Special Offerings

2025-2026 Special Course Offerings 

(Subject to change, check the time schedule for most current information.)

Undergraduate Special Topics

Summer 2025, Math 380 A: Communicating Mathematics Through Sports Data

This course bridges the gap between mathematical analysis and communication, using sports statistics and analytics as our playground. Students will develop technical skills in Excel/R/Python as well as mathematical exposition skills, learning to transform raw sports statistics into compelling narratives for non-technical audiences. Through weekly readings from Mathletics: How Gamblers, Managers, and Fans Use Mathematics in Sports, hands-on work with real-world datasets, and a culminating video project, students will develop fundamental data literacy and practice the crucial skill of making complex mathematical concepts accessible to general audiences. While sports provide our primary lens, the course emphasizes universally applicable skills in data analysis, visualization, and communication that prepare students for bridging technical and non-technical audiences across any sector, an essential skill for any role in modern data-driven careers.

Prerequisites: At least one previous reasoning course and interest in sports or popular mathematics would be great! This course is open to math majors looking to get their feet wet working with real-world data, and to students with a deeper data science background to develop the skill of communicating their analysis to a general audience.


Text and materials:
● Mathletics: How Gamblers, Managers, and Fans Use Mathematics in Sports, Second Edition by Konstantinos Pelechrinis, Scott Nestler, and Wayne L. Winston
● Storytelling with Data: A Data Visualization Guide for Business Professionals by Cole Nussbaumer Knaflic
● Real sports datasets from sportsreference.com

Summer 2025, Math 380 B: The Mathematical behind Quantum Computing


Quantum computing has become a big term that we have always heard in the news nowadays. Due to special properties of quantum states, quantum computing offers strong computational power. For example, classical computers cannot factorize an arbitrarily large integer into primes within polynomial time; however, quantum
computers can do this with Shor’s algorithm. Even though quantum computing offers high computational power, it is currently hard to conduct many classical computational tasks in quantum computers. It is difficult
to build a good quantum computer and to design quantum algorithms, making this topic more attractive and mysterious. In this course, we will explore the basic mathematical principles behind the fancy term “quantum computing”, and we will follow the book "Quantum Computing: From Linear Algebra to Physical Realizations" by Nakahara and Ohmi. While the subject may appear complex, we will take a step-by-step approach, beginning with the basics of quantum bits (qubits, or just some nice vectors with tensor products), quantum gates (unitary matrices), basic quantum physics principles, and quantum algorithms (a series of unitary matrices that multiply to give us meaningful outcome). Towards the end of the quarter, students will break into groups to do a final project on various topics and deliver a brief presentation in class. Potential projects may include more advanced topics like the quantum Fourier transform, quantum error correction codes, or if the student is interested in coding, they can also do some small actual implementations of quantum algorithms on real quantum computers.

Prerequisites: Math 208 (basic linear algebra)

Textbook: Quantum Computing: From Linear Algebra to Physical Realization by Mikio
Nakahara and Tetsuo Ohmi

MATH 380B Registration Request Form Form opens 6:00am (Pacific Time) April 14, 2025.

Autumn 2025, Math 380: Math That Lies, Communicating Why Some Quantitative Arguments Are Misleading or Bogus (a Calderwood Seminar in public writing) 

The Calderwood Seminars started at Wellesley College in 2013 and later spread to a few dozen colleges and universities nationwide. The Calderwood seminars at UW started in 2019. The seminar offers a unique opportunity to work collaboratively with fellow students in the writing and peer-editing process.  The course
requires commitment, curiosity, and a critical mindset. It is open to juniors and seniors, and qualifies for W-credit and DIV-credit.

Although it’s important to educate the public about the ways that our field contributes to technological advance, medical progress, etc., mathematically trained people also have a special role to play in critiquing the fallacious and misleading ways that statistics, equations, and mathematical models are often used. Students in the seminar will learn: 

  • how bogus statistical arguments were used to bolster white supremacist pseudoscience (from a book by paleontologist Stephen Jay Gould); 
  • how proxy data fed into algorithms led to a revival of redlining, a practice that had supposedly been outlawed by the civil rights legislation of the 1960s (from a book by mathematician Cathy O’Neil); 
  • how misuse of probabilistic arguments has repeatedly led to the conviction of innocent people (from a book by Leila Schneps and Coralie Colmez, a mother-and-daughter team of mathematicians);   
  • how mathematical modeling early in the Covid-19 pandemic that was based on false assumptions gave support to Donald Trump’s claims that Covid-19 would be no worse than the flu. 

In a given week half of the class will write commentary on the reading, in the form of a book review, a blog posting, a newspaper article, or a letter to the editor; the other half will be student-editors who meet with the writers to suggest corrections and improvements in their drafts. That will be followed by workshopping all the 2nd drafts during class time. 

