The Washington University Mathematic's Department Directed Reading Program is a Graduate Student run and organized selection of upper-level math courses aimed for undergraduate students. The purpose of these courses is to give undergraduates the opportunity to learn more niche higher math in a low-stakes environment, and to give the math graduates the opportunity to teach an upper-level course.
Current Organizer(s): John Naughton.
| Semester | Title | Instructor | Description |
|---|---|---|---|
| SP24 | Geometry, Euclid and Beyond | Zain Siddiqui | For this directed reading, we will go back to the very beginning of axiom-based mathematics by examining the work of Euclid. Euclid's Elements still stands today as an impressive feat of mathematical reasoning, and it has served as a blueprint for the way modern mathematics is done. That said, in order to put Euclidean geometry on a firm logical ground by modern mathematical standards, some revisions to Euclid's postulates are required. In this course, after pondering Euclid's approach for a while, we will study David Hilbert's more modern approach to axiomatizing Euclidean geometry, which introduces the notions of incidence and betweenness. Time permitting, we can either look at the material on constructability problems, or the material on non-Euclidean geometry (or whatever else in the book we find interesting). |
| SP24 | Introduction to (Toric) Algebraic Geometry | Shibashis Mukhopadhyay | Algebraic varieties are, roughly speaking, geometric objects described by the vanishing locus of a set of polynomials. Because of their rich structure and symmetries, they have enthralled mathematicians of all kinds from as early as the 17th century. Algebraic geometry explores the interplay between geometric properties of these spaces and algebraic properties of the polynomials defining them. We will begin with an introduction to classical algebraic geometry. Later on, we will work togther through many examples for a special class of varieties known as toric varieties. Because toric varieties have a very concrete description and many abstract constructions in the more general setting become reduced to explicit combinatorial computations, these provide a fertile ground for beginning to understand many fascinating phenomena in algebraic geometry. |
| FL23 | Large Language Models | Eric Pasewark | Large language models have become extremely popular, with many people using models such as ChatGPT in their daily lives. In this course, students will get acquainted with the latest LLM research and practice implementing techniques discussed. We will start with a brief introduction to machine learning and the transformer architecture. Then, we will dive into topics including: fine-tuning, RLHF, scaling laws, multimodal models, LLMs as agents, societal impacts, and more as time permits. The main course readings will consist of recent research papers. The math background necessary for this course is minimal, with multi-variable calculus and the ability to perform matrix-vector operations being sufficient. It is recommended to have some prior coding and machine learning experience, although some basics will be reviewed. |
| FL23 | Representation Theory | Nathan Lesnevich | How might we study groups without all the abstraction? Representation theory analyzes groups and other algebraic structures by insisting they act on vector spaces. This makes the group more concrete as its elements can now be represented as matrices, and we can use the tools of matrix algebra to study them. The first half of this course will go over the basics of representation theory for finite groups, and the second half will apply what we have learned to a very special collection: the symmetric groups. |
| FL23 | Introduction to Graph Theory | Cesar Meza | Imagine being tasked with coloring a map where no neighboring countries share the same color. What's the smallest number of colors you need to complete this task? Graph theory provides insights to help us answer this question. Ever wondered how social networks, transportation systems, and even the internet function? Graphs are the perfect tools to understand, analyze, and describe these intricate networks. This course offers a glimpse of math beyond calculus, revealing a new dimension of problem-solving and analytical thinking. Together we will play games, solve puzzles, spark discussions, and solve problems in a collaborative environment. The pace of the course will be determined by the students. This course is for anyone interested in discovering the joy of doing advanced mathematics. There are no prerequisites. All you need is a curious mind and a willingness to explore. We'll work together through the basics and delve into a deeper understanding of graphs and their properties. In addition, this course will give you a sneak peek of material covered in courses like Combinatorics and Graph Theory. |
