Please direct inquiries about our graduate program to:
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Graduate Courses, Summer 2023Spring 2024
Contact
Summer 2023
For Detailed Course Outlines, click on course numbers.
79991 Problem Lab in Algebra—practice for PhD Qualifying Exam in Algebra
 Time:
 Instructor:
 Prerequisite:
 Text:
79992 Problem Lab in Analysis—practice for PhD Qualifying Exam in Analysis
 Time:
 Instructor:
 Prerequisite:
 Text:
79993 Problem Lab in Topology—practice for PhD Qualifying Exam in Topology
 Time:
 Instructor:
 Prerequisite:
 Text:
7999n Assorted Individual Reading Classes
8000n Assorted Sections of MSThesis Research
9000n Assorted Sections of Doctoral Dissertation Research
Fall 2023
For Detailed Course Outlines, click on course numbers.
Core courses are listed in bold.
4997 Vertically Integrated Research: Cluster algebras. Prof Achar
 10:3011:50 TTh
 Instructor: Profs. Achar
 Prerequisite: Math 4200
 Text:
 Description: Cluster algebras are certain commutative rings constructed via a recursive process called "seed mutations". Although they were only introduced by Fomin and Zelevinsky in 2001, they have already played an important role in developments in representation theory, topology, combinatorics and algebraic geometry. The goal of this seminar is provide an elementary introduction to the theory of cluster algebras, emphasizing the combinatorial aspects. We will also discuss some of the most accessible applications. No special background is assumed besides familiarity with the definition of a commutative ring.
4997 Vertically Integrated Research: TACI: Topological, Algebraic, and Combinatorial Interactions. Profs. Bibby and Cohen
 10:3011:20 MWF
 Instructors: Profs. Bibby and Cohen
 Prerequisite: linear algebra (Math 2085 or 2090)
 Text: none
 Description: This course will explore interactions between topology, algebra, and combinatorics. Topics may include, for example, finite groups arising from Latin squares, special families of plane curves, or neighborly partitions.
4997 Vertically Integrated Research: Groups, graphs, and beyond. Profs. Dani and Schreve
 1:302:50 TTh
 Instructors: Profs. Dani and Schreve
 Prerequisite: Math 4200 or consent of instructors
 Text:
 Description: Groups arise naturally as symmetries of a space. This course will explore how features of the space influence the algebraic structure of the group. We will begin by studying symmetry groups of graphs, and then proceed to their higher dimensional generalizations: cube complexes.
7001 Communicating Mathematics I. Prof. Shipman and Dr. Ledet
 TTh 3:004:50
 Instructor: Prof. Shipman and Dr. Ledet
 Prerequisite: Consent of department. This course is required for all firstyear 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.
7210 Algebra I. Prof. Achar
 9:0010:20 TTh
 Instructor: Prof. Achar
 Prerequisite: Math 4200 or its equivalent
 Text: Dummit and Foote, Abstract Algebra
 Description: This is the first semester of the first year graduate algebra sequence. It will cover the basic notions of group, ring, and module theory. Topics will include symmetric and alternating groups, Cayley's theorem, group actions and the class equation, the Sylow theorems, finitely generated abelian groups, polynomial rings, Euclidean domains, principal ideal domains, unique factorization domains, finitely generated modules over PIDs and applications to linear algebra, field extensions, and finite fields.
7230 Topics in Number Theory: Algebraic Number Theory. Prof. Kopp
 12:001:20 TTh
 Instructor: Prof. Kopp
 Prerequisite: Math 7210 and 7211 (or equivalent; students will be assumed to know about principal ideal domains, unique factorization domains, and Galois theory). Math 7510 (particularly metric spaces and the product topology) may be helpful depending on the optional topics covered. Taking Math 7350 concurrently may be helpful for a few topics but isn't required.
 Text: Algebraic Number Theory by Jürgen Neukirch (trans. Norbert Schappacher), Springer
 Description: The course will cover the arithmetic of rings of integers of algebraic number fields. Topics include geometry of numbers (Minkowski space), class groups and class numbers, unit groups and Dirichlet's unit theorem, Dedekind domains, localization and discrete valuation rings, norms of ideals, splitting of primes and ramification, discriminants, regulators, Dedekind zeta functions, and the class number formula. Optional topics, some of which may be covered depending on time and student interest, include nonmaximal orders, padic/local fields, adeles and ideles, introduction to class field theory, Hecke and Artin Lfunctions, continued fractions, Shintani decomposition (Shintani's unit theorem), aspects of the theory of cyclotomic fields, and computational techniques in algebraic number theory.