LEARNING GOALS
• To increase your skill and confidence as writers.
• To learn how to collaborate effectively as editors and workshop participants.
• To learn how to process, analyze, and criticize mathematical arguments related to
socially important controversies.
• To learn how to communicate in clear, crisp, lively, and error-free prose about
the challenges and pitfalls in interpreting quantitative information.

Prerequisite: None

Please note: This is an intensive 5-credit course. Every week you need to (1) do all the
assigned reading (whether you’re a writer or editor), and do it carefully; (2) proofread your
written work and revise it carefully in response to feedback, if you’re a writer; and (3) give
the writer extensive helpful corrections and suggestions, if you’re an editor. During the
first few weeks of the quarter I will check (1) by giving quizzes on the assigned reading
and monitor (3) by viewing Zoom recordings of the editor-writer meetings and by noting
what shape the written work is in when we workshop it in class. In the event that you do
not do a conscientious job as editor in your first Zoom meeting with a writer, I will ask
you to continue making Zoom recordings of your editing sessions so I can check for rapid
improvement in that crucial aspect of the seminar. Of course, (2) will also affect the grade
directly, because the final version of the written work will be a major component of the
grade.

Winter 2026, Math 480/Math 582: Mathematics and AI

Description: The field of mathematics is experiencing transformative changes driven by the confluence of the symbolic reasoning of interactive theorem provers and the statistical reasoning of superpowered machine learning algorithms.  This course will survey the dynamic interplay between mathematics and artificial intelligence (AI). What are the mathematical foundations of AI, and conversely how can AI accelerate mathematical research?

We will learn the foundations of neural networks, build simple neural networks from scratch, and also build advanced models using the Python library PyTorch.  We will cover theoretical results such as the Universal Approximation Theorem (asserting that the functions computed by feed-forward neural networks are dense in the space of continuous functions) as well as recent results on the theoretical limitations of neural networks and transformers.  

We will also survey the history of symbolic AI, otherwise known as good old-fashioned AI, and we will cover one of its greatest successes, namely, interactive theorem provers.  We will learn how to formalize mathematical theorems using the Lean Theorem Prover.  Finally, we will survey the state-of-the-art of autoformalization, which is the automation (or at least partial automation) of the formalization process.  We will cover various search algorithms for proof generation including how various tactics (such as exact?, simp, ring, linarith, and omega) are implemented in Lean as well as explore applications of reinforcement learning algorithms in the style of Alphazero.  A large component of this course are group projects, each focused on an exploratory question within one of the themes of Math AI.

Spring 2026, Math 380: Computational Algebraic Geometry

This course will focus on the algebra, geometry, and algorithms involved in understanding the set of solutions to a system of nonlinear polynomial equations. Topics will include an introduction to polynomial rings and ideals, affine varieties, monomial orderings, Groebner bases, elimination theory, Hilbert's Nullstellensatz, and applications. 


Prerequisite: either MATH 334, or both MATH 208 (or MATH 136) and MATH 300. 

 

Spring 2026, Math 336/480: Differential Topology 

We'll develop multilinear algebra and the theory of differential forms, including exterior differentiation, integration of forms and the generalised Stokes' theorem. We'll introduce smooth manifolds and explore transversality and intersection
 theory. If time permits, we will explore de Rham cohomology and/or applications to physics. 

Prerequisites: Either 441 and 224, or 335 or 425.


Graduate Special Topics 2024-2025  --- 2025-2026 coming soon!

NOTE: courses are restricted to current Mathematics graduate students or those who have completed a minimum of three 500-level core math courses.  If you have questions about this enrollment requirement visit our page about registering for 500-level courses.

504/505/506 McGovern - Modern Algebra

See Syllabus.DVI (washington.edu) for the full list of topics.

 

524/525: Real Analysis

524: We will cover essentially all of Chapters 1 through 3 of "Real Analysis: Modern Techniques and Their Applications" by  Folland (2nd Edition)

525: We will cover Chapter 5, and parts of Chapters 6 - 7 of Folland. If time permits I may do a quick review of point set topology along the way, depending on whether students have seen it.

 

534/535: Rohde - Complex Analysis

The course covers the following topics:

  • The complex exponential and Polar form
  • Complex logarithms and roots
  • Complex differentiation and the Cauchy-Riemann equations
  • The Riemann sphere
  • Linear fractional transformations
  • Contours and line integrals
  • Cauchy's theorem and Cauchy's formula
  • Power series expansions of analytic functions
  • Zeroes of analytic functions
  • Laurent expansions
  • The structure of isolated singularities
  • The residue theorem
  • Integrals over the real line and trigonometric integrals
  • The argument principal
  • Theorems of Rouche and Hurwitz
  • Local inverse theorem
  • Simple connectivity
  • Maximum modulus theorem and Schwarz's lemmma
  • Conformal automorphisms of D and H.
 

