## 2019-2020 Course Special Offerings

### Undergraduate Special Topics

- Autumn Math 480 A: The Mathematical Theory of Knots
- Winter Math 480 A: Partial Differential Equations
- Spring Math 480 A: Fourier Analysis

#### AUTUMN Math 480 A: Arms - The Mathematical Theory of Knots

Why study knots? Partly, just because it's fun to do math with an object you can carry in your pocket! But knot theory also offers a great way to get acquainted the branch of mathematics called topology. Topology cares about how things are connected, but doesn't notice if they are stretched or bent without tearing. Knot theory and topology are relatively young areas of mathematics, and we'll be able to look at some questions that have been answered only recently, perhaps even a few for which the answers are unknown. Topics will include knot and link presentations and Reidemeister moves; prime, composite, and alternating knots; tabulating knots; knot invariants such as colorability, stick number, genus, and knot polynomials; and selected applications, as time permits.

*Prerequisites. 2.0 in Math 327 OR Math 335, OR permission of instructor.*

#### WINTER Math 480 A: Uhlmann - Partial Differential Equations

Course on Partial Differential Equations (PDE) for undergraduate students. We plan to follow Walter Strauss’s book "Partial Differential Equations: An introduction." This is perfectly suited for undergraduate students who have a strong background in several variable calculus.

Syllabus: Chapters 1, 2, 3*, 4, 5 and 6 in [S]. Topics include

– Waves and Diffusions

– Boundary problems

– Fourier series

– Harmonic functions

*Prerequisites: A minimum grade of 3.0 in Math 307, Math 308 and Math 324 or a minimum grade of 2.5 in Math 335; or instructor's permission.*

#### SPRING Math 480 A: Toro - Fourier Analysis

Fourier Analysis and its applications to PDE. This course would deal with those aspects of Fourier Analysis that are useful in physics and engineering. As the author describes: it is a book on applicable mathematics which reflects the emphasis of the course. The course will use "Fourier Analysis and Its Applications," G. B. Folland.

Syllabus: Chapters 1, 2, 3, 4, selected topics from 5, 6, 7 and 8. Topics include

– Fourier series

– Orthogonal sets of functions

– Boundary value problems in differential equations

– Fourier Transform

– Laplace Transform

*Prerequisites: A minimum grade of 3.0 in Math 307, Math 308 and Math 324 or a **minimum grade of 2.5 in Math 335; or instructor’s permission.*

## Graduate Special Topics 2019-2020

- Math 516 A: Convex Analysis and Optimization
- Math 534/524/525: Core Analysis (Complex and Reals)
- Math 561 A: Foundations of Combinatorics - Enumeration
- Math 562 A: Discrete Geometry and Polytopes
- Math 563 A: Foundations of Combinatorics Graphs and Tropical Curves
- Math 564-6: Algebraic Topology
- Math 581 A: Theory of Linear and Nonlinear Second Order Elliptic Equations
- Math 581 B: Algebraic Groups I
- Math 581 C: Optimal Transport I
- Math 581 E: High Dimensional Probability for Data Science
- Math 582 A: Introduction to the Mathematics of Medical Imagining
- Math 582 B: Algebraic Groups II
- Math 582 C: Optimal Transport II
- Math 582 E: Rational Points I
- Math 582 H: Complex Analysis
- Math 583 B: Intersection Theory
- Math 583 C: Optimal Transport III
- Math 583 E: Communicating Math Effectively
- Math 583 H: Complex Analysis
- Math 581 I: Rational Points II

#### 516 A: Drusvyatskiy - Convex Analysis and Optimization

This is an introductory course in convex analysis and optimization, with a focus on applications in data science. Some representative topics include duality, monotone operators, computational complexity, acceleration, splitting methods, stochastic algorithms, and elements of nonsmooth and nonconvex optimization. This course is appropriate for anyone with a working knowledge of linear algebra and mathematical analysis.

Though the course "MATH 581 E: High dimensional probability for data science (Autumn 2019)" is not a prerequisite, it would help to contextualize the material.

