## 2021-2022 Special Course Offerings

### Undergraduate Special Topics

- Autumn 2021 Math 180/380: Radicant Mathematics
- Spring 2022 Math 480: Fourier Analysis and Its Applications

#### AUTUMN 2021 Math 180/380: Athreya - Radicant Mathematics

Overview*Radicant, n. Taking root on, or above, the ground; rooting from the stem, like the trumpet creeper and the ivy.*

Mathematics is a precise language for systematic pattern recognition. It is enriched by contributions and developments in many different cultural contexts, and by diverse groups of people. This course will explore the widely varied roots of mathematics, from Latin America to Asia to Africa and the Mediterranean world, and discuss how mathematics has been enriched by the contributions by a wide range of people and civilizations across history. The term radicant, borrowed from botany, seems to capture some aspects of this multivalent evolution.

Students taking it as Math 180/380 would be expected to include mathematical content in their final paper reflective of their background in mathematics. This could include an explanation of the particular mathematics involved in their research paper, written in a way that is accessible to a current student. For Math 380, this mathematical content should be more advanced, and written in a way that reflects the additional background and maturity of the students, with an exploration of proofs and conjectures in the mathematics that they are discussing.

Textbooks: Our main textbook will be *The Crest of the Peacock: the non-European roots of Mathematics* by George Gheverghese Joseph, supplemented by *Mathematics in India* by Kim Plofker and *African Fractals: Modern Computing and Indigenous Design*, by Ron Eglash, and other books and articles.

Prerequisites: A deep interest in mathematics and completion of any of the College of Arts and Science Q/SR courses.

#### SPRING 2022 Math 480: Toro - Fourier Analysis and Its Applications

Fourier Analysis is a powerful tool for many problems, and especially for solving various

differential equations of interest in science and engineering. The techniques presented in this

course are well suited to study problems such as the vibrations of a stretched string (e.g.

guitar string), the vibrations of a stretched membrane (e.g. drumhead), the waves in an

incompressible fluid, electromagnetic waves (e.g. light or radio waves); the diffusion of heat

or the minimization of certain energies. This is a course in applicable mathematics.

The study of Fourier analysis can serve as a capstone course in undergraduate mathematics,

because it builds on such a variety of mathematical topics. Solving ordinary differential

equations is the most crucial prerequisite, but ideas from many other courses are useful.

Some aspects of Fourier series are best understood by regarding the coefficients in the series

as components of an infinite dimensional vector. Then linear algebra becomes useful, in

particular ideas about eigenvalues and eigenvectors. Convergence properties for sequences

and series enter the course as described above, and vector calculus occasionally makes an

appearance.

Textbook: *Fourier Analysis and Its Applications*, G. B. Folland

Prerequisites: A minimum grade of 3.0 in Math 307 and 308; or a minimum grade of 3.0

in Math 324; or a minimum grade of 2.5 in Math 336; or instructor's permission.

## Graduate Special Topics 2021-2022

- Math 534/524/525: Core Analysis (Complex and Reals)
- Math 562 A: Polytopes
- Math 563 A: Algebraic Combinatorics
- Math 564/5 A: Algebraic Topology
- Math 581 A: Non-Arch Geometry
- Math 581 B: Gaussian Free Field
- Math 581 C: Geometry of Flag Variety
- Math 581 D: Optimal Transport
- Math 581 E: Measure Theory
- Math 581 F: Torsion Points
- Math 582 A: Non-Arch Geometry
- Math 582 B: Algebraic Groups
- Math 582 C: Self-Organized Criticality
- Math 582 D: Linear Algebra
- Math 582 E: Complex Manifolds
- Math 582 F: Semiclassical Analysis
- Math 582 G: Convex Algebraic Geometry
- Math 583 B: Noncommutative Algebra
- Math 583 C Model Theory
- Math 583 D: Linear Algebra
- Math 583 F: Moduli of Curves

#### 524/525/534: 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 and 582 A**: Shokrieh - Non-Arch Geometry

**Description:** The main goal is to introduce Berkovich's theory of analytic spaces. This generalizes the notion of complex manifolds, where complex numbers is replaced by a general valued field, including p-adic numbers or formal Laurent series.

