## 2024-2025 Special Course Offerings

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

## Undergraduate Special Topics

- Autumn 2024
- Winter 2025
- Spring 2025

#### Autumn 2024, Math 180: Mathematics and Democracy

We'll take a mathematical look at problems that arise in a representative democracy. Some questions we may consider:

- How can we determine the winner in a ranked-choice election? What are the advantages and disadvantages of different systems?
- In a voting system with unequal weights like the Electoral College, how can we measure the relative power of different groups?
- In the House of Representatives, each state is assigned representatives according to their population. How exactly does that work?
- What algorithms can we use to divide limited resources, or settle disputes between parties with different goals and priorities?
- Can mathematics help us create fair congressional districts or detect gerrymandering? Along the way, we'll learn about different areas of math that can help us answer these questions.

Prerequisite: None

#### Autumn 2024, 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.

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

#### Winter 2025, Math 180: A Walk in the Garden of Mathematics

This course is aimed at first- and second-year students who are interested in finding out what real proof-based mathematics is all about, and above all, beautiful. The class will cover elementary results from different areas of mathematics together with their beautiful proof – indeed, the availability of a beautiful, insightful, non-technical proof should inform the selection of results. If the only available proof is long and technical and not insightful, maybe the Theorem should be left for another math class.

Motivated students should be able to follow the course content without difficulty provided they put in the time (though not an unreasonable amount of time). The goal for such a class is to help students decide if they would like to major in math. Course topics will include some or all of the following:

- Proof as an explanation why something is true. Euclid and his axioms. Axioms as rules of a game. Direct Proofs. Indirect Proofs. Proofs by Induction.
- The fact that we don’t know everything: mathematics as an evolving science.
- The Pigeonhole Principle and its applications. Kronecker Approximation Theorem as an immediate consequence. Discuss Thue-Siegel-Roth Theorem (without proofs of course). Convergence of the Flint-Hill series.
- Countable Sets & Uncountable Sets.
- Basic Graph Theory.
- Discuss Brouwer’s Fixed Point Theorem
- Importantly: Always show how the simple result we just proved is adjacent to a problem that has not been solved for 50+ years. We don’t know everything, people still discover new math every day, it is an active science!

Prerequisite: None

#### Winter 2025, Math 480: Mathematical Introduction to Data Analysis and Image Processing

This class will discuss theoretical aspects of data analysis and image processing techniques. We will cover principal component analysis and its applications, graph theory, Fourier-based techniques such as compressing and denoising, and (if time permits) an introduction to neural networks. There will be a few Python projects, but no preliminary knowledge of programming is necessary. Lecture notes: https://www-users.cse.umn.edu/~jwcalder/5467Notes.pdf

Prerequisites: A proof-based real analysis class such as Math 327 or 335; and an advanced linear algebra class such as Math 318 or 340.

#### Spring 2025, Math 180: Mathematics in Games

In this course we will explore the mathematics behind games. We will look into winning strategies, probability of winning and patterns the games reveal. This course will connect the best of two worlds: having fun playing and analyzing the situation through the logic lense of a mathematician. Anyone who likes to get a glimpse into proofs and the methodical ways of abstraction is welcome to join. Many ideas will be taken from the book 'Math Games with Bad Drawings' by Ben Orlin.

Prerequisite: None

#### Spring 2025, Math 480 A: The Burnside Problem

The Burnside Problem(s) tantalized the algebra community over the 20th century. The general version, posed in 1902, asks whether a finitely generated group in which every element has a finite order must be finite. It was resolved by Golod and Shafarevich more than 60 years afterwards, discovering fascinating methods in combinatorial ring theory which found surprising applications in algebraic number theory, lattices and more. Later on, dramatic developments in group theory enabled Adian-Novikov and later Olshanskii (using geometric methods) to solve the bounded version of the problem: must a finitely generated group with a bound on the orders (that is, finite exponents) of its elements be finite? Finally, in the early 90s, Efim Zelmanov solved the Restricted Burnside Problem, showing that there is a largest finite group with any given number of generators and a given exponent, using ingenious connections between p-groups, Lie algebras, Jordan algebras and polynomial identities. We will discuss these three versions of the Burnside Problem and overview some of the core ideas behind their solutions. We will focus on the novel concepts that these solutions required and revealed, and on their impacts on different mathematical disciplines.