7260 Homological Algebra. Prof. Hoffman
 8:309:20 MWF
 Instructor: Prof. Hoffman
 Prerequisite: Abstract Algebra, Math 7210. Familiarity with Algebraic Topology, Math 7520 is helpful but not required. Also, an acquaintance with Category Theory will be essential.
 Text: A standard text is Charles Weibel, An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics, Series Number 38) but we will not follow this exclusively.

Description:
Topics:
 Chain complexes, resolutions (projective and injective).
 Examples: Ext, Tor, Group cohomology, Hochschild cohomology.
 Derived functors in abelian categories.
 Introduction to sheaf theory in a topos.
In the “old days” one introduced ideas of triangulated derived categories. The modern thinking is to replace these by infinity categories. We won’t have time for this, but possibly, time permitting, we could introduce ideas of Simplicial Sets.
There will be homework sets assigned at each class. Assignments will not be collected and graded. Instead, at the beginning of each class, a student will present a solution to one of the homeworks.
7290 Seminar in Algebra and Number Theory: Geometric methods in representation theory. Dr. Singh
 11:3012:20 MWF
 Instructor: Dr. Singh
 Prerequisite: Algebraic geometry (scheme theory) and basic Lie theory.
 Text: Lectures on Springer theories and orbital integrals arXiv:1602.01451v1
 Description: This course is about three interesting geometric objects that are important in representation theory. We will discuss the geometry of Springer fibers, affine Springer fibers and Hitchin fibers. We will then construct representations of Weyl groups and their affine version. An interesting aspect of these representations is that these do not arise from actions on such fibers but on the cohomology of these fibers. In the last part of the course, we will discuss the relation of the affine Springer fibers with the Hitchin fibers. A review of background material will be given as needed.
7311 Real Analysis (a.k.a. Analysis I). Prof. Tarfulea
 9:3010:20 MWF
 Instructor: Prof. Tarfulea
 Prerequisite: Undergraduate real analysis
 Text: Stein and Shakarchi : Real Analysis
 Description: 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 Hilbert spaces. We then introduce Lp spaces and discuss the main properties of those spaces. Further topics include functions of bounded variations Lebesgue differentiation theorems, Lp and its dual. Other topics might be included depending on the time.
7350 Complex Analysis. Prof. Yang
 12:301:20 MWF
 Instructor: Prof. Yang
 Prerequisite: Math 7311
 Text:
 Description: Theory of holomorphic functions of one complex variable; path integrals, power series, singularities, mapping properties, normal families, other topics.
7360 Probability Theory: Prof. Ganguly
 11:3012:20 MWF
 Instructor: Prof. Ganguly
 Prerequisite: Math 7311
 Text: Lecture notes

Description: Probability spaces, random variables and expectations, independence, convergence concepts, laws of large numbers, convergence of series, law of iterated logarithm, characteristic functions, central limit theorem, limiting distributions, martingales.
The course is a selfcontained introduction to modern probability theory. It starts from the concept of probability measures, random variables, and independence. Several important notions of convergence including almost sure convergence, convergence in probability, convergence in distribution will be introduced. Wellknown limit theorems for sums of independent random variables such as the Kolmogorov's law of large numbers, Central Limit Theorem and their generalizations will be studied. Sums of independent, centered random variables form the prototype for an important class of stochastic processes known as martingales, and therefore play a major role in probability and other areas of mathematics. A part of the course will be devoted to understanding conditional probability which then forms the foundation for the theory of martingales. Brownian motion is an important example of a continuoustime martingale. If time permits, basic features of martingale theory and Brownian Motions will be briefly discussed. The course will not have any fixed text book, but some books will be cited for reference. It is important that the students take and read the class notes carefully.