#### 534A/524/525: Rohde and Chen - Core Analysis (Complex and Reals)

The first 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.

The second and third quarters of this class ("Math 524 and 525") will be devoted to Real Analysis. Winter 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. Spring 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)\$.

#### 561 A: Billey - Foundations of Combinatorics - Enumeration

Combinatorics has connections to all areas of mathematics and many other sciences including biology, physics, computer science, and chemistry. We have chosen core areas of study which should be relevant to a wide audience. The main distinction between this course and its undergraduate counterpart will be the pace and depth of coverage. In addition we will assume students have a basic knowledge of linear and abstract algebra. We will include many unsolved problems and directions for future research.

Every discrete process leads to questions of existence, enumeration and optimization. This is the foundation of combinatorics. In this quarter we will present the basic combinatorial objects and methods for counting various arrangements of these objects.

Specific topics covered include basic counting methods, sets, multisets, permutations, graphs, inclusion-exclusion, recurrence relations, integer sequences, generating functions, partially ordered sets, basic complexity theory. Grading: Grading will be based on weekly homework.

Textbook: *Enumerative Combinatorics; Volume 1* by Richard Stanley, Cambridge Studies in Advanced Mathematics, 2nd edition.

#### Math 562 A: Novik - Discrete Geometry and Polytopes

Discrete Geometry deals with finite sets of points, lines, planes, circles, and many other seemingly simple geometric objects such as polytopes (defined as the convex hull of a finite set of points in \$Rd\$). It has deep connections to combinatorics, optimization, number theory, and computer science. It is also very rich in simple-to-state yet long-unsolved problems. Some examples include: does every *d*-dimensional centrally symmetric polytope have at least \$3d \$ faces? Can the number of *i*-dimensional faces of a polytope be smaller than both the number of its vertices and the number of its top-dimensional faces? These questions are still open except for a few small values of *d*.

This course will be a sampler of a few of the topics in this vast field. We will start with classical theorems due to Radon, Helly, and Carathéodory as well as Dehn's solution to Hilbert's third problem. Then, depending on time and interest, we'll discuss some of the following topics:

(1) Various extensions of the Radon, Helly, and Carathéodory theorems, including fractional, colorful, and topological versions (and potentially touching on very recent exciting developments due to Florian Frick and others).

(2) Kahn-Kalai's counter-example to Borsuk's problem.

(3) Graphs of polytopes -- "classical" results: Steinitz's theorem for 3-polytopes; Balinski's theorem; reconstructing simple polytopes from their graphs.

(4) Graphs of polytopes -- last decade's results: Santos's counterexample to the Hirsch conjecture; positive results towards the polynomial Hirsch conjecture.

(5) Face numbers of simplicial polytopes: the Dehn-Sommerville relations; the upper and the lower bound theorems; the *g*-theorem and its consequences; face numbers of centrally symmetric polytopes.

(6) Volumes, high-dimensional polytopes, and high-dimensional paradoxes: the weak perfect graph conjecture; the Brunn-Minkowski inequality; almost spherical sections of the cube and the octahedron.

Books: There will be no official textbook. Instead, we'll use quite a few sources (including several recent papers). Some of the textbooks that will be handy are:

(1) *A course in convexity* by Alexander Barvinok, 2002.

(2) *Lectures on Discrete Geometry* by Jirí Matoušek, 2002.

(3) *Lectures on Discrete and Polyhedral Geometry* by Igor Pak, available at www.math.ucla.edu/pak/book.htm

(4) *Lectures on Polytopes* by Günter Ziegler, revised first edition.

In addition, I'll be posting my lecture notes on-line.

#### Math 563 A: Farbod Shokrieh - Foundations of Combinatorics Graphs and Tropical Curves

Graphs (1-dimensional simplicial complexes) appear in a variety of subjects including Computer Science, Physics and Chemistry, Social Sciences, Biology, Optimization, Knot Theory, Algebraic Geometry, Group Theory, and Number Theory.

