# Non-Qual Graduate Course Descriptions Fall 2019 & Spring 2020

## Non-Qual Graduate Course Descriptions Fall 2019

#### MATH 8080 (Lie Algebras) - C-J. Lai

Textbook:

•    J.~E.~Humphreys, Introduction to Lie Algebras and Representation Theory
•    (Additional reference) R.~Carters, Lie Algebras of Finite and Affine Type.

The Lie algebras were originally introduced by Sophus Lie as algebraic structures used for the study of the Lie groups. However, the Lie algebras proved to be of interest in their own right. In this course, we study Lie algebras from a purely algebraic point of view. The structure and representation theory of the semisimple Lie algebras over C is not just beautiful, but comprehensible using mostly just linear algebra. Since then the Lie theory has found many applications throughout mathematics and physics (e.g., differentiable manifolds, algebraic groups, quantum groups, finite groups of Lie type, modern combinatorics).

Our goal is to cover parts of Humphreys' book, including

•    Structures of complex simple Lie algebras
•    PBW theorem on the universal enveloping algebras
•    Classification theorem of complex simple Lie algebras
•    Finite-dimensional representation theory

Carter's appendix is a great reference for working representation theorists.

Disclaimer: there will be no Lie groups in this course.

#### MATH 8210 - Topology of Manifolds - Wu

We will cover the definition of smooth manifolds, de Rham cohomology, Poincare duality.

#### MATH 8330 - Abelian Varieties - Lorenzini

This is an advanced course in algebraic geometry that will assume that participants are familiar with the basic properties of schemes. Ambitious goal: by the end of the course, the participants will have a good grasp of the main topics in the following three chapters in Cornell-Silverman: Group schemes, Jacobian varieties, and Abelian Varieties. A good guide towards learning about abelian varieties is Mumford's book Abelian Varieties. Mumford often works over an algebraically closed field; we will also be interested in fields of arithmetical interest.

#### MATH 8440 - Further Topics in Analytic Number Theory - Magyar

The course will introduce techniques of analytic number theory and discuss both classical and more recent     results, the emphasis being to those related to prime numbers. Topics may include but may not be restricted to,

•    The Prime Number Theorem with explicit error terms
•    The ternary Goldbach problem and  3-term progressions in the primes
•    Large Sieve and the Bombieri-Vinogradov theorem
•    Diophantine equations in the primes
•    Gaps between primes

#### MATH 8500 - Numerical Analysis - M-J. Lai

MATH8500 called Advances in Numerical Analysis is a course for preparation of qualifying examination on "Numerical Analysis". It serves also an introduction to do research on numerical analysis.  The course consists of two parts: one is numerical linear algebra. We will discuss how to solve linear system of equations by using various iterative methods.   Next we shall discuss SVD and solutions to least squares problems. In addition, we shall discuss how to find sparse solution of underdetermined linear systems. The other part is numerical approximation. We shall study polynomial interpolation, spline interpolation and their tensor product version. Next triangulated splines over 2D and 3D will be introduced. In addition, numerical differentiation and integration will be discussed.
Anyway, there are two topics of research which have been carried out for centuries and will be carried out forever: how to solve linear system fast and accurately and how to approximate a high dimensional function based on its values over finitely many points.
The prerequisite of this course is to have any undergraduate numerical analysis taken and a basis knowledge of MATLAB or other programming skill.  Otherwise, talking to me and see if the prerequisite can be waived.

#### MATH 8770 - Partial Differential Equations - Tie

I will use the book by Walter Craig to cover the basic PDEs. Here is the introduction of the book that you may tell the first year students. Currently  two students registered for the courses. Students need to have some background in ODEs and basic real analysis (baby Rudin is enough).

This textbook is intended to cover material for an introductory course in partial differential equations (PDEs), aimed for fourth year undergraduate
or first year graduate mathematics students. The material focuses on the three most important aspects of the subject: namely

(i) theory, which is to say the questions of existence, uniqueness, and continuous dependence on given data or parameters;
(ii) phenomenology, meaning the study of properties of solutions of the PDEs that we examine; and
(iii) applications, by which we mean that we will make an attempt to exercise good scientific taste in choosing the topics in this text, often based on physical or geometrical applications.

