Summer 2018
 MATH 79991: Problem Lab in Algebra —practice for PhD Qualifying Exam in Algebra.
 Instructor:
 Prerequisite: Math 7210.
 Text: Online Test Bank.
 MATH 79992: Problem Lab in Real Analysis—practice for PhD Qualifying Exam in Analysis.
 Instructor:
 Prerequisite: Math 7311.
 Text: Online Test Bank.
 MATH 79993: Problem Lab in Topology—practice for PhD Qualifying Exam in Topology.
 Instructor:
 Prerequisite: Math 7510.
 Text: Online Test Bank.
Fall 2018
 MATH 49971: Vertically Integrated Research: The representation theory of the Lie algebra 𝔰𝔩_{2}(ℂ)
 1:302:50 TTh
 Instructor: Profs. Achar and Sage
 Prerequisites: Math 4200 and 4153
 Text: Lectures on 𝔰𝔩_{2}(ℂ)modules by V. Mazorchuk
 A representation of a Lie algebra L is a Lie algebra map from L to the space of linear maps from a vector space V to itself. The representation theory of semisimple Lie algebras is of fundamental importance in many areas of mathematics and physics, including number theory, differential equations, and quantum mechanics. For example, the finitedimensional irreducible representations of the simplest example, 𝔰𝔩_{2}(ℂ) (the 2x2 complex matrices with trace 0) arise naturally in studying the quantummechanical states of the hydrogen atom. The goal of this course is to provide an introduction to representation theory by focusing exclusively on 𝔰𝔩_{2}(ℂ). In particular, we will give a complete description of the irreducible representations of 𝔰𝔩_{2}(ℂ) (including the infinitedimensional ones). No prior familiarity with Lie algebras or representation theory is required.
 MATH 49972: Vertically Integrated Research: Invariants in LowDimensional Topology
 9:0010:20 TTh
 Instructor: Profs. VelaVick and Wong
 Prerequisites: None
 Text: None. Suggested references will be provided during the course.
 In recent years, many powerful invariants have been invented that allow us to better understand the topology and geometry of lowdimensional spaces. For example, they can help us distinguish different knots and links, and explore the geometry of 3dimensional manifolds. The study of these invariants represents a very active area of current research. In this vertically integrated research seminar, we will study some of these invariants, with a view towards computations and applications.
 MATH 63032: Implementing Curriculum Standards for Mathematics in High School: Statistical Reasoning
 3:306:00 Tuesday. Second Floor of Prescott Hall
 Instructor: Prof. Ferreyra.
 Goal: be ready to teach Statistical Reasoning and AP Statistics in high school. This is also a course for undergraduate and graduate students who never took a statistics course and want to learn it for understanding and multiple applications.
 Prerequisite: The course Statistical Reasoning for high schools only has high school Algebra I as prerequisite for high school students because it is a selfcontained course. The requirements for MATH 6303 are to have sufficient pedagogical, math reasoning skills (as in the Standards of Mathematical Practice), and content knowledge to teach other high school math courses including dual enrollment college algebra. Calculus will not be used/taught in MATH 6303.
 Statistical Reasoning is a new course written for high schools following the Louisiana Student Standards for the domain of Statistics and Probability and the Standards of Mathematical Practice. Approximately 100 detailed lessons of 50 minutes have been written and organized in 7 units. These lessons will be available to students of MATH 6303.
The content of the high school Statistical Reasoning course is approximately the first 75% of the content of the AP Statistics course. The technology taught and utilized in the course is Microsoft Excel.
MATH 6303 will model high school teaching. Class time will be used to go over the lessons that have been written. Students will also “teach” some of these lessons to their peers and will prepare additional materials for the high school classroom such as activities, homework and assessments. Additionally, students will provide editorial and feedback comments on some of the available lessons. Students will also create a few lessons on topics which are in the AP Statistics course but not in the Statistical Reasoning course. Some students may also choose to work on converting Microsoft Excel lessons into lessons using Google sheets.
Students need to purchase an AP Statistics preparation book (around $15).
Tests of the type encountered in Praxis II will count for 50% of the grade. The remaining 50% will be awarded for inclass presentations, and preparation, editing and writing of lessons and assessments. The amount of assigned work will depend on the number of credits the student has registered for.
 MATH 7001: Communicating Mathematics I
 3:004:50 T Th
 Instructor: Prof. Oxley.
 Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for onesemesterhour's credit.
 This course provides practical training in the teaching of mathematics at the precalculus level, how to write mathematics for publication, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both precalculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
 MATH 7210: Algebra I
 12:001:20 TTh
 Instructor: Prof. Tu.