Textbook: * A. I. Kostrikin, "Around Burnside", Ergeb. Math. Grenzgeb. (3), 20 [Results in Mathematics and Related Areas (3)] Springer-Verlag, Berlin, 1990. * A. Yu. Ol'shanskii, "Geometry of defining relations in groups", Kluwer Academic Publishers Group, Dordrecht, 1991.

Prerequisites: Math 402 - familiarity with groups and rings

#### Spring 2025, Math 480 B: Introduction to Mathematical Formalization

This course will explore how mathematical theorems can be formalized in computer proof systems such as the Lean Theorem Prover. We will focus on topics in undergraduate mathematics such as set theory, logic, functions, induction, number theory, and other topics typically encountered in Math 300: Introduction to Mathematical Reasoning. For each topic, we will review the mathematical foundations, and then attempt to formalize main theorems and exercises. As the course progresses, we will learn more and more sophisticated high-level tactics that will assist in the formalization. By the end of the quarter, we will have group projects, where each group chooses a mathematical statement to formalize, depending on the interests of the students.

Prerequisites: Math 300, programming experience

Graduate Special Topics 2023-2024

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 (A/W/Sp): Pevtsova, Pevtsova & McGovern - Modern Algebra
- 524/525 (A/W): Smith - Real Analysis
- 534 A (SPR2023): Rohde - Complex Analysis
- 581 A (AUT2023): McGovern - Linear Algebraic Groups
- 581/582 B (A/W): Y. Yuan -Theory of Linear and Nonlinear Second Order Elliptic Equations
- 581D (AUT2023): Shokrieh - Chip-firing
- 581E (AUT 2023): Shokrieh - Algebraic Number Theory
- 581 F (AUT2023): S. Pal - Gradient Flows in Optimal Transport Framework
- 581/582/583 G (A/W/Sp): Drusvyatskiy - Mathematics of Data Science
- 582 A (WIN2024): J. Lee – Introduction to Bundles
- 582 D (WIN 2024): Kovacs - Grothendieck duality and applications
- 583 A (SPR2024): J. Zhang - Hopf Algebra
- 583 B (SPR2024): Ormsby - Topological Data Analysis
- 583 C (SPR2024): Kovács - Intersection Theory
- 583 E (SPR2024): Uhlmann – Calderón's Inverse Problem

#### 504/505/506 Pevtsova, Pevtsova, McGovern - Modern Algebra

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

#### 524/525: Smith - 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: 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: McGovern - Linear Algebraic Groups

Autumn 2023, MWF 10:30

Description: I will give a low-tech introduction to the theory of linear algebraic groups, building up to the classification of reductive groups via root data. I will follow Springer's 1981 text and assume as little algebraic geometry as possible.

Prerequisites: 504/5/6 sequence or permission of the instructor.

#### 581/582 B: Yuan -Theory of Linear and Nonlinear Second Order Elliptic Equations

Autumn 2023-Winter 2024, MW 9-10:20AM

Linear Theory: Solvability, a priori estimates, Schauder and Calderon-Zygmund estimates, and regularity

Nonlinear Theory: De Giorgi Nash Moser theory for divergence equations (e.g. minimal surface equation), Krylov-Safonov theory for nondivergence equations (e.g. Monge-Ampere equation, special Lagrangian equations, Bellman equations, and Isaacs equations).