| FL23 | Elliptic Curves | Mohao Yi | Given a polynomial with integer coefficients, a classical question in number theory is to find all of its rational solutions. These "Diophantine equations" are simple when our polynomials have a single variable; one can simply use the rational root test to determine all rational solutions. However, things become more interesting when we allow for two or more variables. Elliptic curves are intimately related to this story, for example they were used in Wiles' proof of Fermat's Last Theorem, which states that the Diophantine equation x^n+y^n=z^n has no(non-zero) integral solutions when n is greater than 2. Elliptic curves constitute a major area of research in their own right, as well as through their applications to number theory and cryptography. In this course we will study these amazing objects and some of their applications. |
| FL23 | Analytic Number Theory | Lars Nilsen | In the words of Gauss, "Mathematics is the queen of the sciences and number theory is the queen of mathematics." Number Theory concerns itself with arguably the simplest interesting mathematical structures: The Natural Numbers, or perhaps Integers. As such, questions under the umbrella of Number Theory can arise naturally from other fields of mathematics, or even just out of a curiosity for the question in its own right! It would thus be helpful to have tools to study and hopefully answer these questions, but despite the seeming simplicity of the question, oftentimes efficient answers may need to come from more \textit{complex} methods. In this course, we aim to explore some of these questions, and the surprising methods we can use to approach them. |
| SP23 | Introduction to Algebraic Geometry | RJ Acuna | Systems of polynomial equations and ideals in polynomial rings; solution sets of systems of equations and algebraic varieties in affine n-space; effective manipulation of ideals and varieties, algorithms for basic algebraic computations; Groebner bases; applications. |
| SP23 | Category Theory | Devin Akman | What do unions, intersections, Cartesian products, kernels, images, preimages, direct sums, and tensor products have in common? We'll delve into the dark art of category theory, which unifies these and much more into a single concept. Along the way, we'll decode those mysterious diagrams with dots and arrows that mathematicians are so fond of drawing. |
| SP23 | Introduction to B-series | Sanah Suri | For a differential equation y' = f(y), the B-Series or Butcher series of y is a series involving rooted trees and elementary differentials of f. This is a Taylor series where the elementary differentials are described using concepts borrowed from graph theory. B-Series allow us to develop broader classes of methods and study their properties. We will be looking at numerical methods like Euler's method, Runge-Kutta methods, and so on through the lens of B-Series. This will give us deeper insight into the numerical methods and their analysis. |
| FL22 | Introduction to Matroids | Jodi McWhirter | Matroids, introduced in the 1930s, are a mathematical object that generalize the idea of linear independence in vector space. They have many equivalent definitions, which both make them harder to get a grasp on but also give them the flexibility to show up in many circumstances. In this course, we will learn about matroids and explore their uses. The pace of the course will be determined by the students. During class, we will work through Oxley's Matroid Theory; additional resources include Federico Ardila's notes and videos on matroids. |
| FL22 | Introduction to Fourier Analysis | Ana Colovic and Jeremy Cummings | Fourier series allow us to study functions in terms of their elementary parts. We start our study of Fourier series on the circle, with the goal of eventually extending it to the real line. The course will consist of a mixture of lectures and student presentations. |
| FL22 | Elliptic Curves | RJ Acuna | We will study the work of Abel and Gauss on elliptic integrals and elliptic functions. And then we will develop the theory of elliptic curves over the complex numbers, and finite fields. Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles' proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization. |
| FL22 | Lie Groups | Devin Akman | A Lie group is a mathematical group (collection of symmetries) that also has the structure of a geometric space. Its associated Lie algebra is a vector space which "remembers" some local information about the group. In this course, we will study Lie groups (primarily matrix Lie groups) and Lie algebras along with their representation theories. As an application, we will see how representations of U(1), SU(2), and SU(3) give rise to properties of particles in the Standard Model. |