 Prerequisites: Math 4200 Algebra I, or equivalent
 Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003. References:
 A first course in Abstract Algebra; Fraleigh
 Algebra: An Approach via Module Theory; Adkins and Weintraub
 This is the first semester of the first year graduate algebra sequence, and covers the material required for the Comprehensive Exam in Algebra. It will cover the basic notions of group, ring, module, and field theory. Topics will include GROUPS: finitely generated abelian groups, group actions, and the Sylow theorems; RINGS and MODULES: Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings, and modules over PIDs; FIELDS: vector spaces, applications to Jordan canonical form, field extensions, and finite fields.
 MATH 7230: Algebraic Number Theory.
 1:302:20 MWF
 Instructor: Prof. Mahlburg.
 Prerequisites:
 Text:
 MATH 7240: Algebraic Geometry.
 10:3011:20 MWF
 Instructor: Prof. Casper.
 Prerequisites:
 Text:
 MATH 7311: Real Analysis I.
 10:301150 TTh
 Instructor: Prof. Olafsson.
 Prerequisites: Math 4032 or 4035 or equivalent.
 Text: Real Analysis, Modern Techniques and Their Applications, by G. B. Folland.
 This is a standard introductory course on analysis based on measure theory and integration. We start by introducing sigma algebras and measures. We will then discuss measurable functions and integration of real and complex valued functions. As an example we discuss the Lebesgue integral on the line and ndimensional Euclidean space. We also discuss the Lebesgue integral versus the Riemann integral. Important topics here are the convergence theorems, product measures and Fubini’s theorem and the RadonNikodym derivative. We give a short discussion of Banach spaces and then apply that to the Lp spaces. Further topics include functions of bounded variations Lebesgue differentiation theorems, Lp and its dual, and the RieszMarkovSaksKakutani theorem and Fourier series may be presented if there is sufficient time.
We will mostly follow the book by Folland. We will also hand out notes as needed. There are several other very good books on analysis and measure theory:
 L. Richardson: Measure and Integral: An Introduction to Real Analysis.
 P. R. Halmos: Measure Theory (Graduate Text in Mathematics)
 A. Friedman: Foundations of Modern Analysis. Dover
 MATH 7350: Complex Analysis.
 12:001:20 TTh
 Instructor: Prof. Antipov.
 Prerequisites: Math 7311 or its equivalent.
 Text: Complex Analysis by Elias Stein and Rami Shakarchi, Princeton Lectures in Analysis II.
 Holomorphic and meromorphic functions of one variable including Cauchy's integral formula, theory of residues, the argument principle, and Schwarz reflection principle. Multivalued functions and applications to integration. The Fourier transform in the complex plane. Entire function including Jensen's formula and infinite products. Conformal mapping including the Riemann mapping theorem and the SchwarzChristoffel integral. Riemann surfaces of algebraic functions (this topic is not covered by the book, and notes will be available).
 MATH 73602: Probability Theory.
 11:30012:20 MWF
 Instructor: Prof. Sundar.
 Prerequisites: Math 7311 or equivalent
 Text: Probability and Measure by Patrick Billingsley
 This is a selfcontained introduction to modern probability theory. It starts from the concept of probability measures, and introduces random variables, distributions and independence. A study of various modes of convergence is presented. Next, the course delves into wellknown limit theorems such as the Kolmogorov strong law of large numbers, threeseries theorem and the law of the iterated logarithm. Weak convergence of probability measures will be discussed in detail, which leads us to the central limit theorem and its applications. A main goal of the course is to develop the concept of conditional probability and its basic properties. Martingales and Brownian motion will be studied and their essential features discussed.
 MATH 7380: Topics in Spectral Geometry and Partial Differential Equations.
 1:302:50 MWF
 Instructor: Prof. Zhu.
 Prerequisites: Some knowledge of real analysis Math 7311 is helpful, but it will be reviewed in the course.
 Text: Class notes and references will be provided
 Eigenproblems are the most basic and important equations in the study of partial differential equations (PDEs). Understanding of any useful properties for eigenfunctions and eigenvalues contributes the development of PDEs. This course will present some basic theory of PDEs and some topics in spectral geometry. The first part of the course will discuss the variational and nonvariational techniques for PDEs, including EulerLagrange equation, fixed point theorem, and method of subsolutions and supersolutions. The second part will cover some background knowledge of Riemannian geometry and analysis of LaplaceBeltrami operator. Topics include heat kernel, isoperimetric inequality, local and global properties for eigenfunctions. No prior knowledge of PDEs is required (necessary concepts will be reviewed). Grade in the course will be based on student engagement and a final presentation. This course is suitable for everyone interested in PDEs, spectral theory, harmonic analysis and geometric analysis.
