Please direct inquiries about our graduate program to:
grad@math.lsu.edu
Graduate Courses, Summer 2023 – Spring 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
 No additional information.
8000n Assorted Sections of MSThesis Research
 No additional information.
9000n Assorted Sections of Doctoral Dissertation Research
 No additional information.
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
 No additional information.
8000n Assorted Sections of MSThesis Research
 No additional information.
9000n Assorted Sections of Doctoral Dissertation Research
 No additional information.
Spring 2024
For Detailed Course Outlines, click on course numbers.
4997
Vertically Integrated Research: TACI: Topological, Algebraic, and Combinatorial Interactions. Profs. Bibby and Cohen
 TTh 1:302:50
 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, Latin squares and associated groups, nets, special families of plane curves, block designs, and neighborly partitions, as well as their implications for a variety of subjects, including the theory of hyperplane arrangements and matroids, coding theory, etc.
4997
Vertically Integrated Research: TBA. Profs. VelaVick and Dr. Wu
4997
Vertically Integrated Research: Topics in Machine Learning. Profs. Drenska and Wolenski
7002
Communicating Mathematics II. 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.
7211
Algebra II. Prof. Long
 MWF 12:301:20
 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
 MWF 11:3012:20
 Instructor: Prof. Tu and Dr. Allen

Prerequisites:
1. Math 7210 (Algebra), Math 4036 (complex analysis) or equivalent are very helpful. You should be comfortable with meromorphic and analytic functions, groups, group actions and Galois groups (definitions and examples).
2. Know how to find resources and how to use computer algebra systems (such as Maple, Mathematica, MATLAB, SageMath, or Magma). 
Text: There is no required text, but the following references and web sources are useful.
1. Modular Forms  A Classical Approach by Henri Cohen and Fredrik Stromberg
2. A First Course in Modular Forms (GTM) by F. Diamond and J. Shurman
3. Modular Forms by T. Miyake 
Description:
Modular forms appear naturally in connection with problems arising in many areas and play a central role in the development of number theory. One of the reasons is because they have very tight connections with elliptic curves and Galois representations. All the arithmetic properties of elliptic curves, modular forms, and Galois representations are extremely important in Wiles’ proof of Fermat's Last Theorem.
The aim of this course is to cover the classical theory of modular forms. The course will begin with basic definitions and examples. In the later part of the semester, we will discuss some applications, advanced topics, and open problems, depending on students’ interest.
7320
Ordinary Differential Equations. Prof. Wolenski
 MWF 8:309:20
 Instructor: Prof. Wolenski
 Prerequisite: Undergraduate ODEs and analysis
 Text: I will begin with my notes and then follow the text An Introduction to Dynamical Systems: Continuous and Discrete (Pure and Applied Undergraduate Texts), 2nd Edition, by R. Clark Robinson
 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
 MWF 10:3011:20
 Instructor: Prof. Han
 Prerequisite: Math 7311 or equivalent and undergraduate complex variables
 Text: Methods of modern mathematical physics, vol 1: Functional analysis” by Reed and Simon
 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. Ganguly
 TTh 9:0010:20
 Instructor: Prof. Ganguly
 Prerequisite: Probability theory at the level of Math 7360. Alternatively, a good knowledge of advanced undergraduate probability theory including the concept of stochastic processes and graduate level analysis is also acceptable.
 Text: Stochastic differential equations by Bernt Oksendal
 Description: The course will cover elements of stochastic integrals and stochastic differential equations. A common choice of the integrator is a Brownian motion. A Brownian motion is a continuous random function of time whose paths are very rough. This in particular means that integrals with respect to a Brownian motion cannot be defined in the usual RiemannStieltjes way, and a new approach needs to be introduced to address them. Understanding this approach and the necessary materials behind it is the broad goal of this course. Along the way we will learn about Ito isometry, martingale property of stochastic integrals, Ito's lemma, etc. The last part of the course will focus on stochastic differential equations which are now extensively used in modeling timeevolution of a variety of random systems. If time permits we will also discuss some researchproblems in these areas.
7370
Lie Groups and Representation Theory. Prof. Ólafsson
 TTh 1:302:50
 Instructor: Prof. Ólafsson
 Prerequisite: Math 7311. A basic knowledge of differential geometry is also helpful.

Text:
We will not use any fixed textbook but mainly use our own lecture notes. The notes will be posted and upgraded regularly. Good introductory books on Lie groups include
J. Hilgert and KH. Neeb: Structure and Geometry of Lie Groups
V. S. Varadarajan: Lie Groups, Lie Algebras, and their Representations
N. R. Wallach: Harmonic Analysis on Homogeneous Spaces

Description:
This is an introductory course in Lie groups, homogeneous spaces and representation theory needed for further research in several fields including analysis on Lie groups and homogeneous spaces and representation theory. We will mostly consider linear Lie groups which makes several proofs much easier, but the statement are still be valid for general Lie groups. In fact, every connected, finite dimensional Lie group is locally isomorphic to a linear Lie group.
The course starts with basic definitions of Lie groups and Lie algebras. Several examples are included. Then the exponential map, the Lie algebra of a closed linear groups, actions on manifolds and homogeneous spaces. The rest depends on how much time we have. The topics might include finite dimensional representations and in particular finite dimensional representation of semisimple Lie group, compact Lie groups and homogeneous vector bundles. Further topics might depend on the interests of the participants.
7380
Seminar in Functional Analysis: Harmonic Analysis in Phase Space. Prof. He
 MWF 1:302:20
 Instructor: Prof. He
 Prerequisite: Real Analysis (MATH 7311)
 Text: Harmonic Analysis in Phase Space by Gerald Folland, Princeton University Press.
 Description: We will start with basic idea of quantization, CCR (canonical commutation relation) and discuss the Heisenberg uncertainty principle, Schrodinger model, FourierWigner transform, Fock space, SegalBargmann transform, Weyl calculus, principal symbols,. This will be done with the introduction of Heisenberg group and other linear groups. Our main focus will be the first two chapters. If we have time, we will either get into the wave packets and Wave front set (Ch 3) involving more analysis, or the Weil representation (Ch 4) involving more group representation, or the trace formula involving more operator theory. This course is suitable for students having interests in harmonic analysis, analytic number theory, mathematical physics or the general audience.
7384
Spectral theory and applications. Prof. Shipman (Run as "Topics in Material Science")
 MWF 2:303:20
 Instructor: Prof. Shipman
 Prerequisite: Real analysis and undergraduate complex variables
 Text: Notes of the professor, plus various sources such as the classics "Theory of Linear Operators in Hilbert Space" by Akhiezer and Glazman and the 4volume series of Reed and Simon.

Description: This course concentrates on the spectral theory of Schrödinger operators with a view toward modern research in the field. The huge literature in this field was spawned by nonrelativistic quantum mechanics and has led to rich advances in pure spectral theory and applications. The course first develops abstract spectral theory of selfadjoint operators in Hilbert space with some emphasis on classical ideas of harmonic analysis, namely spectral resolutions induced by symmetry groups. Then we introduce continuous and discrete Schrödinger operators with electric and magnetic potentials and some of the standard theorems. We treat periodic, quasiperiodic, and ergodic operators, in decreasing detail. The treatment of periodic operators will emphasize the connections to commutative algebra centering around the Fermi and Bloch algebraic or analytic varieties. The course will conclude with analysis in physical, momentum (dual), configuration, and reciprocal space.
1. Spectral theory of selfadjoint operators in Hilbert space à la mode de Akhiezer and Glazman
2. Schrödinger operators, continuous and discrete
3. Periodic operators: The Fermi and Bloch varieties and Fourier (FloquetBloch) analysis
4. Quasiperiodic and ergodic operators
5. Analysis in physical, momentum (dual), configuration, and reciprocal space
7390
Seminar in Analysis: Topics in Elliptic and Parabolic Partial Differential Equations. Prof. Zhu
 MWF 9:3010:20
 Instructor: Prof. Zhu
 Prerequisite: Math 7311 or equivalent

Text: Elliptic Partial Differential Equations, Second Edition, by Qing Han and Fanghua Lin.
Reference Texts:
1, Elliptic partial differential equations of second order by David Gilbarg and Neil S. Trudinger. Springer Verlag 2001.
2. Partial Differential Equations: Second Edition by Lawrence C. Evans  Description: Partial differential equations play a central role in modern mathematics, especially in analysis, applied mathematics and geometry. This course presents basic methods to obtain a priori estimates for solutions of second order elliptic and parabolic partial differential equations. Topics covered include weak and viscosity solutions, Hopf and Alexandroff maximum principles, Harnack inequalities, De GiorgiNashMoser regularity theory, regularity (i.e. continuity and differentiability) of solutions, basic theory in homogenization. The course can be viewed as a continuation of MATH 7386 and MATH 7382 but no prior knowledge of PDEs is necessary. It provides some necessary background knowledge for the further study in applied analysis.
7390
Seminar in Analysis: Topic course on convex optimization theory and algorithms. Prof. Zhang
 TTh 1:302:50
 Instructor: Prof. Zhang
 Prerequisite: MATH 2057 Multidimensional Calculus or equivalent and MATH 2085 Linear Algebra or equivalent
 Text: Class notes
 Description: Convex optimization theory and algorithms have played important role in modern optimization. This course will focus on the theory and algorithm development for solving convex optimization problems. Tentative topics include Convex sets, Convex functions, Duality theory, Smooth Convex Optimization, Nonsmooth Convex Optimization, Firstorder and Secondorder methods and their complexities.
7490
Seminar in Combinatorics, Graph Theory, and Discrete Structures: Matroid Theory. Prof. Oxley
 MWF 8:309:20
 Instructor: Prof. Oxley
 Prerequisite: Permission of the Instructor
 Text: J. Oxley, Matroid Theory, Second edition, Oxford, 2011

Description:
What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
7512
Topology II. Prof. Schreve
 TTh 9:0010:20
 Instructor: Prof. Schreve
 Prerequisite: Math 7510
 Text: Algebraic Topology by Hatcher
 Description: This course covers the basics of homology and cohomology theory. Topics discussed include singular and cellular (co)homology, Brouwer fixed point theorem, cup and cap products, universal coefficient theorems, Poincare duality, Alexander duality, Kunneth theorems, and the Lefschetz fixed point theorem.
7550
Differential Geometry. Prof. Zeitlin
 TTh 12:001:20
 Instructor: Prof. Zeitlin
 Prerequisite: Math 7210 and 7510; or equivalent.
 Text: An Introduction to Manifolds by L. Tu.
 Description: Manifolds, vector fields, vector bundles, transversality, deRham cohomology, metrics, other topics.
7590
Seminar in Geometry and Algebraic Topology: Topological groups. Prof. Dani
 MWF 9:3010:20
 Instructor: Prof. Dani
 Prerequisite: Algebra I (7210), Real Analysis I (7311), Topology I (7510) and some familiarity with manifolds.
 Text: Instructor's notes. Optional reference: Hilbert's Fifth Problem and Related Topics by Terence Tao

Description:
A topological group is a group endowed with a topology such that the multiplication and inversion maps are continuous. Topological groups arise naturally in mathematics, for example as groups of automorphisms of graphs or isometries of metric spaces. The course will begin by exploring the interaction of topological properties (eg. connectedness, compactness, metrizability, separation properties) with the group theoretic structure in such groups. Highlights include topological versions of the group isomorphism theorems and the open mapping theorem.
The nicest type of topological group is a Lie group, in which the underlying topological space is a smooth manifold, and the group operations are smooth maps. Hilbert's Fifth Problem (from his highly influential problem list published in 1900) asks whether Lie groups can be characterized purely topologically. This can be interpreted as asking whether there are topological criteria that force a topological group to be isomorphic to (or "almost" isomorphic to) a Lie group. The motivating theme of the course will be the theorems of MontgomeryZippin and GleasonYamabe, which provide solutions to this problem. We will develop the necessary ingredients to prove these theorems, including limits of topological groups, Haar measure, and the structure of locally compact groups. If time permits we will explore further topics such as the HilbertSmith Conjecture and Gromov's Polynomial Growth Theorem.
7590
Seminar in Geometry and Algebraic Topology: Contact Geometry. Prof. VelaVick
 TTh 12:001:20
 Instructor: Prof. VelaVick
 Prerequisite: Math 7550
 Text: Instructor’s notes
 Description: A contact structure on an odddimensional manifold is a hyperplane field which is maximally nonintegrable (i.e., as far from defining a foliation as possible). Contact structures are odddimensional analogues of symplectic structures and, indeed, arise naturally as the boundaries of symplectic manifolds. Over the past decade or so years, contact manifolds and their natural subspaces have played prominent roles in the resolution of many interesting and important problems. In this course, we will develop the foundational toolset used to study contact structures.
7710
Advanced Numerical Linear Algebra. Prof. Walker
 TTh 10:3011:50
 Instructor: Prof. Walker
 Prerequisite: linear algebra, advanced calculus, some programming experience (but not required)
 Text: Fundamentals of Matrix Computations, 3rd edition, by D. S. Watkins.

Description: This course will develop and analyze linear algebra algorithms. Fundamentals will be discussed, i.e.:
Basic tools: norms, projectors, spectral theorem, singular value decomposition. Direct methods: LU factorization, Cholesky, least squares problem, QR factorization. Iterative methods: Jacobi, Richardson, GaussSeidel, successive overrelaxation, steepest descent, conjugate gradient. Eigenvalue problems: power methods, Rayleigh quotient iteration, deflation, QR algorithm.
As time permits, we may do some machine learning problems/examples, e.g. curve fitting, Bayesian viewpoint, classification.
7999n Assorted Individual Reading Classes
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8000n Assorted Sections of MSThesis Research
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9000n Assorted Sections of Doctoral Dissertation Research
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