Contents: Harmonic functions (properties), Schauder for , Weighted norm, solvability for Laplace, Boundary Schauder, L*p* for ; Energy method, capacity, Poincare, Soblev, W1,2 or H1 space, trace; De Giorgi/Nash/Moser, Harnack, Quick applications of Harnack, Minimal surface equations, Viscosity solutions to Nondivergence equations, Alexandrov maximum principle, Krylov-Safonov, Uniqueness and Existence of viscosity solutions, C1,*a* regularity, Evans-Krylov C2,*a* regularity for convex equations, Monge-Ampere and special Lagrangian equations, Bellman equations, and Isaacs equations.

Prerequisites: Advanced calculus/Real Analysis (No previous PDE knowledge is required)

References:

- Han, Qing; Lin, Fanghua, Elliptic partial differential equations. Second

edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of

Mathematical Sciences, New York; ; American Mathematical Society, Prov-

idence, RI, 2011. - Caffarelli, Luis A.; Cabre, Xavier, Fully nonlinear elliptic equations.

American Mathematical Society Colloquium Publications, 43. American

Mathematical Society, Providence, RI, 1995. - Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations

of second order. Reprint of the 1998 edition. Classics in Mathematics.

Springer-Verlag, Berlin, 2001.

Lecture notes will be provided.

#### 581D: Shokrieh - Chip-firing

#### 581E: Shokrieh - Algebraic Number Theory

Dedekind domains, ideal class group, Kummer’s theorem (on factoring ideals), splitting of primes, cyclotomic fields, geometry of numbers, Dirichlet’s unit theorem, p-adic numbers and Hensel’s lemma.

#### 581 F: Pal - Gradient Flows in Optimal Transport Framework

Autumn 2023

Description: The space of probability distributions with finite second moments can be made into a natural metric space, called the Wasserstein space, whose metric is defined by using the optimal transport between distributions. On this metric space one can draw curves that represent motion along the steepest descent (AKA gradient flow) of functionals of probability measures. This is a very fruitful way to view many important families of probability measures that arise from PDEs and stochastic processes. For example, using this geometric framework, one may derive functional inequalities and infer rates of convergence of Markov processes. A striking example is that of the heat equation, whose solution can be interpreted as the family of marginal distributions of Brownian motion. In the Wasserstein space, this curve of probability laws is the gradient flow of the Shannon entropy. We will discuss the theory of Wasserstein gradient flows, including the formal Riemannian calculus due to Otto, and the modern techniques of metric measures spaces. Apart from the classical examples, we will also discuss many modern variations such as Wasserstein mirror gradient flows that come up in applications. A fruitful interaction between probability, geometry, and PDE theory will be developed simultaneously.

More information including class meeting times here: https://sites.math.washington.edu//~soumik/OTgradflowwebpage.html

- This is a part of PIMS online graduate courses.

#### 581/582/583 G: Drusvyatskiy - Mathematics of Data Science

Au/Wi/Sp 2023-2024, MW 1:00-2:20

Description: This is a three-term introductory course sequence in mathematics of data science. The basic question we will address is how does one “optimally” extract information from high dimensional data. The course will cover material from a few different fields, most notably high dimensional probability, statistical and machine learning, and optimization. The topics we will cover include concentration inequalities, geometry of random vectors in high dimensions and isoperimetric inequalities, introduction to statistical inference and maximum likelihood estimation, linear and ridge regression, uniform laws and Rademacher complexity, stability and generalization of learning rules, VC dimension, metric entropy and matrix concentration, minimax lower bounds for estimation, optimization algorithms for learning, kernel learning, Gaussian comparison inequalities and Dudley’s entropy bound, sparse signal and low-rank matrix recovery, exactness of convex relaxations in inference, local averaging methods, and benign overfitting with neural networks.