 MATH 7384: Topics in Material Science:
 10:301150 TTh
 Instructor: Prof. Shipman. and Prof. Junshan Lin (Auburn University)
 Topic: Mathematics of linear wave phenomena
 Prerequisites:
 Text:
 This course will be taught jointly by Profs. Stephen Shipman and Junshan Lin (Auburn University) in a livestreaming setting. Students from any SEC university may enroll. It is intended to be a twosemester course (although the first semester may be taken in isolation). The second semester will be offered in the spring of 2019.
The material in this course builds toward open problems in mathematical physics centered around wave dynamics and scattering in electromagnetics and acoustics. The mathematical topics form a coherent body of theory and techniques. The main components are the partial differential equations (PDE) of electromagnetics and acoustics and other derived phenomena in complex media; integral equations and boundaryintegral representations of solutions to PDE; Fourier analysis and the residue calculus for the study of scattering and resonance; spectral theory of differential and integral operators illustrated and motivated by examples; and asymptotic analysis. The second semester of the course will concentrate on specific problems that are motivated by modern scientific and mathematical research. A goal is that students will be able to understand open problems in this area and be equipped with the basic mathematical tools to begin trying to solve them.
 MATH 7386: Partial Differential Equations.
 2:303:20 MWF
 Instructor: Prof. Lipton.
 Prerequisites: Math 2057 or equivalent
 Text: Partial Differential Equations by Lawrence C. Evans; Chapters 2  10.

Description: Partial differential equations, both linear and nonlinear describe physical laws that govern fluid dynamics, electromagnetics, gravitation, solid mechanics, quantum mechan ics, celestial mechanics, etc. Methods of explicit solution together with the theory of existence and uniqueness of solution are indispensable tools for the rational solution (computational or otherwise) of problems in economics, physics and engineering.
Topics Covered:
(1) Hyperbolic, Parabolic, and Elliptic Partial Differential Equations
(2) Examples: The Wave Equation, The Heat Equation, The Laplace Equation
(3) Sobelev spaces and existence and uniqueness of solution.
(4) Spectral theory
(5) Banach fixed point theorem
(6) Subdifferentials and nonlinear semigroups
(7) Hamilton Jacobi Equations
 MATH 73902: Variational Inequalities.
 9:0010:20 TTh
 Instructor: Prof. Sung.
 Prerequisites: Functional Analysis (MATH 7330)
 Text: An Introduction to Variational Inequalities and Their Applications by David Kinderlehrer and Guido Stampacchia (SIAM 2000)
 The theory of variational inequalities treats optimization problems with inequality constraints. In this course we study the existence, uniqueness and regularity of the solutions of elliptic and parabolic variational inequalities. Applications and numerical methods will also be discussed.
 MATH 73903: Variational Analysis and Dynamic Optimization.
 12:301:20 MWF
 Instructor: Prof. Wolenski.
 Prerequisites: Advanced Calculus, Linear Algebra, or their equivalent; permission of the instructor.
 Text: No formal text, but we will draw from several sources.
 Variational Analysis is a relatively new subject that is in effect the “mathematics of optimization”. It introduces differentiablelike concepts and develops ideals that allow for a systematic treatment of functions and sets that may not be differentiable in the classical sense. The beginning third of the course will introduce normal cones of closed sets and proximal subgradients of lower semicontinuous functions within the background of a review of finitedimensional optimization. The second third of the course will cover the classical calculus of variations, and the final third will combine the first two topics with a modern treatment of optimal control theory.
 MATH 7510: Topology I
 8:309:20 MWF
 Instructor: Prof. Dasbach.
 Prerequisites: Advanced Calculus (Math 4031)
 Text: Topology (2nd ed.) by James R. Munkres.
 This course is a preparation course for the Core I examination in topology. It will cover general (point set) topology, the fundamental group, and covering spaces. We will also introduce simplicial complexes and manifolds, using them often as examples. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online .
 MATH 7520: Algebraic Topology.
 12:301:20 MWF
 Instructor: Prof. Litherland.
 Prerequisites: Math 7510 and 7512
 Text: Allen Hatcher, Algebraic Topology,CUP 2001. A free electronic version is available online.
 This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. While MATH 7510 developed the theory of fundamental groups and MATH 7512 developed homology theories for topological spaces the focus of this course will be cohomology theory which is dual to homology theory. However, one of the advantages of developing cohomology theory of spaces is that they are naturally equipped with a ring structure.
 MATH 7590: Heegaard Floer Homology.
 1:302:50
 Instructor: Prof. Wong.
 Prerequisites: 7510 Topology I recommended, and 7550 Differential Geometry recommended
 Text: None. Suggested references will be provided during the course.
