The DRP is running in Spring Term 2024!
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Analytic Combinatorics
Graduate Mentor: Matt Ellison
We will begin by learning to set up "generating functions" - formal power series which count combinatorial structures (e.g. molecular arrangements, structured permutations, parenthesizations, ...). Then, treating these generating functions as functions of a complex variable, we will see how singularity analysis yields insight into the asymptotics of the combinatorial structure under consideration.
Suggested References: Analytic Combinatorics, by Philippe Flajolet and Robert Sedgewick
Prerequisites: A first course in combinatorics / discrete math
Analytic Number Theory
Postdoc Mentor: Tristan Phillips
The integers are perhaps one of the most fundamental objects in all of mathematics. We learn about them when we are little and we realize that we do not really understand them when we are big. For example, seemingly simple questions like "How many primes have the property that p-2 is also a prime?" and "Is every even number greater than 2 a sum of two primes?" have not yet been answered. Though there are many approaches to studying the integers, this reading course will focus on analytic methods. The goal will be to understand the prime number theory, which gives estimates for the number of prime numbers up to a given bound.
Suggested References: "Problems in Analytic Number Theory" by Ram Murty, "Introduction to Analytic Number Theory" by Tom Apostol
Prerequisites: Courses in real analysis and complex analysis would be helpful, but not necessary (assuming some enthusiasm).
Differential Geometry
Graduate Mentor: Friedrich Bauermeister
We will discuss semi-Riemannian geometry, which generalizes both Riemannian geometry ("the geometry of space") and Lorentzian geometry ("the geometry of spacetime"), from the ground up. The book used will be Barret O'Neill's Semi-Riemannian geometry with applications to relativity, which is a really good book. This DRP is an excellent chance to (1) first learn about smooth manifolds or to strengthen ones familiarity with them, (2) gain insights into how to "properly" generalize notions of straight lines, distances, and angles to the setting of smooth manifolds, and (3) and to learn about the mathematics underlying general relativity in a rigorous way.
Suggested References: Barret O'Neill's "Semi-Riemannian geometry with applications to relativity".
Prerequisites: (proof-based) experience with linear algebra and with at least one of analysis or point-set topology. Experience with smooth manifolds is not required but advantageous.
Algebraic Geometry
Postdoc Mentor: Sarah Frei
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 structures 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
Commutative Algebra
Postdoc Mentor: Sarah Frei
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.
Suggested References: Undergraduate Commutative Algebra (Reid), Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud)
Prerequisites: Math 31 or the equivalent
Category Theory
Postdoc Mentor: Sarah Frei
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 willfully 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 or the equivalent
Elliptic Curves
Postdoc Mentor: Sarah Frei
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.
Suggested References: Rational Points on Elliptic Curves (Silverman)
Prerequisites: Math 31 or the equivalent
Representation Theory
Postdoc Mentor: Sarah Frei
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
geometric numerical integration
Postdoc Mentor: Lizuo Liu
Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are important to physics and astronomy. This DPR project might go through several algorithms that preserve the geometry, including symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators.
Suggested References: Ernst Hairer , Gerhard Wanner , Christian Lubich Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
Prerequisites: Ordinary differential equation, numerical methods for ODE
Representation Theory of Finite Groups
Graduate Mentor: Beth Anne Castellano
Representation theory allows us to study abstract algebraic structures such as groups by "linearizing" them. In particular, a representation of a group is a vector space equipped with a linear action of that group. This allows us to associate matrices to group elements. As it turns out, representations of finite groups can be completely understood by studying their building blocks, called "irreducible representations." Moreover, "characters," which are traces of matrices, give us all of the data we need to distinguish representations of the same group. In this reading course, we'll marry our knowledge of linear and abstract algebra to explore the fundamental ideas of this theory and gain a new perspective on groups! (Depending on the interests of the student(s), we could also focus on the representation theory of the symmetric group, which is very combinatorial.)
Suggested References: Representations and Characters of Groups (James and Liebeck), and/or The Symmetric Group (Sagan)
Prerequisites: Math 31 or 71
Fractal Geometry
Graduate Mentor: Rohan Kapoor
In the past, mathematics has been concerned with smooth sets and functions, to which calculus can applied. Sets that are not smooth or regular have been ignored as pathological. However, a few decades ago, with the pioneering work of Benoit Mandelbrot and others, it has been realized that both a great deal can be said about the mathematics of non-smooth sets, and that they provide a much better representation of natural phenomena that some figures of classical geometry. Fractal geometry is a framework to study these ideas, and we will read this. A student will need a previous course in Real Analysis, as well as some basic programming experience.
