and Applications (Gueant, Lasry, and Lions)

Prerequisites: Math 53 or 63 is recommended

Abstract Nonsense

Mentor: Luke Askew

Come join a book club! This reading project can serve as an introduction to higher mathematics through its preferred language: category theory. We'll see how the very human ideas of objects, relationships, and analogies are made formal mathematically before embarking on a whirlwind tour of beautiful abstract algebraic structures. We'll be using a fresh off the press book which gives us the opportunity to participate in a book club with the author through the Topos Institute!

Suggested References: The Joy of Abstraction: An Exploration of Math, Category Theory, and Life by Eugenia Cheng

Prerequisites: None

Statistical Learning

Mentor: Zhen Chen

The program will focus on statistical/machine learning basics, including linear regression and classification, to more advanced topics, such as deep learning.

Suggested References: “The Elements of Statistical Learning: Data Mining, Inference, and Prediction” by Jerome H. Friedman, Robert Tibshirani, and Trevor Hastie

Prerequisites: Math 13, Math 22, and Math 20 or equivalent

Mathematical Ecology

Mentor: Brian Mintz

Eugene Wigner once noted that Mathematics is often unreasonably effective in the natural sciences. One great example of this is how math can be used to understand ecosystems. With tools from across mathematics, including Game Theory and Differential Equations, we will learn how to think about ecology from a mathematical perspective.

Suggested References: Elements of Mathematical Ecology by Mark Kot, Theoretical Ecology: Principles and Applications, An Illustrated Guide to Theoretical Ecology by Ted J. Case, A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Sarah P. Otto and Troy Day.

Prerequisites: Math 8, Differential equations would be helpful, but is not necessary.

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

Matroids

Mentor: Ben Adenbaum

Matroids are objects which provide a framework in which to think about the abstract notion of independence. The general theory of these objects can provide a unified way to understand aspects of graph theory, linear algebra, and combinatorics.

Suggested References: Matroids: A Geometric Introduction (Gordon and McNulty)

Prerequisites: Math 22 or 28 or 38.

Quantum information theory

Mentor: Travis Russel

Classical information theory concerns how one can efficiently transfer information from point A to point B. Often, information is transmitted over noisy channels which can corrupt the intended message. The basic techniques for correcting these random errors were developed in an elegant theory by Claude Shannon in the 1940s, and these techniques remain a fundamental ingredient of modern computing and communication devices. With the development of quantum computing and quantum communication devices, it has become necessary to consider the efficient transfer of quantum information in the form of qubits. While qubits can store more information than classical bits, they are notoriously fragile and can be easily corrupted by noise. The fragility of qubits is the main obstacle preventing the development of powerful quantum computers today. In this reading course, we will learn what quantum information is, how it is affected by noise, and how "quantum" errors can be corrected. Along the way, we will learn the basics of quantum mechanics and quantum computing.

Suggested References: Quantum Computation and Quantum Information, Nielsen & Chaung

Prerequisites: Math 22 or Math 24, No physics background required.

Quantum Computation and Quantum Information

Mentor: Casey Dowdle

You may have heard about recent hype around quantum computing in the news. This is your chance to study the mathematics behind quantum computing and quantum information with a graduate student who is actively doing quantum computing research. We will be reading the classic "Quantum Computation and Quantum Information" by Nielsen and Chuang. No prior background of quantum mechanics is necessary! We will cover topics such as the Bloch sphere, measurement of quantum states, quantum gates and algorithms, and quantum error correction. Along the way you will also gain a better understanding of classical computation and information theory. We will also program simple quantum algorithms using quantum circuit simulator libraries such as Braket or Qiskit.

Suggested References:"Quantum Computation and Quantum Information" - Nielsen and Chuang

Prerequisites: Math 13, Math 24, (Math 40 and some familiarity with programming would be helpful but is not strictly required)

Deep reinforcement learning

Mentor: Mark Lovett

Reinforcement learning has been described as a key element to solving true AI. Deep reinforcement learning a relatively new and hot field in machine learning. Deep reinforcement learning has been applied to many of the current problems and recent advances in machine learning; including but not limited to, autonomous cars, generalizability of AI, and chat GPT. This project will introduce the student to some of the basics of reinforcement learning and explore how the mathematics of deep reinforcement learning and its implementation in python.