 Heegaard Floer homology is an invariant for 3manifolds, as well as knots and links in them, that has proved to be very powerful since its inception in 2001, capturing plenty of geometrical information. Although they were originally defined analytically, combinatorial algorithms for effective computation have been found in many cases. In this course, we will develop the invariant from both a theoretical and a computational perspective, with a view towards open problems in the subject.
Spring 2019
 MATH 49972: Vertically Integrated Research:
 Instructor: Profs. VelaVick and Wong
 Prerequisites:
 Text:
 MATH 7002: Communicating Mathematics II
 3:004:50 T Th
 Instructor: Prof. Oxley .
 Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for onesemesterhour's credit.
 MATH 7211: Algebra II.
 Instructor: Prof. Yakimov.
 Prerequisites:
 Text:
 MATH 7230: Topics in Number Theory
 Instructor: Profs. Hoffman. and Long.
 Prerequisites: Basic Algebra and Topology; this is covered in Math 7210, 7510. Complex Analysis  familiarity with analytic functions; this is covered in Math 7350. It is helpful, but not required to have some exposure to Number Theory and Algebraic Geometry. There will be courses taught on these in Fall 2018 (Math 7230, 7240).
 Text: No required text, but much of the material will be drawn from
 A First Course in Modular Forms (Graduate Texts in Mathematics, Vol. 228), Diamond & Shurman
 A Classical Introduction to Modern Number Theory, Ireland &Rosen, (Graduate Texts in Mathematics) (v. 84)
 An Introduction to the Langlands Program, Bernstein& Gelbart (eds)
 The focus is on Lfunctions. These are at the center of much research today. Themes: Euler products; analytic continuation and functional equations; special values; relations among different kinds of Lfunctions. Briefly there are two broad classes of Lfunctions: those that arise from automorphic forms and those that arise from geometry. There are conjectural relations between the Lfunctions of the first kind with those of the second kind. We start with the classical Dirichlet Lfunctions and move on to the Lfunctions defined by modular forms. We give an exposition of Tate's thesis which gives a unified account of the Lfunctions that arise from abelian characters of Galois groups. We also give an introduction to Lfunctions arising from algebraic varieties, for instance algebraic curves. We then describe relations, proven and conjectural among these. This course is more like a seminar. The topics and level will depend on the students. Each student will have a project to do, with a written report and a lecture exposition.
 MATH 7250: Representation Theory
 Instructor: Prof. Zeitlin.
 Prerequisites:
 Text:
 MATH 7290: Langlands Program
 Instructor: Prof. Sage.
 Prerequisites:
 Text:
 MATH 7320: Ordinary Differential Equations.
 Instructor: Prof. Shipman
 Prerequisites:
 Text:
 MATH 7330: Functional Analysis.
 Instructor: Prof. Estrada.
 Prerequisites:
 Text:
 MATH 7366: Stochastic Analysis.
 Instructor: Prof. Kuo.
 Prerequisites:
 Text:
 MATH 7380: Topics in Elliptic Partial Differential Equations
 Instructor: Prof. Nguyen.
 Prerequisites:
 Text:
 MATH 73841: Topics in Material Science:
 Instructor: Prof. Shipman.
 Prerequisites:
 Text:
 MATH 7390: Topics Course (TBA)
 Instructor: Prof. Zhang.
 Prerequisites:
 Text:
 MATH 7410: Graph Theory.
 Instructor: Prof. Oporowski.
 Prerequisites:
 Text:
 MATH 7490: Tutte Polynomial for Matroids and Graphs.
 Instructor: Prof. Oxley.
 Prerequisites: Math 7410 and 7490 (Matroid Theory) or permission of the department.
 Text: Matroid Applications edited by Neil White (Chapter 6: The Tutte Polynomial and its Applications by Thomas Brylawski and James Oxley)
 The theory of numerical invariants for matroids is one of many aspects of matroid theory having its origins within graph theory. Most of the fundamental ideas in matroid invariant theory were developed from graphs by Veblen, Birkhoff, Whitney, and Tutte when considering colorings and flows in graphs. This course will introduce the Tutte polynomial for matroids and will consider its applications in graph theory, coding theory, percolation theory, electrical network theory, and statistical mechanics.
 MATH 7512: Topology II.
 Instructor: Prof. Gilmer.
 Prerequisites:
 Text:
 MATH 7550: Differential Geometry.
 Instructor: Prof. Baldridge.
 Prerequisites:
 Text:
 MATH 75901: Geometric Topology: Symplectic Geometry.
 Instructor: Prof. VelaVick.
 Prerequisites:
 Text:
 MATH 75902: Bounded Cohomology
 Instructor: Prof. Dani
 Prerequisites:
 Text:
 MATH 7710: Advanced Numerical Linear Algebra.
 Instructor: Prof. Walker.
 Prerequisites:
 Text: