# Special Offerings

## 2023-2024  Special Course Offerings

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

#### AUTUMN 2023 MATH 198A: Math 124 Support

This course is designed to support students on their journey through the first quarter of college-level calculus. During weekly meetings, we will review relevant algebra skills and investigate various aspects of effective engagement in Math 124 and beyond. These include classroom and study habits, preparation strategies for exams, effective group work, methods to keep up (or catch up) with course material, and how to use UW resources (like the Math Study Center).

#### WINTER 2024  MATH 180: S. Steinerberger - 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:

(1) 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.

(2) The fact that we don’t know everything: mathematics as an evolving science.

(3) 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.

(4) Countable Sets & Uncountable Sets.

(5) Basic Graph Theory.

(6) Discuss Brouwer’s Fixed Point Theorem

(7) 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 2024  MATH 380: N. Koblitz - Math That Lies, Communicating Why Some Quantitative Arguments AreMisleading 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 2024  MATH 480: Z. Su - Topology and It's Application

Topology studies the properties of geometric objects that are preserved under continuous deformation. In recent decades, topological techniques have been applied to analyze the shape of point cloud data. The first half of the course will be an introduction to fundamental concepts and techniques in algebraic and geometric topology, including: topological spaces and manifolds, fundamental group, covering space, complexes, homology group. The second half of the course will focus on applications, discussing tropics in topological data analysis, including filtered simplicial complexes, persistent homology, discrete Morse theory. Course projects will involve implementation of the algorithm and visualization in Matlab or Python.

Suggested Textbooks: Algebraic Topology, by Allen Hatcher and Computational Topology: An Introduction, by Herbert Edelsbrunner and John Harer; course notes will be provided.

Prerequisites: MATH 208 and MATH 300  OR  MATH 334, with minimum grades of 2.0. Programming experience in Matlab or Python is recommended but not required.

#### Spring 2024  Math 480A: N. Naehrig - Mathematics of Public Key Cryptography

Cryptography is ubiquitous in today’s daily digital life. It enables secure internet connections to bank accounts, allows to privately communicate via an end-to-end encrypted messaging app, and authenticates software updates to mobile phones. To achieve these security features, cryptography makes use of abstract mathematical structures.

This course introduces students to the mathematical concepts used in public key cryptography and their implementation in real world applications. Topics include key exchange and public key encryption schemes, digital signatures, zero-knowledge proofs, the discrete logarithm and integer factorization problems, elliptic curves, and post-quantum cryptography. A special focus will be on how these techniques can be used to facilitate end-to-end verifiable elections. This course will be structured with an emphasis on structured group work.

Textbook: An Introduction to Mathematical Cryptography by Hoffstein, Pipher, Silverman

Prerequisites: Math 402. Students who enroll for this course are expected to have basic knowledge in abstract algebra, including ring, group, and field theory.

#### Spring 2024  Math 480B: R. Liu - Combinatorial Game Theory

Combinatorial game theory is the study of games of no chance and perfect information, such as chess, checkers, or go. We will study such games from a mathematical perspective starting from first principles. In particular, we will discuss the mathematical framework for partisan games developed by Berlekamp, Conway, and Guy; the theory of impartial games as developed by Sprague and Grundy; and Conway’s construction of surreal numbers. We will apply these ideas to investigate many examples of combinatorial games with simple rules but complex behavior (such as Nim, Dots and Boxes, Hackenbush, and Domineering).

Textbook: Lessons in Play: An Introduction to Combinatorial Game Theory, AK Peters, Ltd., 2nd edition, 2019 by Albert, Nowakowski, Wolfe

Prerequisites: Math 402. Students who enroll for this course are expected to have basic knowledge in abstract algebra, including ring, group, and field theory.

#### 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, Lp 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

Chip-firing (sandpile model) is a one-person game played on finite or metric graphs. It is closely related to harmonic analysis (potential theory) on graphs. As such, it has connections to combinatorics, tropical geometry, probability, analysis, statistical mechanics, and arithmetic geometry (!).
This course will be self-contained and starts from some basic graph theory, but also discusses some current research topics in the area.
Prerequisite:

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

We study integers, rational numbers, and their generalizations, using techniques from abstract algebra as well as (Euclidean) lattices.
The tentative plan is to cover the following topics (and the surroundings):
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.
If time permits, there could also be some more advanced topics discussed, in which case the audience's preferences will be taken into account.
The plan is to present the material in a concrete way with many examples.
Prerequisite:
The graduate algebra sequence (or equivalent), or permission from the instructor.

#### 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.

• 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 applicationsWinter 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.

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.

Topics may include:
• 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
Prerequisites: Familiarity with abstract algebra and topology; prior exposure to simplicial or singular homology would be beneficial, but we will develop the classical theory in parallel to persistent methods.
Grading: Grades will be based on class participation, occasional homework exercises, and/or projects.
Books: None. I will provide course notes and original references throughout the course.

#### 583 D: Kovacs - Higher Dimensional Singularities

Spring 2024, MWF 11:30-12:20

In this class we will study singularities that are relevant for the moduli theory of higher dimensional varieties of general type. In particular, we will study rational and Du Bois singularities as well as the singularities that come up in the minimal model program and their non-normal variants.

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, but students with a somewhat more limited background may also attend.

#### 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.