581 A: Wilson - Classical & Multilinear Harmonic Analysis 

This course will focus on the study of classical and multilinear harmonic analysis. In particular, the main goal of the class will be to build the theory of wave packet analysis developed by Carleson and used to solve problems related to convergence of Fourier series and elliptic boundary value problems. We will follow the texts of Muscalu and Schlag ("Classical and Multilinear Har- monic Analysis: Volume 1 and Volume 2") as well as the text of Thiele ("Wave packet analysis"). If time permits, we will end with more recent methods of decoupling due to Bourgain and Demeter.

Prerequisites: Math 524 and Math 525

 581 B: Novik - Discrete Geometry Potporrui

Discrete Geometry deals with finite sets of points, lines, planes, circles, and many other seemingly simple geometric objects. It has deep connections to combinatorics, optimization, number theory, and computer science. It is also very rich in simple-to-state yet long-unsolved problems. This course will be a sampler of a few of the topics in this vast field. We will start with classical theorems due to Radon, Helly, and Caratheodory as well as the Steinitz theorem and Dehn's solution to Hilbert's third problem. These 100-year-old results very quickly lead to much more modern developments such as (1) fractional, colorful, and topological versions of some of these theorems, (2) Kahn--Kalai's counterexample to Borsuk's problem, (3) Paco Santos' disproof of 53-year-old Hirsch's conjecture, (4) Florian Frick's counterexamples to the topological Tverberg's conjecture, (5) high-dimensional paradoxes, (6) face enumeration on polytopes. We will cover some of these topics (and perhaps a few others). We will not follow any particular textbook. Instead, we'll use quite a few sources (including several recent papers). Some of the textbooks that might be handy are:

  • Imre Barany, ``Combinatorial convexity'', 2021
  • Jiri Matousek, ``Lectures on Discrete Geometry'', 2002
  • Igor Pak, ``Lectures on Discrete and Polyhedral Geometry'', available at https://www.math.ucla.edu/~pak/book.htm

Prerequisites: good understanding of core 400-level math classes.

581 C: Billey - Schubert Calculus of Flag Varieties and Grassmannian Varieties

The study of Schubert calculus of Grassmannians and flag manifolds and the associated structures in algebraic geometry, algebraic topology, combinatorics and representation theory inspired some of the great advances in mathematics in the twentieth century. Inherently, the subject rests firmly on the study of matrices, determinants, intersections of linear spaces (meaning vector spaces and their translates), symmetry, and computation. We will introduce the flag variety from a modern perspective and present some of the tools used in intersection theory and the cohomology rings mostly from an algebraic and combinatorial perspective.
 
Textbooks:
  1. "Introduction to the Cohomology of the Flag Variety" by Sara Billey, Yibo Gao, and Brendan Pawlowski. To appear 2024. This article is intended to be the first chapter of a book entitled ``Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety'' that is a compendium of topics in this area. The book is being edited by Erik Insko, Martha Precup, and Ed Richmond. The other chapters will each be written by different authors. Selected experts from other chapters will also be covered.
  2. "Young Tableaux; With Applications To Representation Theory And Geometry" by William Fulton. Cambridge University Press. 1997.
Prerequisites: Math 504/505/506 and 561 or permission from the instructor are required.

581 D: Kovacs - Moduli Theory of Higher Dimensional varieties

 
Topics to be included:
 
  • Overview and some history of moduli theory, statements of the main results
  • Comparison to M_g. Why is this a lot harder? (difficulties, obstacles, long time roadblocks)
  • Stable and related singularities and their deformations
  • Moduli theory without a boundary
  • Moduli of pairs I: flat divisorial part, ``large'' coefficients
  • Moduli of pairs II: K-flatness, marked pairs.

581 E: Viray - Algebraic Points on Curves 

This course will give an overview of the arithmetic of curves over global fields, with a focus on results on points of higher degree and on the arithmetic of modular curves. Students should have background in Galois theory and algebraic geometry.

582 A: Sanmarco - Symmetric Tensor Categories

 

582 B:  Steinerberger - Graduate-level Linear Algebra

Graduate Level Introduction to Linear Algebra, no prerequisites.

 

582 C: Liu, R - Dynamical Algebraic Combinatorics

Dynamical algebraic combinatorics is the study of group actions on objects commonly studied in algebraic combinatorics such as words, partitions, or Young tableaux. In this course we will study various types of actions, most commonly symmetric group or cyclic group actions, often generated by simple local operations, and study the resulting behavior and emergent phenomena using techniques from combinatorics and representation theory. Particular emphasis will be placed on concrete examples.

