## 2020 - 2021 Course Special Offerings

### Undergraduate Special Topics

#### AUTUMN 2020 Math 380: Ostroff - Mathematics of Democracy (non-majors welcomed)

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

- How can you determine the winner in a ranked-choice election? What are the advantages and disadvantages of different methods? Is there a perfect system?
- In a voting system with unequal weights like the Electoral College, how much power does each voter have?
- The Constitution says that representatives should be apportioned to states "according to their respective numbers". What exactly does that mean?
- 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?

**Prerequisites: None**

#### SPRING 2021 Math 3480: Thomas - The Power of Polynomials (open to non-majors period II)

Polynomials are perhaps the most well known functions in all of mathematics with surprising applications to a wide array of areas. This course will be an introduction to the theory, computation and applications of multivariable polynomial systems. The geometry behind polynomials is the foundation of algebraic geometry, while solving polynomial systems is all about algorithms, both symbolic and numerical, and underlie a number of practical real world problems. They have also become an important tool in combinatorics, number theory, theoretical computer science, and many many more parts of pure and applied mathematics. This course will be a tour through some of these topics, and will also include working with software such as Macaulay2.

Topics:

1. Ideals and Varieties

2. Groebner bases & Homotopy Continuation Methods

3. Elimination theory

4. The algebra-geometry dictionary

5. Polynomial Optimization

6. Applications

Suggested texts:

Ideals, Varieties and Algorithms by Cox, Little and O’Shea, Springer UTM

An Invitation to Nonlinear Algebra, Michalek and Sturmfels

Prerequisites: 3.0 in MATH 308, MATH 318, or MATH 136

Open to non-majors period II.

## Graduate Special Topics 2020-2021

- Math 534/524/525: Core Analysis (Complex and Reals)
- Math 581 A: Dynamics of Evolutionary Equations
- Math 581 C and D: Algebraic Geometry Topics
- Math 581 E: Inverse Geometric Problems
- Math 581 H: Polyhedral Combinatorics (tentative)
- Math 581 I: An introduction to Variational Analysis
- Math 600: Topics in Applied Mathematics I: Optimal Transport + Economics
- Math 582 B: Hyperbolic geometry and Teichmueller theory
- Math 582 C and D: Algebraic Geometry Topics
- Math 582 F: Asymptotic Convex Geometry
- Math 583 A: Ergodic Theory
- Math 583 B: Hyperbolic geometry and Teichmueller theory
- Math 583 C and D: Algebraic Geometry Topics
- Math 583 E: Geometric Measure Theory
- Math 583 H: Real and Complex Reflection Groups

#### 524/525/534: Toro and Rohde (Reals and Complex)

The first two quarters of this class ("Math 524 and 525") will be devoted to Real Analysis. Autumn quarter will cover the fundamentals of measure theory and Lebesgue integration. Topics include functions of bounded variation and absolute continuity, the fundamental theorem of calculus, and the Radon-Nikodym theorem. Winter quarter will cover elements of the theory of functional analysis. Topics include the fundamental theorems for Banach and Hilbert spaces; L^p spaces; and the Riesz representation theorem for L^p and C(X).

The third quarter of this class ("Math 534") will concentrate on Complex Analysis. It will cover the basic theory of analytic functions from complex numbers to power series to contour integration, Cauchy's theorem and applications.

**581 A**: Wilson - Dynamics of Evolutionary Equations

In this course, we will study the dynamics of a class of partial diﬀerential equa-tions as they evolve in time. Our major focus will be on two important concepts in dynamics and dispersive PDEs: integrability and KAM for infinite-dimensional Hamiltonian systems. Integrability is the concept of a Hamiltonian system ad-mitting a diﬀeomorphism which decouples the equation into a set of independent harmonic oscillators in such a way that preserves the Hamiltonian structure of the system. KAM developed as a method of analysis of intractable problems in celes-tial mechanics. This method was first suggested by A. Kolmogorov and cultivated by V. Arnold and J. Moser as a way to demonstrate that certain perturbations of integrable equations preserve integrability.

We will use the classical Korteweg-de Vries (KdV) equation as a medium through which to explore these ideas. First, we will show that KdV is integrable, and then we will perturb KdV and prove a KAM theorem.

**References:**

[1] P. Deift, C. Levermore, and C. Wayne, *Dynamical Systems and Probabilistic Methods in**Partial Differential Equations: 1994 Summer Seminar on Dynamical Systems and Probabilistic Methods for Nonlinear Waves,* *June 20-July 1, 1994, MSRI, Berkeley, CA.* Lectures in applied mathematics. American Mathematical Soc. ISBN 9780821897003.