7380 Seminar in Functional Analysis: Singular Integrals. Prof. Nguyen
 1:302:20 MWF
 Instructor: Prof. Nguyen
 Prerequisite: Math 7311

Text: Loukas Grafakos, Classical Fourier Analysis, Third Edition, GTM 249, Springer, New York, 2014. xviii+638 pp. ISBN: 9781493911936.
Recommended Reference: Elias Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.  Description: This is an introductory course on Fourier Analysis with an emphasis on the boundedness of Singular Integral Operators of convolution type. Such a boundedness property plays a fundamental role in various applications in pure and applied analysis. The course also covers such classical topics as Interpolation, Maximal Functions, Fourier Series, and possibly LittlewoodPaley Theory if time permits.
7382 Introduction to Applied Mathematics. Prof. Massatt
 10:3011:20 MWF
 Instructor: Prof. Massatt
 Prerequisite: Simultaneous enrollment in Math 7311
 Text: Lecture notes
 Description: Overview of modeling and analysis of equations of mathematical physics, such as electromagnetics, fluids, elasticity, acoustics, quantum mechanics, etc. There is a balance of breadth and rigor in developing mathematical analysis tools, such as measure theory, function spaces, Fourier analysis, operator theory, and variational principles, for understanding differential and integral equations of physics.
7384 Topics in Material Science: Free fracture theory and nonlocal differential equations. Prof. Lipton
 10:3011:50 TTh
 Instructor: Prof. Lipton
 Prerequisite: Math 7311 and Math 7386
 Text:

Description: The mechanics of fracture propagation provides essential knowledge for the risk tolerant design of devices, structures, and vehicles. Ideally fracture should emerge naturally from a field theory described by an initial boundary problem. Nonlocal approaches to fracture modeling are formulated along these lines, coupling fracture caused by breaking bonds at the atomic scale with continuous and discontinuous deformation at the macroscopic scale. Both localization and emergent behavior is the hallmark of theory and simulations using nonlocal formulations. The numerical solution of the nonlocal differential equation provides for spectacular results. On the other hand the field theory while promising is in its early stages and needs to recover established results in a mathematically rigorous and systematic way.
In this course we introduce the nonlocal initial boundary value problem and prove convergence to a sharp fracture theory in the limit of vanishing nonlocality. This is a highly nonlinear problem involving methods introduced in the past 15 years. We introduce the generalization of Sobolev space given by the Special Functions of Bounded Variation (SBV) and Special Functions of Bounded Deformation (SBD). We review their fine properties over Lebesgue measurable sets. We define and show Gamma convergence of the nonlocal energy to the classic Griffith fracture energy. Here we apply nonlocal methods used in the proof of the de Giorgi’s conjecture for the convergence of nonlocal approximations of the Mumford Shah functional developed by Gobbino. These are used in conjunction with slicing decompositions of SBV and SBD together with the integralgeometric measure of geometric measure theory. Convergence of the dynamics of intact material away from the crack is established with the aid geometric measure theory.
We conclude the course, showing how the kinetic relations of classic fracture theory, relating the speed of propagation of the crack tip singularity to the energy flowing into it are derived directly from the nonlocal differential equation and then passing to the local limit.
This course represents the beginning of the story. For example in the nonlocal differential equation the notion of regularity of solution, including the rigorous origins of crack branching, remains unanswered. These questions also lie on the frontiers of the physics describing the fracture process.
7386 Theory of Partial Differential Equations. Prof. Bulut
 9:0010:20 TTh
 Instructor: Prof. Bulut
 Prerequisite: Math 7330
 Text: Partial Differential Equations by L. C. Evans
 Description: Sobolev spaces. Theory of second order scalar elliptic equations: existence, uniqueness and regularity. Additional topics such as: Direct methods of the calculus of variations, parabolic equations, eigenvalue problems.
7390 Seminar in Analysis: Iterative Methods for Sparse Linear Systems. Prof. Sung
 12:001:20 TTh
 Instructor: Prof. Sung
 Prerequisite: Math 7710
 Text: Iterative Methods for Sparse Linear Systems (Second Edition) by Yousef Saad (This book can be downloaded for free through the LSU library.)
 Description: Basic Iterative Methods, Projection Methods, Krylov Subspace Methods, Preconditioning.