Suggested References:Deep Reinforcement Learning-Plaat

Prerequisites: Math 13, Math 24, Math 23, Programing experience in python is encouraged

Winter 2023

Mathematics and Machine Learning

Mentor: Atticus McWhorter

Mentee: Andrew Koulogeorge

What's going on under the hood of machine learning? In this course we will read up on some relevant concepts from functional analysis and linear algebra. Applications to machine learning include supervised and unsupervised learning, neural networks, optimization, and algorithmic fairness. Projects in Python will aid in our understanding, so some coding experience is required. Additionally, a strong background in Linear Algebra and some familiarity with real analysis will help.

Suggested References: Machine Learning: A Probabilistic Perspective (Murphy), Deep Learning by (Goodfellow, Bengio, and Courville)

Prerequisites: Math 22, Real analysis would be helpful

Game-theoretic Models in Biology

Mentor: Alina Glaubitz

Mentee: Damini Kohli

Why does the peacock have a large and beautiful tail? Why is the sex ratio of so many species close to a half? Why do animals (and, in particular, humans) cooperate? These and many other questions can be answered by using evolutionary game theory. Depending on interest, this project can focus on various topics, including infectious disease modeling, the evolution of cooperation, and evolutionary cancer modeling.

Suggested References: Game-Theoretic Models in Biology (Broom and Rychtár), Game Theory in Biology: Concepts and Frontiers (McNamara and Leimar)

Prerequisites: Math 22, Math 23, and Math 20 are recommended

Magical Mathematics

Mentor: Alex Wilson

Mentee: Sofia Goncalves

Card tricks are not only fun to watch and to perform, but there is often sophisticated mathematics associated with them. In this project, we will learn about the math underlying various card tricks and take detours into whatever related math is appropriate for your interests and mathematical experience. For example, the study of card tricks can even lead to questions in graph theory, group theory, and probability.

Suggested References: Magical Mathematics (Diaconis and Graham)

Prerequisites: Math 8 or Math 11

Ergodic Theory

Mentor: Longmei Shu

Mentee: Evan Barrett

What is a dynamical system? Depending on who you ask, you can get very different answers. It can be continued fractions, billiards, irrational rotation on a circle, Hamiltonian systems, or geodesic flows on Riemannian manifolds. We will read through various examples in the book Introduction to Ergodic Theory by Yakov Sinai and can spend more time on specific problems of interest.

Suggested References: Ergodic Theory (Sinai)

Prerequisites: Math 13 and 23, Math 20 would be helpful

Category Theory

Mentor: Ciaran Schembri

Mentee: Paul Shin

Suggested References: Category theory in context (Riehl), Basic Category theory (Leinster)

Prerequisites: Math 31

Mathematical Bioeconomics

Mentor: Matthew Ellison

Mentee: Amya Luo

It’s easy to hate overfishing, deforestation, and other unsustainable uses of the world’s renewable resources. But how should we do things right? We’ll read from Colin Clark’s Mathematical Bioeconomics: The Mathematics of Conservation to see how mathematical models can be developed and analyzed as a tool for making informed policy choices.

Suggested References: Mathematical Bioeconomics: The Mathematics of Conservation (Clark, 3rd edition)

Prerequisites: Calculus, prior exposure to probability and programming useful but not necessary.

The Rebel Women of Mathematics

Mentor: Sarah Frei

Mentee: Reshmi (Mia) Anwar

We will learn about the contributions of 27 women to mathematics by following the book “Power in Numbers: The Rebel Women of Mathematics”. This book not only features the biographies of known classical mathematicians like Sofya Kovalevskaya and Sophie Germain; but it also talks about people doing mathematics right now, for example Chelsea Walton or Tatiana Toro. The end goal of this project will be to will focus on understanding the work of a woman mathematician of your choice.

Suggested References: Power in Numbers: The Rebel Women of Mathematics.

Prerequisites: Math 8

Bayesian Inverse Problems

Mentor: Dylan Green

Mentee: Ian Gill

Given some indirect data, like the backscatter from a radar transmission or the magnetic response from an MRI machine, how do we recreate the image that gave us this data? This reading course will explore some of the methods used to tackle these and other inverse problems. We will specifically investigate the role Bayesian inference can play in not only scene reconstruction, but in quantifying how certain we are in the reconstruction we have. We will begin the course with an introduction to inverse problems and regularization methods before moving into statistical inversion theory.

Suggested References: Statistical and Computational Inverse Problems (Kaipio and Somersalo)

Prerequisites: Math 11 or 13, Math 22 or 24, and some probability

Knot Theory

Mentor: Lizzie Buchanan

Mentee: Calvin George

A knot is an embedding of the circle into 3D space. It is just a tangled up circle sitting in space. Despite this simple description, it can be very hard to tell two such knots apart. Classically, mathematicians used easy-to-describe but hard-to-compute invariants of knots to tell them apart: crossing number, genus, unknotting number, slice genus, etc. Eventually, they progressed to associating polynomials to knots: the Alexander polynomial, Jones polynomial, HOMFLY polynomial, etc. The current state of knot theory is to associate even more complicated invariants to knots to try to tell even more of them apart - and in many cases, these new invariants can also give us information about the more classical invariants. We can take a tour of these classical invariants, explore knot polynomials and newer invariants, and perhaps discuss ideas of how one might try to come up with invariants of their own. We will see (and draw) lots of pictures and diagrams along the way!

Suggested References: The Knot Book (Adams), Knot Theory (Livingston)

Prerequisites: Math 31 would probably be useful, but any interested students should reach out

Matrix Groups: An Introduction to Groups and Topology Through Linear Algebra

Mentor: Richard Haburcak and Melanie Ferreri

Mentees: Mimi Grozeva, Heather Wang, and Daniel Carstensen

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

Spring 2022

The Probabilistic Method

Graduate Mentors: Kathy Lin

Mentee: Michael Gonzalez

Is it possible to choose n points in d-dimensional space such that all angles determined by three of them are strictly smaller than 90 degrees? Is it possible to 2-color the edges of the complete graph on n vertices such that no complete subgraph on r vertices is monochromatic? The Probabilistic Method, pioneered by Erdös, is a powerful approach for attacking such existence results. The core idea is to investigate when a random construction will have positive probability of success — in which case one can be sure a solution exists! We’ll cover the basics and then go into applications and more advanced theory according to interest.

Suggested References: The Probabilistic Method (Alon and Spencer)

Prerequisites: You should have a solid foundation in discrete math and probability, such as from Math 20 (Probability) or CS30 (Discrete Math in Computer Science).

Integer-Point Enumeration in Polyhedra

Graduate Mentor: Alex Wilson

Mentee: Calvin George

A polytope is a higher-dimensional analog to a polygon, some examples being boxes, tetrahedra, crystals, and any convex object whose faces are all flat. The discrete volume of a polytope is the number of grid points that lie inside it, and computing this value is an interesting combinatorial problem which has connections to geometry through e.g. the continuous volume and to number theory through e.g. Dedekind sums and finite Fourier series. I think any student with a little background in abstract algebra and some interest in combinatorics, geometry, or number theory could get a lot out of this project.

Suggested References: Computing the Continuous Discretely (Beck and Robins)

Prerequisites: A course in abstract algebra, Math 31 or the equivalent

Sheaf Cohomology

Graduate Mentor: Richard Haburcak

Mentee: Varun Malladi

Sheaves are a generalization of continuous functions on spaces, but with a more intricate local behavior. Meanwhile, sheaf cohomology is a theory that helps to measure how the local behavoir of functions deviates from their global behavior. We'll study the notion of sheaves on topological spaces, with many of our examples coming from algebraic geometry, and then focus on sheaf cohomology in the context of Riemann surfaces to work towards a proof of the Riemann—Roch theorem. Time permitting, we'll also explore sheaves in a more categorical context, with a view towards Grothendieck topologies.