Proofs from the BOOK
Graduate Mentor: Rohan Kapoor
Paul Erdos often said that there was a Book in which God kept the most elegant proof of every theorem in mathematics. He was quoted to have said in a lecture that "[to do mathematics] you don't have to believe in God, but you should believe in The Book." Martin Aigner and Gunter Ziegler wrote a book dedicated to the memory of Paul Erdos compiling a few proofs from the book. This course would be a relaxed exploration of a few of these proofs. No particular mathematical knowledge is required, but experience with proof-based mathematics is necessary.
AI Safety and Alignment
Graduate Mentor: Rohan Kapoor
Artificial Intelligence poses an existential risk to society. In this project, we would do a survey of topics in research aiming to prevent this, and better encode safety features into this. We would discuss the Alignment Problem: getting AI systems to behave exactly how we instruct them to, and understand how this fails, as well as have a bird's eye view of research fields aiming to prevent this. The course would end with either a project replicating a modern paper in the field, or a presentation on general ideas in the field depending on where the student wants to take it. This course will not, however, spend a lot of time on government policy/regulation, and rather focus on technical challenges. We won't follow a single textbook, but rather this curriculum (free for public use).
Quantum Mechanics for Mathematicians
Graduate Mentor: Casey Dowdle
Quantum Mechanics encompasses a rich variety of mathematical subjects from functional calculus and spectral theory to PDEs and representation theory. It also has many practical applications including chemistry, condensed matter systems, particle physics, and quantum computing. We will be reading "Quantum Theory for Mathematicians" by Brian Hall. We will start with a brief but formal review of classical and quantum mechanics. We will then cover spectral theorems for self adjoint operators. Finally, depending on the students interest we can cover either representation theory of angular momentum or the path integral formulation of quantum mechanics.
Suggested References: "Quantum Theory for Mathematicians" - Brian C. Hall
Prerequisites: Math 11 or 13 (Calculus), Math 22/24 (Linear Algebra), some familiarity with physics would be helpful but is not required.
Category Theory and Further Mathematical Abstraction
Advanced Undergraduate Mentor: Benjamin Singer
When first learning linear and abstract algebra, we repeatedly encounter concepts that transcend sets, groups, rings, vector spaces, and algebras. Notions like injectivity, surjectivity, isomorphism, "sub-____s", "quotient ____s", and kernels are ideas that we can consider with all algebraic structures. These notions also satisfy certain "universal" properties that make using them convenient. Furthermore, there seem to be ways we can obtain algebraic structures from other algebraic structures, such as taking the abelianization of a group, matrix algebras over a field, or the free vector space of a set. All of these observations can be made rigorous through the lens of category theory! In this program, we will go over the basics of category theory: how many basic constructions using categories abstract what we already know, how certain types of categories (specifically abelian and monoidal categories) become extremely useful in further fields of math such as homological algebra and algebraic geometry, and how we can prove many difficult theorems much more easily using category-theoretic results. Students will build upon the abstract thinking that their first abstract algebra course first formed, alongside gaining familiarity with concepts that are prevalent in higher fields of math like universal properties, diagram chasing, functoriality, adjunction, and representability.
Suggested References: Saunders MacLane, Categories for the Working Mathematician
Prerequisites: Math 31 or 71
Mechanism Design and Algorithmic Game Theory
Graduate Mentor: Brian Mintz
What policies should the government set to incentivize specific behaviors, like switching to electric cars or following the law? Which auction format is the most efficient? Can we "solve" traffic? Mechanism is an exciting field at the intersection of economics, mathematics, and computer science that attempts to rigorously answer questions like these by formulating computational approaches to analyze and optimize rational decision making. This project will explore the theory of this field and some of its many practical applications.
Suggested References: Algorithmic Game Theory 1st Edition by Noam Nisan, Tim Roughgarden, Eva Tardos, Vijay V. Vazirani
Prerequisites: None.
Complex Systems Theory
Graduate Mentor: Brian Mintz
From physical matter and the economy to societies and our brains, many systems are characterized by a large number of simple parts. However fundamentally different rules seem to govern the macroscopic trends that emerge from microscopic interactions. Understanding how to translate between these scales is the core of an innovative approach reshaping our scientific models of the universe, while informing existential questions like whether free will is possible in a deterministic universe. This project will walk through the fundamentals of this fascinating field while keeping a perspective towards its profound implications.
Suggested References: Introduction to the Theory of Complex Systems by Stefan Thurner, Rudolf Hanel, and Peter Klimek
Prerequisites: None, though basic probability (MATH 20) is preferred.