7490 Seminar in Combinatorics, Graph Theory, and Discrete Structures: Graph Minors. Prof. Ding
 1:302:20 MWF
 Instructor: Prof. Ding
 Prerequisite: Math 4171 or equivalent
 Materials: Lecture notes
 Description: This is an introduction to the theory of graph minors. We will discuss many problems of the following two types: determine all minorminimal graphs that have a prescribed property; determine the structure of graphs that do not contain a specific graph as a minor. We will focus on connectivity and planarity.
7510 Topology I. Prof. Bibby
 8:309:20 MWF
 Instructor: Prof. Bibby
 Prerequisite: Advanced Calculus (Math 4031)
 Text: Topology (2nd ed.) by James R. Munkres.
 Description: 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. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online.
7520 Algebraic Topology. Prof. Cohen
 2:303:20 MWF
 Instructor: Prof. Cohen
 Prerequisite: MATH 7512
 Text: Algebraic Topology by Allen Hatcher (freely available online)
 Description: This is a continuation of Math 7512 (for students from any prior offering of this course). We will discuss various forms of duality involving homology and cohomology, basic homotopy theory, and related topics potentially including the theory of fiber bundles, spectral sequences, etc.
7590 Seminar in Geometry and Algebraic Topology: Gauge Theory and SeibergWitten Invariants. Prof. Baldridge
 10:3011:50 TTh
 Instructor: Prof. Baldridge
 Prerequisite:
 Text:

Description:
Gauge theory is often staged on 4dimensional manifolds, both in physics and in the creation of smooth invariants for 4manifolds. In this course, we will investigate the SeibergWitten equations, which are a pair of nonlinear partial differential equations on the space of connections of a line bundle and the space of sections (spinors) of a Spin^c bundle. Solutions to these equations turn out be gaugeinvariant, which leads, after several steps, to the celebrated SeibergWitten invariants of smooth 4manifolds—invariants that can (sometimes) detect when two homeomorphic smooth 4manifolds are not diffeomorphic (exotic manifolds).
Along the way, we will study examples of smooth 4manifolds and how to construct exotic 4manifolds. This will include (fun) calculations with fundamental groups, symplectic 4manifolds, and ``tying knots’’ into smooth 4manifolds (FintushelStern construction). For the algebraminded student, we will show how Clifford algebras and spin representations can be and are used in gauge theory. For the analysisminded student, we will work show how to apply Sobolev norms and Sobolev embedding theorems to understand the moduli space of solutions.
Low dimensional topologists look for new invariants by studying different gauge theories. SeibergWitten, in a sense, is one of the easiest gauge theories to study because it is a based upon an abelian U(1)theory. This course will give students exposure to how such invariants are discovered and investigated. It should be interesting to students who want an overview of 4manifold constructions and want to learn how to use advance techniques in topology, geometry, and global analysis to study smooth manifolds.
7999n Assorted Individual Reading Classes
8000n Assorted Sections of MSThesis Research
9000n Assorted Sections of Doctoral Dissertation Research
Spring 2024
For Detailed Course Outlines, click on course numbers.
4997 Vertically Integrated Research: TBA. Prof Achar
 Time TBA
 Instructor: Profs. Achar
 Prerequisite:
 Text: none
 Description:
4997 Vertically Integrated Research: TACI: Topological, Algebraic, and Combinatorial Interactions. Profs. Bibby and Cohen
 Time TBA
 Instructors: Profs. Bibby and Cohen
 Prerequisite: linear algebra (Math 2085 or 2090)
 Text: none
 Description: This course will explore interactions between topology, algebra, and combinatorics. Topics may include, for example, finite groups arising from Latin squares, special families of plane curves, or neighborly partitions.
4997 Vertically Integrated Research: TBA. Profs. VelaVick and Dr. Wu
4997 Vertically Integrated Research: Topic in Machine Learning. Profs. Drenska and Wolenski
7002 Communicating Mathematics II. Prof. Oxley
 TTh 3:004:50
 Instructor: Prof. Oxley
 Prerequisite: Consent of department. This course is required for all firstyear 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.
7211 Algebra II. Prof. Long
 Time TBA
 Instructor: Prof. Long
 Prerequisite: Math 7210 or equivalent
 Text:
 Description: Fields: algebraic, transcendental, normal, separable field extensions; Galois theory, simple and semisimple algebras, Wedderburn theorem, group representations, Maschke’s theorem, multilinear algebra.