Suggested References: Lectures on Algebraic Geometry I (Harder), Algebraic Geometry (Hartshorne)

Prerequisites: Familiarity with commutative algebra, some homological algebra, and some algebraic geometry. Math 74, Math 104, and Math 101 are highly recommended.

Bayesian Modeling and Computation

Graduate Mentor: Jonathan Lindbloom

Mentee: Ivy Yan

Bayesian computational methods provide a general, powerful toolkit for statistical modeling and uncertainty quantification. We will take a tour of the Bayesian modeling methodology, with our goal being to learn the relevant mathematical theory behind important computational tools (Markov chain Monte Carlo, variational inference, model comparison, etc.) while also gaining practical experience using them to model data of all sorts using probabilistic programming languages. We will take a "learn by doing" approach to guide our theoretical understanding. This program would be a great jumping-off point for anyone interested in the intersection of applied mathematics, computational science, and statistics.

Suggested References: Bayesian Modeling and Computation in Python (Chapman and Hall)

Prerequisites: Math 20, Math 60 or a course in Probability Theory may be useful, familiarity with Python.

Decision Theory

Graduate Mentor: Grant Molnar

Mentee: Harrison Fells

What makes a decision “rational” or “irrational”? In this reading course, we approach this problem from a foundational perspective by making a thorough study of Leonard Savage’s epoch-making axiomatization of expected utility theory. Along the way, we will discuss the philosophical limits of expected utility theory, and its competitors in the theory of normative and prescriptive decision-making.

Suggested References: The Foundations of Statistics (Savage)

Prerequisites: None

Winter 2022

Game Theory

Graduate Mentor: Matt Jones

Mentee: Love Tsai

The world is full of interesting interactions between individuals. One of the most powerful and versatile tools for analyzing these interactions is game theory, which can be applied to almost any scenario. We will take a deep dive into the rigor and notation that makes game theory work from a mathematical perspective, but that gets skipped over in most introductory courses in game theory.

Suggested References: A Course in Game Theory (Osborne and Rubinstein)

Prerequisites: Previous experience with game theory and a proof-based class like 31 or 35 would be good to have but are not essential.

Nonlinear Dynamics, Chaos, and Ergodicity

Graduate Mentor: Jonathan Lindbloom

Mentee: Andrew White

A deterministic dynamical system is one whose state evolves in time according to some fixed, non-random rule typically given by a differential equation or iterated map. In principle, knowing the rule for such a system is enough to completely characterize its behavior for all time. However, for a special class of systems known as chaotic systems, knowledge of the rule turns out to be insufficient (in practice) and we must instead settle for a statistical description of their behavior. We will start by surveying the basic principles of nonlinear dynamical systems and chaos, including stability, bifurcations, entropy, fractals, and strange attractors. We will then revisit these ideas with a probabilistic lens, building up to the notion of ergodicity and its applications in various settings.

Suggested References: Chaos: An Introduction to Dynamical Systems (Alligood, Sauer, and Yorke)

Prerequisites: Math 11, Math 23, and some probability

Elliptic Curves

Graduate Mentor: Juanita Duque-Rosero

Mentees: Jenny Song, Athina Avrantini

Elliptic curves are algebraic objects that are important in number theory, algebraic geometry, and applications. They consist of solutions to equations of the form y^2=f(x), where f(x) is a polynomial of degree 3. It turns out that one can put the structure of a group on the solutions for the equation. This gives rise to a rich theory that touches areas like diophantine equations, groups, lattices, number theory and algebraic geometry. In addition, elliptic curves can be applied to cryptography, so doing computations with them is essential. In this program, we can study elliptic curves focusing on the perspective that is preferred by the student. This could be a purely theoretical project, or it could also include some computations.

Suggested Reference: Rational Points on Elliptic Curves (Silverman)

Pre-requisites: Math 31 or the equivalent (knowledge of groups)

Knot Theory

Graduate Mentor: Sam Tripp

Mentee: Katherine Woolfolk

A knot is an embedding of the circle into three-dimensions. It is just a tangled up circle sitting in space. Despite this simple description, it can be very hard to tell two such knots apart. Classically, mathematicians used easy to describe but hard to compute invariants of knots to tell them apart: crossing number, genus, unknotting number, slice genus, etc. Eventually, they progressed to associating polynomials to knots: the Alexander polynomial, Jones polynomial, HOMFLY polynomial, etc. The current state of knot theory is to associate even more complicated invariants, categorifications of these polynomials, to knots, to tell them apart, and in many cases, give us information about the more classical invariants. We can take a tour of these classical invariants, then through polynomials and onto categorifications. We will see lots of pictures and diagrams on the way.