In this course we will discuss some fundamental results on both finite and metric graphs (aka abstract tropical curves). We will also present a few open problems along the way, some of which might be within reach for strong undergraduate or graduate students.

Topics: Classical topics can include: Fundamentals, Matching, Connectivity, Planarity, Trees, Coloring, Extremal Problems, Ramsey Theory, Random Graphs.

Modern topics can include: Metric Graphs and Tropical Curves, Chip-Firing Games, Abel-Jacobi and Riemann- Roch Theory on Graphs, Potential Theory, and maybe even some recent applications in Algebraic Geometry and Number Theory.

The exact emphasis on the material will be decided based on the target audience.

Grading: Grading will be based on weekly homework and class participation.

Books: There will be no official textbook. We will use various sources, including

several research papers. Some of the references (for the more classical topics) are:

(1) Reinhard Diestel, *Graph theory*, Fifth, Graduate Texts in Mathematics, vol.173, Springer, Berlin, 2018. MR3822066

(2) Béla Bollobás, *Modern graph theory*, Graduate Texts in Mathematics, vol. 184, Springer-Verlag, New York, 1998. MR1633290

(3) Norman Biggs, *Algebraic graph theory*, Second, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. MR1271140

(4) Chris Godsil and Gordon Royle, A*lgebraic graph theory*, Graduate Texts in Mathematics, vol. 207, Springer-Verlag, New York, 2001. MR1829620

*Prerequisites: We will only assume a basic knowledge of linear algebra.*

#### Math 564-6: Devinatz - Algebraic Topology

This is intended to be a one year introduction to algebraic topology. The first two quarters will cover simplicial complexes and CW-complexes; singular, simplicial, and cellular homology; and products and universal coefficient theorems. We will end with Poincaré Duality, which gives a very strong result on the behavior of the homology and cohomology of a compact manifold. The third quarter will be devoted to characteristic classes; these are cohomological invariants of vector bundles. We will cover Stiefel-Whitney and Chern classes, as well as cohomology operations and some background information on vector bundles. Applications will be made to the study of manifolds.

*Prerequisites: The first year courses in algebra and manifolds are more than sufficient. A lot less is really necessary, so if you haven't completed these sequences, please see me if you are interested in registering for the course.*

**581 A**: Yuan - Theory of Linear and Nonlinear Second Order Elliptic Equations Part II

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

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

Contents: Harmonic functions (properties), Schauder for Δ, Weighted norm, solvability for Laplace, Boundary Schauder, Lp for Δ; Energy method, capacity, Poincare, Soblev, *W¹‚²* or *H¹* space, trace; De Giorgi/Nash, 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, *C²′ª* regularity for convex equations, Monge-Ampere and special Lagrangian equations, Bellman equations, and Isaacs equations.

**References:**

- Han, Qing; Lin, Fanghua, Elliptic partial di¤erential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; ; American Mathematical Society, Providence, 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.

*Prerequisites: Advanced calculus/Real Analysis (Comparable to Rudin's Principles of Mathematical Analysis; Previous PDE knowledge is welcome, but not required.)*

**581 B: Alper - Algebraic Groups I and 582 B: Pevtsova- Algebraic Groups II**

The topic of algebraic groups is a rich subject combining both group-theoretic and algebro-geometric-theoretic techniques. Examples include the general linear group GLN, the special orthogonal group SON or the symplectic group SpN. Algebraic groups play an important role in algebraic geometry, representation theory and number theory.

In this course, we will take the functorial approach to the study of linear algebraic groups (more generally, affine 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.

**First quarter: Algebraic Groups I, Jarod Alper**

We will use the language of schemes following the excellent introductory texts of W. Waterhouse and J.S. Milne. We will also reference the classic texts by A. Borel, J. Humphreys and T.A. Springer.

Topics at a glance:

-Group schemes over an arbitrary base

-Affine group schemes vs Hopf algebras;

-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);

-Barsotti-Chevalley Theorem on Algebraic Groups;

-Existence of quotients of algebraic groups;

-Detour on descent and algebraic spaces;

-Actions of algebraic groups on schemes and G-torsors;

-Geometric properties: connectedness, irreducibility, smoothness;

-Jordan decompositions;

-Tannaka duality

**References:**We will use the language of schemes following the excellent introductory texts of W. Waterhouse and J.S. Milne. We will also reference the classic texts by A. Borel, J. Humphreys and T.A. Springer.

**Second quarter: Algebraic Groups II, Julia Pevtsova**

Upon developing tools for working with group schemes and Hopf algebras we will pursue the goal of classifying semisimple algebraic groups. Our approach will be based partially on the observation due to Weil that every adjoint simple algebraic group of classical type occurs as a connected component of an automorphism group of a central simple algebra with involution. This will complement the classical approach due to Borel.

Topics at a glance:

-Infinitesimal theory: differential and Lie algebras of affine group schemes;

-Borel fixed point theorem, parabolic subgroups, flag varieties;

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

-Root datum and classication theorem;

-Representations: Highest weight theory (if time allows).

**References:**(1) Introduction to Affine Group Schemes, W. Waterhouse;

(2) Representations of Algebraic Groups, J. Jantzen;

(3) The book of Involutions, M-A Knus, A. Merkurjev, M. Rost, J-P. Tignol;

(4)

*Algebraic groups: the theory of group schemes 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!)

*Prerequisites. Modern Algebra 504/5/6. The course will be suitable for a 2nd year or above graduate student leaning towards an algebra-related field (understood broadly: combinatorics, representation theory, algebraic geometry, algebraic topology). The second year graduate algebra course "Algebraic structures" is desirable**but not required.*

**581 C/582 C/583 C:** **Pal **- Optimal Transport I, II, III

The modern theory of Monge-Kantorovich optimal transport is barely three decades old. Already it has established itself as one of the most happening areas in mathematics. It lies at the intersection of analysis, geometry, and probability with numerous applications to physics, economics, and serious machine learning.

This year-long graduate topics course will serve as an introduction to this rich and useful theory.

The textbooks will be [1] and [2]. This will be supplemented often with papers,especially on the very recent approach by the so-called Schrodinger problem. We will roughly follow the following outline.**Fall:** State the transport problem with a general cost function. Analytic description of solutions. Duality. Displacement convexity. Special examples.**Winter:** The geometry of Wasserstein space. Benamou-Brenier formulation.Otto calculus. Gradient flow of entropy for the heat equation.**Spring:** Schrodinger problem. Particle systems and the probabilistic view of optimal transportation. Gradient flow of entropy for Fokker-Planck equations.

**References**

[1] Cedric Villani. Optimal transport, old and new. Grundlehren der mathematischen Wissenschaften.

Springer. 2009.

[2] Filippo Santambrogio. Optimal Transport for Applied Mathematicians. Calculations of Variations, PDEs, and Modeling. Progress in nonlinear differential equations and their applications, 87. Birkhauser. 2015.

*Prerequisites: Measure theory and some topology and functional analysis. Basically **the graduate real analysis sequence. It is useful to have some knowledge **of probability (large deviations, Brownian motion) and Riemannian manifolds, but **not necessary, since we will develop some of these notions on the way.*

**581 E: Drusvyatskiy - High Dimensional Probability for Data Science**

This is an introductory course in high-dimensional probability with a view towards applications in data science. The main focus will be on the concentration phenomenon in high dimensions. In parallel, we will use the developed techniques to analyze algorithms for various statistical inverse problems, such as community detection, low-rank matrix completion, phase retrieval, robust principal component analysis, etc. This course is appropriate for anyone with a working knowledge of linear algebra and mathematical analysis.

Textbook: Roman Vershynin, "High-dimensional probability: An introduction with applications in data science."