[2] T. Kappeler and J. Poschel, *KdV & KAM*, Ergeb. Math. Grenzgeb. 3, Springer, Berlin, 2003

[3] G. R. Sell and Y. You, *Dynamics of Evolutionary Equations*, Springer, New York, 2002

*Prerequisites: Math 524 and 525*

**581 C and D/582 C and D/583 C and D:** Lieblich, Alper, Kovacs - Algebraic Geometry Topics

In academic year 2020–2021, Aise Johan de Jong will be spending his sabbatical at the University of Washington. De Jong is a Cole Prize winner, the discoverer of alterations (the only real progress in the resolution of singularities problem in positive characteristic in several centuries), and the founder of the Stacks Project (which is rapidly becoming a standard reference in algebraic geometry), among many other things. He’s also a truly wonderful person to talk to about algebraic geometry and arithmetic geometry. We propose a cluster of activity for all of next year to celebrate and take full advantage of his visit.

A year-long sequence of topics courses, culminating in the Spring quarter when Alper and de Jong will teach a course on Alterations. De Jong’s theory of alterations is a replacement for resolution of singularities in contexts where we don’t know that resolutions must exist (mainly: in positive charactersitic). Instead of relying on invariants measuring the complexity of arbitrary singularities, the theory of alterations makes very elegant use of the moduli space of stable curves to immediately produce a variety with very simple singularities. The price one pays for this elegance is that the smooth variety one produces is not birational to the original, but only dominates it by a generically ﬁnite map. This turns out to be suﬃcient for many of the classical applications of resolution of singularities (for example, the construction of mixed Hodge structures), as well as many other fascinating results (for example, applications to anabelian geometry, the theory of motives, etc.).

**Fall 2020**:**581 C Resolution of singularities (Lieblich)**

We will discuss one of the oldest problems in algebraic geometry: resolution of singularities. After learning about classical approaches to resolution for curves and surfaces, we will discuss a clean and simple proof of embedded resolution of singularities in characteristic 0. If time permits, we may also brieﬂy discuss some of the ideas in the new simple proofs of functo-rial resolution (still in characteristic 0) due to Abramovich–McQuillan–Temkin–W lodarczyk. This is suitable as a second course in algebraic geometry.

**Text:**

J. KollÂar: *Lectures on resolution of singularities*

*Prerequisites:Basic knowledge of algebraic geometry on the level of 567/8/9, or permission of the instructor.*

**581 D Advanced Topics in Algebraic Geometry: Greatest Hits (Lieblich) **This course will focus on reading important papers in algebraic geometry. Each week, a participant will select from a menu of key papers and give a presentation after consultation with (other) faculty. One goal will be to mentor younger participants as they learn to absorb and communicate diï¬ƒcult and important pieces of mathematics. Another goal is to give participants a sense of the scope of the subject, and to enjoy some wonderful ideas from its history.

**Text:**

Seminal papers in the subject in the last century.

*Prerequisites:Basic knowledge of algebraic geometry on the level of 567/8/9, or permission of the instructor.*

**Winter 2021**:**582 Introduction to moduli (Alper)**The primary goal of this coarse is to understand and establish the following statement: the moduli space parameterizing stable curves of genus g is represented by an irreducible, smooth and proper Deligne-Mumford stack with a projective coarse moduli space. Assuming the background of a ï¬rst course in algebraic geometry, we will begin by introducing the language of algebraic spaces and algebraic stacks. Using this language, we will then proceed to construct the moduli space of stable curves as a projective variety. Topics at a glance:

- Grothendieck topologies and sites
- Categories fibered in groupoids
- Descent
- Algebraic spaces and stacks
- Deligne-Mumford stable curves
- Semistable reduction for curves: properness
- Irreducibility
- Existence of a coarse moduli space: the Keel-Mori theorem â€¢ Projectivity of moduli

**Text:**

Olsson, *Algebraic spaces and stacks*

Harris and Morisson, *Moduli of curves*

*Prerequisites:Basic knowledge of algebraic geometry on the level of 567/8/9 or permission of the instructor.*

**582 Advanced Topics in Algebraic Geometry: Recent Developments (Alper)**

In this course, we will focus on reading recent papers and preprints. This has three main goals: first, to learn about recent progress in the field; second, to practice digesting the main ideas of recent papers (even if we don't yet understand all of the details); third, to communicate the essence of the paper to an audience of peers.

**Text:**

Recent papers in algebraic geometry, decided upon together by the instructor and the participants.

*Prerequisites:Basic knowledge of algebraic geometry on the level of 567/8/9, or permission of the instructor.*

**Spring 2021:****583 Alterations (Alper and Aise Johan de Jong)**Using the material discussed in the previous two quarters, we will show how any projective variety over an algebraically closed ﬁeld can be dominated by another variety which is smooth and projective. This result will be proven using the theory of stable reduction of curves (which provides us with proper moduli spaces/stacks of stable curves) combined with an induction on the dimension. We will also discuss a more arithmetic version of the theorem which is used in the proof of many comparison theorems between various p-adic cohomologies.

**Text:**

A. J. de Jong: *Smoothness, semi-stability, and alterations*

*Prerequisites:Basic knowledge of algebraic geometry on the level of 567/8/9, and participation in the Fall and Winter courses described above, or permission of the instructor.*

**583 Advanced Topics in Algebraic Geometry: Current Research (KovÂcs)**Each week, one participant will speak about his or her current work, leading to a blocking diï¬ƒculty. The goal will be to learn about this participant's work, encourage him or her to persevere, and (if possible) make helpful suggestions to deal with the blockage.

**Text:**

Notes provided by participants as desired.

Basic knowledge of algebraic geometry on the level of 567/8/9, or permission of the instructor.

**581 E: Uhlmann - Inverse Geometric Problems**

The study of geometric inverse problems is typically motivated by inverse problems in PDEs, geophysics and medical imaging. The main goal is the reconstruction of geometric structures (metrics, connections, vector bundles etc.) from either boundary measurements or local measurements. The course will describe recent developments in the area with an emphasis on the 2D picture.

We will consider two geometric inverse problems. The ﬁrst one is to determine a Riemannian metric of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map associated to the Laplace-Beltrami operator. Physically this corresponds to deter-mining the anisotropic electrical conductivity of a domain by making voltage and current measurements at the boundary. The second inverse problem consists on determining a Riemannian metric of a Riemannian manifold with boundary from the distance function between points in the boundary. Physically this corresponds to determining the anisotropic index of refraction or sound speed of a medium by measuring the travel times of waves going through the medium. We will also discuss the relation between these two problems.

**Text:** We will use a book written jointly with Gabriel Paternain an Mikko Salo that is entitled *Geometric 2D Inverse Problem*. I intend to go through the book in this course.

*The only prerequisite is the Topology and Manifolds ﬁrst year course or its equivalent and the ﬁrst year Analysis course or its equivalent. No previous knowledge of Riemannian manifolds or PDE’s will be assumed. All the material from these subjects that are needed will be introduced in the class.*

581 H: Gaku Liu - Polyhedral Combinatorics - tentative

This course is on the combinatorics of polytopes and polyhedral complexes. Poly- topes are one of the oldest and most ubiquitous objects in math and appear at the intersection of geometry, topology, and combinatorics. The first part of the course will cover general polytope theory, with an emphasis on polyhedral subdivisions and regular subdivisions. The second part will cover topics on lattice polytopes, with an eye toward connections to other areas of math.

Tentative list of topics

- Polytopes, face lattice, normal fan
- Realizability, Steinitz’s theorem
- Polyhedral subdivisions, regular subdivisions, secondary fan
- Zonotopes, hyperplane arrangements, zonotopal tilings
- Special polytopes (permutohedron, associahedon, product of simplices, matroid polytopes)
- Polyhedral complexes, shellability,
*f*-vectors - Lattice point counting, normality
- Unimodular triangulations

References:

*T**riangulations*, Jesu´s De Loera, J¨org Rambau, Francisco Santos*Handbook of Discrete and Computational Geometry, Third Edition*, Jacob Goodman,

Joseph O’Rourke, Csaba D. T´oth*L**e**ctures on Polytopes*, Gu¨nter Ziegler

*No prerequisites are required, although students should be familiar with basic concepts from combinatorics and topology. Suitable for advanced undergraduates.*

#### 581 Drusvyatskiy - An introduction to Variational Analysis

Abstract: This course will serve as an introduction to Variational

Analysis. The origin of the subject dates back to 1970-1980s, when the first systematic tools for working with nonsmooth sets and functions in optimization and control were developed. Some of the most important results on the subject have a strong connection to UW, with the work of Tyrell Rockafellar (Emeritus Professor) and Francis Clark (Alumnus). Recent years have seen a growing interest in the subject, influenced by new modeling techniques in data science and engineering disciplines. The course will focus on both variational analytic theory and algorithms for nonsmooth and nonconvex optimization problems. Some representative topics include the following.

Theory: generalized differentiation of nonsmooth functions, transversality, variational principles (Ekeland and Borwein-Preiss), error bounds, metric regularity and conditioning, elements of semi-algebraic geometry, desingularization with Kurdyka-Lojasiewicz (KL) inequality, and generalized Sard's theorem.

Algorithms: (stochastic) Gauss-Newton, proximal point, and subgradient methods for weakly convex problems, iterate convergence of descent methods under the Kurdyka-Lojasiewicz inequality, differential inclusions and asymptotic convergence of the subgradient algorithm, random search/gradient sampling algorithm.

We will cover a number of recent articles and some material in the textbooks:

Francis H. Clarke, Yuri S. Ledyaev, Ronald J. Stern, and Peter R. Wolenski, *Nonsmooth Analysis and Control Theory.*R. Tyrrell Rockafellar and Roger J-B. Wets,

*Variational Analysis*.

Alexander Ioffe,

*Variational Analysis of Regular Mappings*.

The only prerequisites are a strong background in real analysis and linear algebra. Some familiarity with convex analysis will be helpful but is not necessary.

#### Math 600: Pal - Topics in Applied Mathematics I: Optimal Transport + Economic - BEGINS SEPT 1, 2020

**Course Description:** formulation of the optimal transport problem, Kantorovich duality

theory, existence and uniqueness theory, c-monotonicity and structure of solutions,

discrete optimal transport. Economic applications including: transferable utility models

in matching theory, hedonic and discrete choice models in contract theory, inference in

incomplete models and multi-variate analogues of quantiles in the analysis of

dependence structures in econometrics, the analysis of profit maximization and

screening problems, optimal portfolio rebalancing and robust option pricing.

**Students who wish to sign up for credit, should contact Prof. Soumik Pal for MATH 600 registration.** Link to PDF for course details.

**582 B ****and 583 B: Athreya/Rohde - Hyperbolic geometry and Teichmueller theory**

The aim of this two-quarter sequence is to provide an introduction to the deformation theory of compact Riemann surfaces. After reviewing the basic theory of Riemann surfaces (as covered in the third quarter of a year-long complex analysis graduate course), we will discuss the hyperbolic geometry of Riemann surfaces (Fuchsian groups, pants decomposition), quasiconformal maps, quadratic differentials, Teichmueller spaces, their analytic and geometric structures and compactifications. We will draw from several sources, particularly Katok's "Fuchsian Groups," Farb-Margalit's "Primer on Mapping Class Groups," Hubbard's "Teichmuller theory," and Ahlfors' "Lectures on Quasiconformal Mappings."

*Prerequisites are the complex analysis graduate sequence as well as some basics from manifolds and real analysis (which will be reviewed if necessary).*

** ****582 F: Rothvoss - Asymptotic Convex Geometry**

This course covers the modern theory of convex bodies in high dimensions. It turns out that convex bodies are becoming surprisingly regular in a certain sense if the dimension is growing. This will provide a better understanding for norms and objects in higher dimensions. Many of the arguments will be probabilistic and geometric. As a reference, we will use the textbook: Artstein-Avidan, Giannopoulos, Milman: *Asymptotic geometric analysis. *Part I. Mathematical Surveys and Monographs, 202. AMS, 2015. xx+451 pp. ISBN: 978-1-4704-2193-9. However, a full set of lecture notes with (sometimes simplified arguments) will be provided.

#### Content

- Chapter 1: Basics of Convex Geometry. We cover basic definitions and tools such as polarity, Steiner symmetrization, Brunn-Minkowski Inequality, Prekopa-Leindler Inequality and the inequality of Rogers and Shepa
- Chapter 2: John’s Theorem. We prove the full version of John’s theorem and see applications such as the Theorem of Kadets-Snobar and the Dvoretzky-Rogers Theorem.
- Chapter 3. Isoperimetric inequalities. Arguably the main tool in convex geometry is the concentration of measure in its various forms. We will derive concentration inequalities for volume and Gaussian measure. Additionally, we will prove Talagrand’s Inequality, Khintchine’s Inequality and Kahane’s Inequality.

- Chapter 4. Covering numbers. We study the quantity N (A, B) which is the number of translates of B to cover A. We will prove Sudakov’s Inequality and the Dual Sudakov Inequality. The chapter culminates in the beautiful result that any convex bodies A, B, satisfy N (A, B) ≈ Voln(A − B)/Voln(B).
- Chapter 5. Almost Euclidean Subpaces. We will prove the Theorem of Dvoretzky which says that any symmetric convex body in Rn has a subspace of logarithmic dimension where the body is close to a Euclidean ball.

- Chapter 6. Pisier’s Inequality and the ℓℓ◦-estimate. We will prove a very powerful result: any convex body in Rn admits a linear map so that the transformed body has the volume of a Euclidean unit ball and mean width at most O(log n).
- Chapter 7. Quotient of Subspace Theore We will prove a surprising theorem due to Milman: every symmetric convex set has a quotient subspace of dimension n in which it is approximately ellipsoidal.

- Chapter 8. M-ellipsoids and applications. We will prove one of the most powerful results in convex geometry: every convex body K ⊆ Rn has an ellipsoid — called the M -ellipsoid — so that the body and the ellipsoid can cover each other with only 2O(n) translates and the same holds for the respective polars. One immediate application is that the product (Voln(K) · Voln(K◦))1/n is in a constant range, if K is symmetric.
- Chapter 9. The Gaussian approach. We will see several comparison inequalities for Gaussians which allows us to upper and lower bound the supremum of a Gaussian process in terms of covering numbers.
- Chapter 10. Volume distribution in convex bodies and the isotropic positio This chapter discusses convex bodies in isotropic position with known results and presents several equivalent forms of the Slicing Conjecture. The latter conjecture suggests that for a convex body in isotropic position, every (n − 1)-dimensional hyperplane through the origin has volume upper and lower bounded by a constant. We will also prove Bourgain’s estimate of LK ≤ O(n1/4 log(n)) for this question.

Grading: There will be weekly homework which determines the grade.

*Prerequisites: A good understanding of probability and convexity is strongly recommended.*

**583 A: Hoffman - Ergodic Theory**

Ergodic theory is the study of dynamical systems from a measurable or statistical point of view. Starting with Poincare’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

**583E: Toro - Geometric Measure Theory**

Geometric Measure Theory (GMT) is a classical subject in geometric analysis which in recent years has seen a new revival. Tools introduced to study perimeter minimizers and minimizing surfaces have found applications in areas such as metric geometry, harmonic analysis, free boundary problems and theoretical computer sciences. The goal of this course is to introduce the subject including some of the fundamental results concerning perimeter minimizers. This course can be seen as a continuation of Math 524 and can be taken in parallel with Math 525.

The goal is to cover the following topics:

- Advanced Measure Theory
- Covering Theorems
- Differentiation of Radon Measures
- Riesz Representation Theorem
- Weak Convergence

- Hausdor↵ Measures
- Area and Coarea Formulas
- Lipschitz Functions, Rademacher’s Theorem
- The Area Formula
- The Co-area Formula
- First and Second Variation formulae

- Sobolev Functions
- Functions of Bounded Variation, Sets of Finite Perimeter
- Definitions, Structure Theorem
- Approximation and Compactness
- Co-area Formula for BV Functions
- Isoperimetric Inequalities
- The Reduced Boundary
- Gauss-Green Theorem
- Perimeter minimizers: first results

**References:**

- [EG] Measure Theory and Fine Properties of Functions, Revised Edition, C. Evans & R. F. Gariepy
- [M1] Sets of Finite Perimeter and Geometric Variational Problems, Maggi.
- [M2] Geometry of sets and measures in Euclidean spaces : fractals and rectifiability. P. Mattila
- [S] Lectures on geometric measure theory, M. Simon.

*Prerequisites: Math 524 or instructors approval.*

#### 583 H: Billey - Real and Complex Reflection Groups

The topic of finite reflection groups and Coxeter groups appear at the intersection between combinatorics, Lie theory, geometry and representation theory. These topics are beautifully described in the proposed textbooks and are very relevant to current research. The goal of this course is to introduce students to the basic material in this area and connect with some of the current open problems. As time permits, we will explore finite complex reflection groups as well.

**Outline:**

- Root systems and finite reflection groups
- Classification of finite reflection groups using Dynkin diagrams/ Coxeter graphs.
- Affine Weyl groups
- Polynomial invariants of finite reflection groups
- Pattern avoidance in Coxeter groups
- Chip firing games on Coxeter groups.
- Complex reflection groups.
- Open problems and recent research.

**Texts:** *Reflection Groups and Coxeter Groups by* James Humphreys and/or *Combinatorics of Coxeter Groups *by Anders Bjorner and Francesco Brenti

*The prerequisites for the course are just a full year long course in algebra similar to 402-3-4. No combinatorics or representation theory is required. Therefore, an advanced undergraduate could also benefit from this course. Students taking Lie algebras will see how these topics grows out of that theory.*