7230 Topics in Number Theory: Modular forms. Prof. Tu and Dr. Allen
7320 Ordinary Differential Equations. Prof. Wolenski
 Time TBA
 Instructor: Prof. Wolenski
 Prerequisite: Undergraduate ODEs and analysis
 Text:
 Description: Existence and uniqueness theorems, approximation methods, linear equations, nonlinear equations, stability theory; other topics such as boundary value problems.
7330 Functional Analysis (a.k.a. Analysis II). Prof. Han
 12:001:20 TTh
 Instructor: Prof. Han
 Prerequisite: Math 7311 or equivalent
 Text:
 Description: Banach spaces and their generalizations; Baire category, BanachSteinhaus, open mapping, closed graph, and HahnBanach theorems; duality in Banach spaces, weak topologies; other topics such as commutative Banach algebras, spectral theory, distributions, and Fourier transforms.
7366 Stochastic Analysis. Prof. Chen
 Time TBA
 Instructor: Prof. Chen
 Prerequisite: Math 7360
 Text:
 Description: Wiener process, stochastic integrals, stochastic differential equations.
7370 Lie Groups and Representation Theory. Prof. TBA
 Time TBA
 Instructor: Prof. TBA
 Prerequisite: Math 7311, 7210, and 7510 or equivalent.
 Text:
 Description: Lie groups, Lie algebras, subgroups, homomorphisms, the exponential map. Also topics in finite and infinite dimensional representation theory.
7380 Seminar in Functional Analysis: TBA. Prof. He
 Time TBA
 Instructor: Prof. He
 Prerequisite:
 Text:
 Description:
7384 Topics in Material Science: Spectral theory and applications. Prof. Shipman
 Time TBA
 Instructor: Prof. Shipman
 Prerequisite:
 Text:
 Description: Pure and applied spectral theory. I will include periodic, quasiperiodic, and ergodic Schrödinger operators and connections to commutative algebra, and perhaps some C^{*} algebra techniques.
7390 Seminar in Analysis: TBA. Prof. Zhu
 Time TBA
 Instructor: Prof. Zhu
 Prerequisite:
 Text:
 Description:
7390 Seminar in Analysis: TBA. Prof. Zhang
 Time TBA
 Instructor: Prof. Zhang
 Prerequisite:
 Text:
 Description:
7410 Graph Theory. Prof. Wang
 12:001:20 TTh
 Instructor: Prof. Wang
 Prerequisite: Math 2085 and Math 4039, or equivalent.
 Text:
 Description: Matchings and coverings, connectivity, planar graphs, colorings, flows, Hamilton graphs, Ramsey theory, topological graph theory, graph minors.
7490 Seminar in Combinatorics, Graph Theory, and Discrete Structures: Matroid Theory. Prof. Oxley
 Time TBA
 Instructor: Prof. Oxley
 Prerequisite:
 Text:
 Description:
7512 Topology II. Prof. Schreve
 Time TBA
 Instructor: Prof. Schreve
 Prerequisite: Math 7510
 Text:
 Description: Theory of the fundamental group and covering spaces including the SeifertVan Kampen theorem; universal covering space; classification of covering spaces; selected areas from algebraic or general topology.
7550 Differential Geometry. Prof. Zeitlin
 Time TBA
 Instructor: Prof. Zeitlin
 Prerequisite: Math 7210 and 7510; or equivalent.
 Text:
 Description: Manifolds, vector fields, vector bundles, transversality, deRham cohomology, metrics, other topics.
7590 Seminar in Geometry and Algebraic Topology: TBA. Prof. Dani
 Time TBA
 Instructor: Prof. Dani
 Prerequisite:
 Text:
 Description: TBA
7590 Seminar in Geometry and Algebraic Topology: TBA. Prof. VelaVick
 Time TBA
 Instructor: Prof. VelaVick
 Prerequisite:
 Text:
 Description: TBA
7710 Advanced Numerical Linear Algebra. Prof. Walker
 Time TBA
 Instructor: Prof. Brenner
 Prerequisite:
 Text:
 Description: TBA