## Non-Qual Graduate Course Descriptions Spring 2020

#### MATH 8020 - Commutative Algebra - Clark

A first course in commutative algebra: rings, ideals, modules, localization, integral closure, the Nullstellensatz.  Some further topics according to student need and interest.  The text will be http://math.uga.edu/~pete/integral2015.pdf

This is a good course for most AGANT students to take at some point.  The material in this course is used in the algebraic geometry sequence and is essentially a prerequisite for Math 8305 (Intro to Schemes).  It would be helpful to students taking Math 8400 and Math 8410 (Algebraic Number Theory I and II), although the extent to which it is used depends on the particular iteration of the course and the instructor.  In algebra, it is useful for students studying algebraic groups, among other things.

The only prerequisite for the course is Math 8000.  Of course, prior exposure to other topics in algebra couldn't hurt and may make the material more interesting.  It is reasonable course for a first year student with a strong background and interest in algebra to take.

#### MATH 8030 - Representations of Semisimple Lie Algebras: Category O - Boe

Textbook: "Representations of Semisimple Lie Algebras in the BGG Category O" by James Humphreys (AMS Graduate Studies in Mathematics, vol. 94, 2008).

If you've taken a course on Lie Algebras (e.g. MATH 8080) and have seen some elementary homological algebra, then you have all the background you need. Alternately, if you’ve had a course in Algebraic Groups or Quantum Groups or Geometric Representation Theory, plus a course in Homological Algebra, you should be fine. I'll start with a synopsis (Chapter 0 of the text) of some basic material on structure of semisimple Lie algebras g over C, and will review homological algebra as needed, for those who may not have seen exactly those topics.

The category of all g-modules is too large to be studied algebraically. Fortunately, many important representations satisfy appropriate finiteness conditions, which led to the definition in the early 1970s of the Bernstein-Gelfand-Gelfand category O. This is the setting for the famous 1979 Kazhdan-Lusztig Conjecture on the characters of irreducible highest weight g-modules, proved in 1981 using deep methods from algebraic geometry.

The ideas and methods developed for category O have become pervasive in many other settings, including Kac-Moody Lie algebras, Lie superalgebras, quantum groups, algebraic groups in prime characteristic, finite groups of Lie type, symplectic reflection algebras, Schur algebras, and beyond. So what you learn in this course will be a useful foundation for work in almost any area of modern representation theory.

This will be a rare opportunity to take this course at UGA, so if you think you might be interested, I hope you'll sign up.

#### 8170 - Functional Analysis - Wang

Functional analysis is a vastly developed subject of mathematics with broad applications in pure and applied mathematics, as well as in other disciplines such as physics, statistics, economics, engineering and operations research, to name a few. It also provides foundations for active research areas such as operator algebras, operator theory, noncommutative geometry and quantum groups. This course will be of interest for those interested in delving deeper into the subject, as well as those who wish to use the subject as a tool in other fields. The usual prerequisites for the course is Lebesgue integral, but it can be accessible to those with background in the theory of calculus and linear algebra, such as the theory of limits and continuity, linear transformations over vector spaces, as long as they are willing to take on faith some basic facts. Topics of this one semester course will consists of three parts, mostly chosen from Math 8170/8180. Part 1 is on basic material on functional analysis:   Banach spaces and examples, uniform bounded theorem, open mapping theorem, closed graph theorem, various versions of the Hahn-Banach theorem. Weak *-topologies, Alaoglu theorem.  Hilbert spaces and examples, Riesz representation theorem, projection operators, unitary operators and Plancherel theorem, self-adjoint operators, normal operators. Part 2 focuses on spectral theory: Commutative Banach algebras and examples including L^1(G) of a locally compact abelian group (the circle group, group of integers, the real line, etc), characters and maximal ideal space, Gelfand-Fourier transform, function algebras, Stone-Weierstrass theorem, Gelfand-Naimark theorem for commutative C*-algebras, various versions of the spectral theorem for normal operators. Part 3 is on compact operators and Fredolm index:   Compact and trace class operators and examples, Freholm operators, Fredholm index, non-commutative L^p spaces Other or alternative topics may be included, such as introduction to noncommutative C^*-algebras and von Neumann algebras, depending on interests of the class and/or availability of time. Grading of the course will be based on homework during the semester. There will be no exams. Different arrangements on work load for this course may also be made by discussing the situations with me. Textbook for the class will be 1. A Course in Functional Analysis, 2nd Ed. Graduate Texts in Mathematics, Vol. 96, John B. Conway