| SP22 | Lie Algebras | Nathan Lesnevich | Lie algebras arise naturally by giving the set of invertible matrices a new operation: the "Lie bracket". This bracket operation is often given by the commutator, [X,Y] = XY-YX. This construction will allow us to show that essential linear algebra operations such as finding an upper-triangular form and Jordan decomposition can be applied to entire collections of transformations all at once, rather than a single matrix. Root systems arise naturally by taking vectors stabilized by a group of reflections. These groups include some of the most well-studied groups, such as Dihedral and Symmetric groups. It is a fundamental result that there is a one-to-one correspondence between simple Lie Algebras and irreducible root systems. The goal of this class is to introduce enough of the theory of Lie algebras and root systems to show (NOT prove) this correspondence, and state the classification of semi-simple Lie algebras. |
| SP22 | Nonlinear Time Series | Jaiqi Li | Nonlinear time series are developed to investigate the nonstandard features well-observed in many real-life data that cannot be precisely modeled by linear time series, which include, for example, non-normality, nonlinear relationship between lagged variables, variation of prediction performance over the state-space, etc. Therefore, an extension to linear time series models is in great desire. In particular, we will introduce the definitions and characteristics of nonlinear processes. Both parametric and nonparametric approaches in analyzing time series data shall be covered briefly, which preserves the essence of the classical theory and methodology and also provides a state-of-the-art in data-analytic nonparametric techniques that have proven useful for analyzing real time series data. Fun examples of the application on real-life data will be presented using R codes to provoke the practical use. |
| FL21 | Erhart Theory | Jodi McWhirter | The volume of a polytope, a geometric object with flat sides in any dimension, is useful to know and often difficult to compute. Ehrhart theory focuses on the integer, or lattice, point count in polytopes and remarkably also yields the "continuous" volume. In this course, we will learn about polytopes and begin to explore the Ehrhart theory of integral and rational polytopes. |
| SP21 | Young Tableaux | Nathan Lesnevich | In this course, we will study the combinatorics of Young Tableaux, which are special assignments of integers to a diagram constructed from an integer partition. They have wide applications in a variety of topics and are fundamental in constructing the representation theory of symmetric groups and general linear groups. We will focus on construction and combinatorial operations for these tableaux, with the goal of understanding two of the best-known results: RSK correspondence and the Littlewood-Richardson rule. |
| SP21 | The Bergman space on the disc and Toeplitz operators | Nathan Wagner | The Bergman space is a fundamental and well-studied space of analytic functions on the unit disc. Toeplitz operators are a rich class of operators or linear transformations that act on functions in the Bergman space as compressions of multiplication operators. In particular, the function-theoretic properties of the so-called symbol of the Toeplitz operator dictate its operator-theoretic properties. The Hardy space on the disc and Toeplitz operators acting on the Hardy space are perhaps more familiar objects of study, but the Bergman theory is equally interesting. In this course, we will use ideas from operator theory, function theory, and real and complex analysis to study the Bergman space on the disc and the theory of Toeplitz operators, including Schatten class membership, boundedness, and compactness. |
| SP21 | Introduction to Chip-Firing | Hyojeong Son and Adeli Hutton | In this class, we will introduce the dollar game, which is a discrete dynamical system and an example of a chip-firing problem. The dollar game can model a number of natural phenomena; in particular, it can represent the distribution and movement of wealth in a community. In the context of this example, the goal of the dollar game is to make all members of the community debt-free through the lending of other members of the community. Considering this example gives rise to the setting of a graph: The dollar game is played on a finite, undirected, connected graph and consists of lending and borrowing moves along the edges. A graph's configuration is the number of chips at each vertex of the graph, which naturally translates to the notion of a divisor in this setting: a linear combination of the vertices based on the graph's configuration. Various linearly equivalent divisors can be obtained through sequences of lending and borrowing moves at the vertices, which we will study in the class. The theory of divisors on graphs is parallel to the theory of divisors on Riemann surfaces, and we will introduce a graph-theoretic version of the classical Riemann-Roch Theorem provided by Baker and Norine. This course will also provide examples of applications of this discrete version of the Riemann-Roch Theorem to several areas of mathematics. |