The general theory has applications in various fields, including tropical geometry, arithmetic/Arakelov geometry, combinatorics, complex geometry, dynamics, local Langlands program, etc. If time permits, some of these applications will be discussed (especially the first three in this list, which are close to the instructor's research). Students can also study some of these applications as a course project. In any case, we hope to cover enough of the basic concepts and constructions so that participants can start studying advanced research papers

on the subject.**Topics:** Some general theory of valued fields and the basics of Berkovich's theory will certainly be covered. Other topics could include rigid analytic spaces, formal schemes, tropical geometry, as well as some applications. The exact emphasis on the material will be decided based on the audience.

References: There will be no official textbook. We will use various sources, including some research papers.

Prerequisites: The graduate algebra sequence (preferably also some basic algebraic geometry background), or permission from the instructor.

**581 B:** Chen - Gaussian Free Field

Gaussian free field is a Guassian random field. It is a prominent model for random surfaces. Its discrete version can be defined on any graph. The continuum Gaussian free field, which is a natural generalization of Brownian motion defined on the positive halfline, can be thought as Brownian random surface defined on Rd or its subdomains.

The Gaussian free field (GFF) has been one of the main building blocks in mathematical physics, especially in the theory of Quantum Fields. It becomes a focus of intense research in mathematics during the past 15 years due to its various connectionsto diverse areas including Schramm-Lowner evolutions, random planar maps, Liouville quantum gravity and random matrices.

In this course, we will give a gentle introduction to the mathematical theory of Gaussian free fields as well as some of its recent advances. The following is a tentative list of the topics that this course may cover. It contains more material than what we can cover in one course. The final selection of the topics may depend on the interests of the audience.

1. Discrete Gaussian free field on graphs

2. Continuum Gaussian free field on Euclidan spaces

3. Properties of GFF: Markov property, conformal invariance and local sets

4. Liouville measure and random surfaces

5. KPZ relation of GFF

6. Liouville Brownian motion and Liouville heat kernel

7. Liouville quantum gravity metric

Prerequisites: Measure theory and basic knowledge in Probability theory at undergraduate

level (such as M424/5/6 and M394/5/6). Basic knowledge about Brownian motion

will be helpful.

**581 C:** McGovern - Geometry of Flag Variety

The course is based on Fulton’s book “Young Tableaux” together with some of my own research. I will cover the Schur-Weyl construction of irreducible polynomial representations of GL_n via semistandard Young tableaux and its applications to the coordinate ring of the flag variety, together with Young’s construction of irreducible representations of symmetric groups. I will then talk about singularities of various subvarieties of the flag variety

and combinatorial methods for detecting and analyzing them.

Prerequisite: Knowledge of the basic structure theory of complex semi simple Lie algebras, as covered for

example in the course devoted to them in the second-year algebra sequence.

**581 D:** Pal - Optimal Transport

This is an online PIMS class on Optimal Transport + Machine Learning. This is a part of a sequence of courses on Optimal Transport and applications that PIMS has approved and is open to all PIMS campuses and folks from the industry. In fall 2020, the first iteration, OT + Economics, was taught by Brendan Pass from U. Alberta. Students from UW who took this course got credit as in math 600.

#### 581 E: Toro - Measure Theory

This course will be a continuation of the Introduction to Geometric Measure Theory course taught in the Spring of 2021 by the same instructor. In this course we will build on the material from the previous course, we will introduce the notion of rectifiable varifolds. Their development was motivated by a classical problem in calculus of variations, namely the existence of minimal submanifolds. Varifolds play a crucial role in the study of

questions concerning existence and regularity of minimal submanifolds. In some sense they provide the

starting point for analyzing this problem in higher co-dimensions. The study of the dimension one case will be discussed in the Spring 2021 course. The monotonicity formula and several of its applications will be presented. Allard's regularity theorem with be discussed in detail. One of the aims of the course is to present some of the recent results concerning anisotropic energies. The guiding principle behind most of the material to be discussed in this course is that measure theoretic properties combined with the variational attributes of an object imply smoothness.

The course will loosely follow F. Maggi's book *Sets of Finite Perimeter and Geometric Variational Problems* and L. Simon's book: *Lectures in Geometric measure theory*. Some of the applications will come from recent papers.

Tentative plan

1. Anisotropic surface energies

2. Regularity theory and analysis of singularities for perimeter minimizers (co-dimension

1 case)

3. Theory of rectifiable varifolds - Monotonicity formulae & Applications

4. Allard regularity theorem

Prerequisites: Math 524-525, Introduction to GMT (Spring 2021) or instructor's approval.

#### 581 F: Vogt - Torsion Points

The goal of this course is to study the arithmetic of the torsion points on elliptic curves. We will start from preliminaries about elliptic curves over Q, and the Galois structure of Q and Qp, and build up to an area of active research today.

The main content of the course will center around some classic papers in arithmetic geometry.

The most important one being:

(1) Proprietes Galoisiennes des points d'ordre fini des courbes elliptiques. J.P. Serre, 1972.

Time-permitting, we will also discuss some combination of the following papers:

(2) Rational isogenies of prime degree. B. Mazur, 1978.

(3) Bornes pour la torsion des courbes elliptiques sur les corps de nombres. L. Merel, 1996.

(4) Serre's uniformity problem in the split Cartan case. Y. Bilu and P. Parent, 2011.

Most of the topics will be covered through student presentations.

Prerequisites: Some familiarity with Q, and number fields more generally, will be helpful. Familiarity with algebraic geometry will be assumed.

#### 582 B: Pevtsova - Algebraic Groups

Linear algebraic groups, such as the general linear group GLN, the special orthogonal group SON or the symplectic group SpN can be introduced in a number of different ways. In this course, we will take the functorial approach to the study of linear algebraic groups (more generally, affne group schemes) equivalent to the study of Hopf algebras. The classical view of an algebraic group as a variety will come up as a special case of a smooth algebraic group scheme. Our algebraic approach will be independent (even complementary) to the analytic approach taken in the course on Lie groups. The first part of the course will follow an excellent introductory text by W. Waterhouse.

Topics at a glance:

-Group schemes and Hopf algebras, and translating between the two;

-Representations: modules vs. comodules;

-Examples and special cases: abelian group schemes and Cartier duality, etale group schemes, matrix groups, groups of multiplicative type (tori), unipotent groups, nilpotent and solvable groups (i.e. Borel subgroups of semisimple groups);

-Lie algebras;

-Geometric properties: connectedness, irreducibility, smoothness;

-Faithful flatness, quotients, and factor groups.

Upon developing tools for working with group schemes and Hopf algebras we will pursue the goal of classifying semisimple algebraic groups.

-Structure theory for split semisimple algebraic groups: maximal tori, root systems, Weyl groups and Dynkin diagrams;

-Root datum and classification theorem.

Prerequisites: Modern Algebra 504/5/6. The second year graduate algebra course \Algebraic structures" is desirable but not required. The course will be suitable for a 2nd year or above graduate student leaning towards an algebra-related field (understood broadly: combinatorics, number theory, representation theory, algebraic geometry, algebraic topology). The book by Waterhouse is pretty much self-contained, with an appendix covering the necessary fundamental facts from algebra.

References.*Introduction to Affine Group Schemes*, W. Waterhouse;*Representations of Algebraic Groups*, J. Jantzen;*The book of Involutions*, M-A Knus, A. Merkurjev, M. Rost, J-P. Tignol;*Algebraic groups: the theory of group sschemes of finite type over a field*,

J.S.Milne; *(5-7) Linear Algebraic groups*, A. Borel; J. Humphreys; T. Springer (these are

three different books!)

**582 C****: Hoffman - Self-Organized Criticality**

** ****582 D and 583 D: Thomas - Linear Algebra**

This is two quarter graduate linear algebra. Linear Algebra is one of the most important skills a student going into industry can have these days, so I believe this is an important basic sequence for us to maintain. It will be

accessible to first year graduate students, needing only undergraduate linear algebra as a

prerequisite.

Topics to be covered:

1. Eigenvalues

2. Unitary and normal matrices

3. Canonical forms

4. Hermitian and symmetric matrices

5. Norms of vectors and matrices

6. Perturbation theory

7. Positive semidefinite matrices

8. Nonnegative matrices

These are the 8 chapters of Horn and Johnson’s book. In addition, I might cover a few selected

topics from the book “Numerical Linear Algebra” by Trefethan and Bau.

Recommended Book: *Matrix Analysis* by Horn and Johnson

**582E: J. Lee - Complex Manifolds**

Definition and examples of complex manifolds, almost complex structures, and integrability;

holomorphic vector bundles; line bundles and hypersurfaces; Hermitian connections; Hermitian and

Kähler metrics; sheaves and cohomology; Hodge theory; the Kodaira embedding theorem; and (if time

permits) a brief introduction to Kähler-Einstein metrics and Calabi-Yau manifolds.

Prerequisites:

Topology, smooth manifolds, Riemannian geometry, vector bundles, and undergraduate complex analysis.

**582 F: Drouot - Semiclassical Analysis**

Description: Semiclassical analysis studies the relation between classical and quantum mechanics. Its key achievement explains mathematically how Newton's law emerges from the Schrodinger equation. This class will have two essential components:

1. Semiclassical techniques: symplectic geometry, WKB approximation, stationnary phase lemma, pseudodifferential operators.

2. Applications: tight-binding approximations, Hamiltonian dynamics, coherent states, spectral asymptotics.

Textbook: Dimassi-Sjostrand, *Spectral asymptotics in the semi-classical limit*, London

Prerequisite: Math 525-527, preferably Math 557.

Mathematical Society Lecture Note Series, 268.

#### 582 G: Vinzant - Convex Algebraic Geometry

Convex algebraic geometry involves the study of convex objects defined by real polynomial inequalities using the interplay of their convex and algebraic structure, with a special emphasis on those appearing in convex optimization. This class will introduce basic notions and techniques in real algebraic geometry,

convexity, and conic optimization. Topics include semidefinite programming, sums of squares, moment problems, hyperbolic and stable polynomials, and applications in combinatorics and optimization.

References:

1) *Semidefinite Optimization and Convex Algebraic Geometry*, edited by Grigoriy Blekherman, Pablo A. Parrilo and Rekha R. Thomas.

2) *A Course in Convexity*, by Alexander Barvinok

3) *Hyperbolicity and stable polynomials in combinatorics and probability*, by Robin Pemantle

Prerequisites: The only prerequisite is a solid background in linear algebra. Some familiarity with concepts from convexity and algebra will be helpful but is not required.

#### 583 B: Zhang - Noncommutative Algebra

This course is an introduction to noncommutative algebra. In addition to basic material, we will present some latest developments in this area.

1. Artin-Schelter regular algebras, duality and Calabi-Yau property.

2. Noncommutative invariant theory, Chevalley-Shephard-Todd theorem on regularity, Watanabe theorem on Gorenstein property.

3. Classification of prime Hopf algebras of low Gelfand-Kirillov dimension.

4. Quiver representations, finite/tame/wild representation types, Frobenius-Perron dimension.

There will be three or four homework assignments and students are encouraged

to do homework.

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

#### 583 C: Lieblich - Model Theory

Course description coming soon.

#### 583 F: Larson - Moduli of Curves

The goal of this course is to explore the geometry of moduli spaces of algebraic curves in projective space. The beginning of the course will focus on a potpourri of topics, including stable reduction, the cotangent complex

and its role in deformation theory, the theory of admissible covers, and Clifford's and Castelnuovo's theorems. The bulk of the course will then focus on Brill-Noether theory, which describes the geometry of the moduli space of maps from a suitably generic curve to projective space.

Prerequisites: The suggested prerequisite is familiarity with algebraic geometry at the level of a year-long introductory graduate course.