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.

 

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 2025-2026

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 - 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: Steinerberger - 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: Rothvoss - Analysis of boolean functions

The analysis of boolean functions deals with functions of the form f : {−1, 1} n → R where the main tool of choice is Fourier analysis. We will see a rich set of applications to combinatorics and theoretical computer science, in particular hardness of approximation. The course is inspired by the terrific textbook by O’Donnell from 2014 where we add some more recent topics. The chapters planned to be covered are:

  •  Introduction
  •  Linearity testing
  • The Goldreich-Levin algorithm
  •  Hardness of Approximation I (via PCP Theorem + Parallel Repetition)
  • Hypercontractivity
  • The invariance principle
  • The Majority is Stablest Theorem
  • Hardness of Approximation II — The Unique Games Conjecture and Hardness for
    MaxCut
  • Induced subgraphs of hypercubes
  • The Aaronson-Ambainis Conjecture

Prerequisites: Solid skills in linear algebra, analysis and probability

 581 B: Drouot - Mathematics of topological phases of matter

Classifying atomic arrangements is a central topic in quantum mechanics. We will study the mathematical theory behind this classification, as well as its implication on the Schrodinger equation. Indicative plan:

  • Tight-binding models. Graph Laplacians, Wallace model for graphene. Fourier transform.
  • Wavepacket evolution. Schrodinger and Dirac equation in graphene.
  • Spectral theory. Trace-class operators. Resolvent and Combes--Thomas inequality. Functional calculus via Helffer--Sjostrand formula. 
  • Topological insulators.  Eigenbundles and Chern numbers for periodic systems. Kubo formula and extensions to non-periodic models.
  • Dynamics of electrons. Gap-filling result: truncated topological insulators are conductors. Bulk-edge correspondence: the edge conductance is the Chern number. 

Prerequisites: Math 524-525. I will not assume any knowledge of quantum mechanics.

581 C: Pal - Optimal Transport + Stochastic Analysis

The modern theory of Monge-Kantorovich optimal transport is a very popular area right now in mathematics and its applications to statistics and ML. This is due to the interaction of the geometric intuition of classical OT with random particle systems and statistical analysis, mostly via what is known as entropic regularization. This two quarter long graduate topics course will serve as an introduction to this rich and useful theory.

Prerequisites: Basic measure theory, topology, functional analysis. Some knowledge of probability, including Brownian motion, will be very useful. Notes will be provided.     

581 D: Shi - Stable homotopy theory, equivariant homotopy theory, and the Kervaire invariant problem

 
TBD

 

Autumn 2025 Math 581: 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.

581 F: Wilson - Geometric Measure Theory: Fractals and Structure

This course will be based in geometric measure theory with a particular emphasis on questions concerning fractals, structure and geometric combinatorics.

Specifically, we will use iterated function systems as a medium through which to study the various notions of dimension of sets and measures and their interplay. This will include the study of Hochman's results using Shannon Entropy to approach the dimension drop conjecture. Furthermore, we will discuss projection theorems including Marstrand's projection theorem and Favard length (If there is time, we will discuss generalized projections of Peres and Schlag). Finally, we will discuss topics of pairwise distances and Kakeya sets including recent progress in these directions. 

The texts of Falconer (Techniques in Fractal Geometry), B\'ar\'any-Simon-Solomyak (Self-Similar and Self-Affine Seats and Measures), and Guth (Polynomial Methods in Combinatorics). Prerequisites: MATH 524 and Math 525.

582 A: Mikulincer - Localization techniques in High-dimensional geometry

The goal of this course is for the students to first become familiar with the concept of concentration of measure in different settings (Euclidean, Riemannian and discrete), and the main open problems surrounding it. The students will later become familiar with the proof techniques that involve the different types of localization and obtain expertise on the ways to apply the localization techniques. We will also explore various applications of these techniques, ranging from geometry and probability to statistics and theoretical computer science.

After taking the course, students will have the necessary background in order to both conduct research around open problems in concentration of measure, find new applications to existing localization techniques and perhaps also develop new localization techniques.

582 C: Pal - Optimal Transport + Stochastic Analysis

The modern theory of Monge-Kantorovich optimal transport is a very popular area right now in mathematics and its applications to statistics and ML. This is due to the interaction of the geometric intuition of classical OT with random particle systems and statistical analysis, mostly via what is known as entropic regularization. This two quarter long graduate topics course will serve as an introduction to this rich and useful theory.

Prerequisites: Basic measure theory, topology, functional analysis. Some knowledge of probability, including Brownian motion, will be very useful. Notes will be provided.     

582 D: Shi - Stable homotopy theory, equivariant homotopy theory, and the Kervaire invariant problem

TBD

582 E: Alper - Stacks

We will introduce the foundations of stacks with applications to moduli problems.  We will discuss topological and differentiable stacks, but the primary focus will be on algebraic stacks and their use to study moduli problems in algebraic geometry.

583 A: Hoffman - Ergodic theory

This course is an introduction to ergodic theory. Ergodic theory is the study of dynamical systems from a measurable or statistical point of view. Starting with Poincar ́e’s recurrence theorem and the ergodic theorems of Birkoff and von Neumann ergodic theory in the early twentieth century. The field has applications to many other areas of mathematics including probability, number theory and harmonic analysis. Among the topics covered will be
• examples of ergodic systems
• the mean and pointwise ergodic theorems 
• mixing conditions
• recurrence
• entropy and
• the Ornstein’s Isomorphism Theorem.
 

 583 B: Inchiostro - Algebraic surfaces

The aim of this course is to study smooth algebraic surfaces over the complex numbers, with the goal of covering Castelnuovo’s theorem and some of the classification results for surfaces in the book of Beauville. The prerequisites are our algebraic geometry sequence.

 583 C: Uhlmann - Microlocal Analysis and Semiclassical Analysis

 

Microlocal analysis comprises techniques developed from the 1950s on-ward based on the Fourier transform related to the study of variable coefficients, linear and non- linear partial differential equations. This includes distributions, pseudodifferential operators, wave front sets, Fourier integral operators, and paradifferential operators. It has applications in many fields including inverse problems, general relativity, quantum mechanics, quantum field theory, spectral theory etc. 

Semiclassical analysis is a branch of mathematical physics and analysis that studies the behavior of quantum systems in the limit where Planck’s constant goes to zero. It serves as a bridge between classical mechanics and quantum mechanics, offering approximations of quantum phenomena using classical concepts.

We will give applications of Microlocal Analysis and Semiclassical Analysis to Inverse Problems.

Textbook We will use the new book by Peter Hintz ”An introduction to Microlocal Analysis, Springer Verlag, GTM 304.

Prerequisites: Math 524 and Math 525

 

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