Topics will include:

  • group actions on sets and posets, Polya counting, and cycle index polynomials
  • the q=-1 and cyclic sieving phenomena
  • homomesy and resonance phenomena
  • Bender-Knuth involutions and crystal operators, promotion, and evacuation
  • toggles and rowmotion on order ideals and plane partitions
  • piecewise-linear and birational analogues.
  • Additional topics will be covered depending on time and interest.

Prerequisites: Some familiarity with algebra (504/505/506) and combinatorics (561/562), or permission of the instructor

582 D: Vinzant - Tropical Geometry

Description: Tropical geometry is the study of certain combinatorial shadows of solutions to systems of polynomial equations. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These combinatorial structures retain a surprising amount of information about their classical counterparts. This course will introduce and survey some topics in tropical geometry, including Puiseux series and valuations, Grobner complexes, tropical varieties, hyperplane arrangements and matroids, Bernstein's theorem, Viro's patchworking, and combinatorial Hodge theory.

Some familiarity with ideals and varieties will be assumed. Grades will be based on occasional homework assignments and a final project.

582 E: Paternain - Microlocal methods in dynamical systems

Description: In recent years there has been notable progress in the application of microlocal techniques to hyperbolic dynamics, showing for example that the Ruelle zeta function admits a meromorphic continuation to the entire complex plane (a conjecture of Smale from the 60s) or giving an alternative proof to the ergodicity of the geodesic flow in negative curvature. The objective of the course would be to introduce students to this circle of ideas.

No background is necessary in either microlocal analysis or dynamics; it will be provided during the course. Familiarity with manifolds, Fourier transforms and ODEs is important.

583 A: Zhang - Invariant Theory of Noncommutative Algebras

We will cover some of recent developments in noncommutative invariant theory. Here is a list of potential topics:

  1. Local cohomology and Artin-Schelter (AS) regular algebras.
  2. Group actions, Hopf algebra actions, and Homological determinant.
  3. Shephard-Todd-Chevalley Theorem for AS regular algebras.
  4. Gorenstein property and Watanabe's Theorem.
  5. Auslander theorem, pertinency and McKay quiver.

Prerequisites: First year graduate algebra sequence

 583 B: Steinerberger - Graduate-level Linear Algebra

 

Graduate Level Introduction to Linear Algebra, no prerequisites.

 583 C: Athreya - Translation Surfaces and their Moduli Spaces

This will be a course translation surfaces and their moduli spaces, with a focus on associated dynamical systems, counting problems, and group actions. In recent years, translation surfaces and their moduli spaces have been the objects of extensive study and interest, with connections to widely-varied fields including (but not limited to), geometry and topology; Teichmuller theory; low-dimensional dynamical systems; homogeneous dynamics and Diophantine approximation; and algebraic and complex geometry.

Recommended (but not required) prerequisites would be complex analysis and measure theory, and background in manifolds may be helpful.

 583 D: Thomas - Symmetry Reduction in Optimization

Techniques from representation theory can be used to reduce the size of optimization problems that are amenable to group actions. This is sometimes the only way to make these optimization problems tractable and there have been many successful applications of this idea in recent years. This course will develop the needed techniques via an array of examples from graph optimization, sum of squares optimization, geometric (sphere) packing, graph homomorphism inequalities etc.

Prerequisites: Some background in group theory and optimization will be helpful. No representation theory background is needed.

 583 E: Uhlmann - Mathematics of Medical Imaging

In this course we will describe the Mathematics of several medical imaging techniques currently used to probe the body including computed tomography which led to the development of CT scans, Positron Emission Tomography (PET), Single Positron Emission Tomography (SPECT) and ultrasound. The Mathematics of these methods involve the inversion of Radon transforms and generalizations that we will study in detail. The following are suggested references, we will follow mostly 6) that is a book that is being proofread now. This will be made available for the students taking the class.

References.

  1. C. Epstein: Introduction to the Mathematics of Medical Imaging.
  2. S. Deans: The Radon transform and S0me of its Applications.
  3. F. Natterer The Mathematics of Computed Tomography.
  4. P. Kichment. The Radon Transform and Some of Its Applications.
  5. S. Helgason. The Radon Transform.
  6. P. Stefanov and G. Uhlmann. Microlocal Analysis and Integral Geometry.

Prerequisites. Math 524 and Math 535, the Real-Analysis course.

 583 F: Zhu: Minimal Surfaces

Course description: The study of the minimal surface spanning a wire is a central problem in differential geometry, dating back to Lagrange in the 1760s but still bringing forth deep mathematical challenges and innovations today. We'll cover the basic theory of minimal surfaces, towards characterisations of stability. If time permits, we'll discuss some recent breakthroughs related to the positive mass theorem and the classification of stable minimal surfaces.

Prerequisites: Manifolds sequence, some familiarity with Geometric Structures/PDE

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