## Winter 2023

### Mathematics and Machine Learning

#### 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

#### 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

#### 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

#### 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

#### Mentee: Paul Shin

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

### Mathematical Bioeconomics

#### 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

#### 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

#### 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

#### 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

#### Mentees: Mimi Grozeva, Heather Wang, and Daniel Carstensen

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

## Spring 2022

### The Probabilistic Method

#### 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

#### 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

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

#### 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

#### 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

#### 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

#### 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

#### 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

#### 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

#### Mentee: Jessica Liu

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).

### Fractional Calculus

#### 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

#### Mentees: Jacob Swenberg, Katie Woolfolk

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)

Prerequisites: Math 31 or the equivalent

### Representation Theory

#### 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

#### 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

#### Mentee: Ahmed Naveed

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.