Instead of considering pure math, we could also look for applications of knots, particularly to protein folding. There is a relatively new way of considering proteins in the language of knots, and we could learn about that and see what we can see. Whatever interests us the most.

Suggested References: The Knot Book (Adams), Knots and Links (Rolfsen), An Introduction to Knot Theory (Lickorish)

Prerequisites: Math 31 would probably be useful, and maybe Math 54, but don’t let those stand in the way of you learning interesting math.

The Probabilistic Method

Graduate Mentors: Kathy Lin

Mentee: Jessica Liu

Suggested References: The Probabilistic Method (Alon and Spencer)

Prerequisites: You should have a solid foundation in discrete math and probability, such as from Math 20 (Probability) or CS30 (Discrete Math in Computer Science).

Fractional Calculus

Graduate Mentor: Grant Molnar

Mentee: Gayeong Song

Multiplication was defined originally as repeated addition, but as we grow more mathematically sophisticated we become comfortable multiplying by negative numbers, fractions, and even real numbers. We can differentiate repeatedly a function repeatedly, just as we can add a number to itself repeatedly: so what meaning might we give to “half” of a derivative? Enter the world of fractional calculus, where it makes perfect sense to differentiate a function like x^4 pi times. Fractional calculus has applications in fluid dynamics and optimization.

Suggested References: Fractional Differential Equations (Podlunby)

Prerequisites: Math 3

Winter 2021

Algebraic Geometry

Graduate Mentors: Richard Haburcak and Juanita Duque-Rosero

Mentees: Jacob Swenberg, Katie Woolfolk

Suggested References: Elementary Algebraic Geometry (Hulek, 2003)

Prerequisites: Math 31 or the equivalent

Representation Theory

Graduate Mentor: Alex Wilson

Mentee: Elizabeth Crocker

Representation theory takes the perspective that the right way to understand algebraic structures is to understand what they “do.” More precisely, a representation of a group is a vector space on which the group acts by linear transformations. This perspective has applications in physics and opens up a beautiful mathematical theory of its own. We will focus on the representation theory of finite groups and how it can be understood by looking at what are called “irreducible” representations, which act like atoms making up other representations.

Suggested References: Representations and Characters of Groups (James and Liebeck)

Prerequisites: Math 31 or the equivalent

Math for Social Systems

Graduate Mentor: Matt Jones

Mentees: Sara Catherine Cook, Alex Bakos

How do people make decisions? How do people in a group interact? How and why do friendships form and break? These are just a few questions that can be answered using mathematical models. This project could take many different paths, depending on interest. We could study networks, game theory, evolutionary dynamics, voting systems, coordination, etc. Before starting, we will discuss what topic and text would be best for you.

Suggested References: Newman's Networks, Nowak's Evolutionary Dynamics, Arrow's Social Choice and Individual Values, etc

Prerequisites: Linear algebra recommended for some topics

Knot Theory

Graduate Mentor: Sam Tripp

Mentee: Ahmed Naveed

Suggested References: The Knot Book (Adams), Knots and Links (Rolfsen), An Introduction to Knot Theory (Lickorish)

Prerequisites: Math 31 would probably be useful, and maybe Math 54, but don’t let those stand in the way of you learning interesting math.

Lambda Calculus and the Curry-Howard Isomorphism

Graduate Mentor: Zach Winkeler

Mentee: Dylan Fridman

Proofs are hard, and checking proofs can be taxing. The Curry-Howard isomorphism is a direct relationship between mathematical proofs and computer programs.

Suggested References: Lectures on the Curry-Howard Isomorphism (Sørensen and Urzyczyn)

Prerequitites: Math 31, or a proof based class; some computer science may be useful as well.