** ****582 A: Uhlmann - Introduction to the Mathematics of Medical Imagining**

In the first part of the course we will study in detail the mathematics of several medical imaging technique like CT (computerized tomography), PET (positron emission tomography), SPECT (single positron emission tomography) and MRI (magnetic resonance imaging). These techniques revolutionized medicine and the development of CT, PET and MRI were awarded the Nobel Prize in Physiology. The underlying mathematical objects that are used in these imaging techniques are the Radon transform and the Fourier transform. The Fourier transform and the Radon transform will be studied in detail. Both of them have applications in several other areas besides medical imaging.

In the second part of the course we will study the mathematics of a different medical imaging technique: Optical Tomography. In this case it is measured the response of the body to light in order to create an image of the optical properties of the medium. This problem can be recast as determining the coefficients of a partial differential equation from some properties of the solutions measured at the boundary of some domain. No background is necessary on partial differential equations. What we will need will be developed in the course.

**References:**

- C. Epstein, An Introduction to the Mathematics of Medical Imaging.
- F. Natterer: The Mathematics of Computerized Tomography
- S. Deans, The Radon transform and some of its applications.

*Prerequisites: The Real Analysis Course or its equivalent.*

**582 E/583 I: Viray - Rational Points I, II**

Fix a field *k* and an algebraic variety *X ⁄ k.* This course is concerned with the fundamental question: what can we say about the *k*-rational points on* X*. We will give an overview of this question in general, but the bulk of the class will focus on fields of arithmetic interest, namely global fields, local fields, and finite fields. Special topics include: Galois descent, the Brauer group, the Weil conjectures, cohomological obstructions, and arithmetic of rational

surfaces.*Prerequisites: 504/5/6, familiarity with basic algebraic number theory and algebraic geom**etry.*

**582 H: Rhode ****and 583 H: Athreya - Complex Analysis**

This two quarter topics sequence is a continuation of the first quarter of the core Analysis (Math 534) class. **Winter quarter** covers the residue theorem, the Riemann mapping theorem, product- and series representations, analytic continuation and the monodromy theorem, and hyperbolic geometry. **Spring quarter** covers Riemann surfaces (examples and basic properties; analytic, geometric, and algebraic perspectives; uniformization theorem) as well as selected topics.

**583 B: Levinson - Intersection Theory**

Bezout's Theorem says that two plane curves of degrees *d* and *e* intersect in *d • e* points (with caveats). That statement, and its far-reaching generalizations, are the basis of intersection theory. Intersection theory provides tools to study the intrinsic geometry of a variety; better yet, when carried out on a moduli space, intersection

theory allows us to study enumeration of the underlying objects (lines, curves, sheaves...), connecting to combinatorics, representation theory and more.

Intersection theory relies not only on theoretical buildup but a body of intuition. Accordingly, we'll start with examples to suggest how intersection theory `should' work. Then we'll introduce the Chow ring of a variety and see (some of) the results needed to define and use intersection products. Then we'll visit some major topics of intersection theory: vector bundles, projective bundles and Chern classes; and Grassmannians (touching on Schubert calculus). Other topics we might see depending on time and interest: blowups and excess intersection formulas; K-theory; the intersection theory of *Mg* and related moduli spaces (which have seen many discoveries in recent years).

*Prerequisites: familiarity with schemes and varieties. Beneficial, but not required, to have seen cup products in algebraic topology.*

**583 E: Hoffman - Viray - Communicating Math** Effectively

This course is about learning and developing strategies for giving clear, motivated, and informative talks. While some of the course will focus on basic skills like good boardwork and eye contact, the majority of the course will be focused on how to effectively convey mathematical ideas to people outside of one's particular specialty. There will be multiple class sessions where faculty and students will give feedback on talks and discuss concrete strategies for preparing and giving talks.

There are no prerequisites, but students should be far enough along in their graduate studies that they have an understanding of a topic that is suitable for a research seminar (e.g., a result that one would learn from a paper, rather than a textbook). Students are encouraged to sign up in groups of 3-5, with the members of the group being representative of a target audience.