#### 582 A: J. Lee – Introduction to Bundles

Winter 2024, MWF 1:30

Description: Introduction to vector bundles and fiber bundles, and their applications to differential geometry and topology. Vector bundles, fiber bundles, structure groups, principal bundles, associated bundles, classifying spaces, characteristic classes, connections, Chern-Weil theory, the Chern-Gauss-Bonnet theorem. If time permits, we'll touch on flat bundles and representations of the fundamental group, spin bundles, and sheaf theory.

Prerequisite: Manifolds (Math 544/545/546); Riemannian Geometry (Math 547).

References: Draft textbook will be provided by the instructor

#### 582 D: Kovacs - Grothendieck duality and applications

Winter 2024, MWF 11:30-12:20

Grothendieck duality is a very powerful tool. We will discuss it with all the necessary ingredients. In particular, derived categories and their properties will be discussed in detail as well as applications of Grothendieck duality in algebraic geometry. *Prerequisites:* Knowledge of algebraic geometry is essential for this class. Ideally students will be familiar with the first three chapters of Hartshorne's famous book on algebraic geometry, but students with a somewhat more limited background may also attend.

#### 583 A: J. Zhang - Hopf Algebra

Spring 2024, MWF 10:30

Description: This course is an introduction to Hopf algebra. In addition to basic material, we will present some latest developments in Hopf algebra and quantum groups. One of main topics is homological properties of Noetherian Hopf algebras of low Gelfand-Kirillov dimension.

Topics will include:

- Classical theorems concerning finite dimensional Hopf algebras.
- Infinite dimensional Hopf algebras and quantum groups.
- Duality and Calabi-Yau property.
- Actions of Hopf algebras and invariant theory.

Prerequisites: Math 502/3/4 (and the second-year graduate algebra is also helpful).

Grades: There will be three or four homework assignments and students are encouraged to do homework.

Reference for the first two topics is the book: ``Hopf Algebras and Their Actions on Rings'' by Susan Montgomery.

#### 583 B: Orsmby - Topological Data Analysis

Topological data analysis (TDA) uses techniques from algebraic topology to study global and metric properties of data. Mathematically omnivorous, the field incorporates techniques from homology, representation theory, statistics, and combinatorics with the ultimate goal of providing stable, parameter-free summaries of structural properties of data suitable for statistical analysis. This course will focus on theory, implementation, and applications of persistent (co)homology, one of the preeminent techniques in TDA.

- Theory and software for simplicial and cubical persistent (co)homology
- Discrete Morse theory and algorithms for persistent (co)homology
- Quiver representations, multiparameter persistence modules, and wild type representation theory
- Nonlinear dimensionality reduction via persistent cohomology and Brown representability
- Applications to image analysis, neuroimaging, vascular structure, disease progression, etc.
- Machine learning and statistical methods for assessing TDA outputs

#### 583 C: Kovács - Intersection Theory

Spring 2024, MWF 1:30-2:20

Intersection theory is an important tool in algebraic geometry with applications within algebraic geometry as well as in other fields. It plays a central role in enumerative geometry, and for instance in Schubert calculus.

In this class we will discuss the basics of intersection theory. An important aspect of the class will be presentations by the students. We will also devote ample time to discussions of the concepts involved.

#### 583 E: Uhlmann – Calderón's Inverse Problem

Spring 2024, MW 2:30-3:50

Description: Calderón's inverse problem consists of determining the electrical conductivity of a medium by making electrical measurements at the boundary. It is an important inverse boundary problem that has many applications to medical imaging, geophysics, nondestructive testing etc. In mathematical terms the problem consists in determining the coefficient(s) of an elliptic equation by measuring the so-called Dirichlet-to-Neumann map that maps Dirichlet data to Neuman data. We will develop the mathematics of this problem that has been developed in the last 42 years or so since it was proposed by Calder ́on in 1980.

Prerequisites: No PDE background will be assumed, only the Analysis courses Math 524 and 525. The PDE that we will need will be developed in the course.

References: We will use the notes of a book in preparation on the subject that will be distributed to the students.