Most material of the course can also be found in the following books: 2. A Short Course on Spectral Theory, Graduate Texts in Mathematics, Vol. 209, William Arveson 3. Analysis Now, Graduate Texts in Mathematics, Vol. 118, Gert K. Pedersen

#### 8230 - Geometry of Morse Theory - Wu

We will mostly follow Milnor's book. We will introduce non-degenerate smooth functions on a manifold, and discuss homotopy types in terms of critical values. Then the course could go towards the definition of Morse homology, or the finite dimensional approximation of the path spaces, depending on students' interests.

#### 8330 - Moduli Spaces - Bakker

A salient feature of algebraic geometry is the fact that parameter spaces of algebraic objects can often themselves be given the structure of an algebraic scheme. This course aims to make this idea precise, and develop tools to construct such moduli spaces algebraically. We will start with an in-depth look at Grothendieck's classical construction of the Hilbert scheme before systematically studying deformation theory and representability criteria of moduli functors. There will be lots of examples of the constructions of specific moduli spaces, which depending on the interests of the students may include: the Picard scheme, the moduli space of stable curves, moduli spaces of stable sheaves, etc. Further topics may include a brief introduction to algebraic stacks and Artin's representability theorems

#### 8430 - Elliptic Curves II - Clark

This course will cover further topics in elliptic curves: here "further" includes anything not covered in Dr. Lorenzini's Fall 2018 first course in elliptic curves. Particular emphasis will be placed on elliptic curves over global fields: height functions, Mordell-Weil Theorem, Selmer groups and Tate-Shafarevich groups, structure of the torsion subgroup.

#### 8550 - Mathematical aspects of Deep Neural Networks - Petukhov

The introductory part of the course (25-30%) will be devoted to the classical methods of Harmonic Analysis and systems for data (especially images) representation.
In particular, we will discuss Fourier, wavelet, and curvelet systems with their properties. We also will touch topics of sparse data representations.

The core part of the course will be devoted to data analysis and processing with the Neural Networks (NN).

A special consideration will be given to the similarities and differences of classical signal/image processing based on applied
harmonic analysis and NN algorithms.

We will start with shallow NN for general data classification problems followed by Deep Convolutional NNs oriented to image and video related tasks.

In AI part of the course, we will partially follow the book:

Deep Learning (Adaptive Computation and Machine Learning series)
Author: Ian Goodfellow, Yoshua Bengio, Aaron Courville
ISBN# 9780262035613
Publisher: The MIT Press

However, a big part of the course will be based on recent publications.

Moreover, this is a “reading course,” meaning that students will read and review classic and recent papers for the rest of the class.
Students will also have a chance to get experience with optimization problems and deep learning software through an individual/group project.

While no special knowledge on the graduate level is required, understanding and interest in probability theory, harmonic analysis, and numerical
methods is very desirable. The knowledge of computer languages like Python / Matlab.

#### 8850 - VRG on Tilings - Engel

This course will explore recent areas of research about the enumeration of tilings, Hurwitz theory, and quasimodular forms. We will give a quick introduction to these types of modularity results, such as work of Eskin and Okounkov showing that generating functions of branched covers of the torus lie in a very special class of q-series---the quasimodular forms for SL_2(Z). There are lots of projects in this area which are best approached computationally, such as determining these quasimodular forms explicitly, finding patterns amongst all such generating functions, analyzing their asymptotic behavior, computing contour integrals, and understanding connections to representation theory through Fock space and the operator formalism.