LSU College of Science
LSU
Mathematics

Graduate Course Outlines, Summer 2018-Spring 2019

Contact


All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu

Summer 2018

  • MATH 7999-1: Problem Lab in Algebra —practice for PhD Qualifying Exam in Algebra.
  • Instructor:
  • Prerequisite: Math 7210.
  • Text: Online Test Bank.
  • MATH 7999-2: Problem Lab in Real Analysis—practice for PhD Qualifying Exam in Analysis.
  • Instructor:
  • Prerequisite: Math 7311.
  • Text: Online Test Bank.
  • MATH 7999-3: Problem Lab in Topology—practice for PhD Qualifying Exam in Topology.
  • Instructor:
  • Prerequisite: Math 7510.
  • Text: Online Test Bank.

Fall 2018

  • MATH 4997-1: Vertically Integrated Research: The representation theory of the Lie algebra 𝔰𝔩2(ℂ)
  • 1:30-2:50 TTh
  • Instructor: Profs. Achar and Sage
  • Prerequisites: Math 4200 and 4153
  • Text: Lectures on 𝔰𝔩2(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 finite-dimensional irreducible representations of the simplest example, 𝔰𝔩2(ℂ) (the 2x2 complex matrices with trace 0) arise naturally in studying the quantum-mechanical states of the hydrogen atom. The goal of this course is to provide an introduction to representation theory by focusing exclusively on 𝔰𝔩2(ℂ). In particular, we will give a complete description of the irreducible representations of 𝔰𝔩2(ℂ) (including the infinite-dimensional ones). No prior familiarity with Lie algebras or representation theory is required.
  • MATH 4997-2: Vertically Integrated Research: Invariants in Low-Dimensional Topology
  • 9:00-10:20 TTh
  • Instructor: Profs. Vela-Vick 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 low-dimensional spaces. For example, they can help us distinguish different knots and links, and explore the geometry of 3-dimensional 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 7001: Communicating Mathematics I
  • 3:00-4:50 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 one-semester-hour's credit.
  • This course provides practical training in the teaching of mathematics at the pre-calculus 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 pre-calculus 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
  • 12:00-1:20 TTh
  • Instructor: Prof. Tu.
  • Prerequisites: Math 4200 Algebra I, or equivalent
  • Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003. References:
    1. A first course in Abstract Algebra; Fraleigh
    2. 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.
  • 1:30-2:20 MWF
  • Instructor: Prof. Mahlburg.
  • Prerequisites:
  • Text:
  • MATH 7240: Algebraic Geometry.
  • 10:30-11:20 MWF
  • Instructor: Prof. Casper.
  • Prerequisites:
  • Text:
  • MATH 7311: Real Analysis I.
  • 10:30-1150 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 n-dimensional 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 Radon-Nikodym 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 Riesz-Markov-Saks-Kakutani 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:

    1. L. Richardson: Measure and Integral: An Introduction to Real Analysis.
    2. P. R. Halmos: Measure Theory (Graduate Text in Mathematics)
    3. A. Friedman: Foundations of Modern Analysis. Dover
  • MATH 7350: Complex Analysis.
  • 12:00-1:20 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 Schwarz-Christoffel integral. Riemann surfaces of algebraic functions (this topic is not covered by the book, and notes will be available).
  • MATH 7360: Probability Theory.
  • 9:00-10:30 TTh
  • Instructor: Prof. Sundar.
  • Prerequisites: Math 7311 or equivalent
  • Text: Probability and Measure by Patrick Billingsley
  • This is a self-contained 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 well-known limit theorems such as the Kolmogorov strong law of large numbers, three-series 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 7365: Applied Stochastic Analysis.
  • 1:30-2:50 TTh
  • Instructor: Prof. Ganguly.
  • Prerequisites: A good knowledge of a graduate level course in Probability theory (Math 7360) will be very useful. But familiarity with Measure theory (Math 7311) along with strong knowledge of a course in stochastic processes at an undergraduate level is sufficient.
  • Text:
  • The course is an introduction to stochastic analysis and its applications. It will cover some core topics in stochastic analysis including Brownian motions, martingales, Markov processes, stochastic integrals, Ito's lemma, stochastic differential equations, Girsanov's theorem. Then connections between stochastic analysis and PDE theory will be covered. In particular, probabilistic tools for solving PDEs will be discussed. Depending on time and the interests of students, the course will also try to explore other interesting topics like ergodicity of Markov processes, large deviation theory, stochastic filtering theory, simulation algorithms, etc.
  • MATH 7380: Topics in Spectral Geometry and Partial Differential Equations.
  • 1:30-2:50 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 non-variational techniques for PDEs, including Euler-Lagrange equation, fixed point theorem, and method of subsolutions and supersolutions. The second part will cover some background knowledge of Riemannian geometry and analysis of Laplace-Beltrami 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:
  • 10:30-1150 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 live-streaming setting. Students from any SEC university may enroll. It is intended to be a two-semester 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 boundary-integral 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.
  • 2:30-3:20 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 7390-1: Variational Analysis and Dynamic Optimization.
  • 12:30-1:20 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 differentiable-like 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 finite-dimensional 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 7390-2: Variational Inequalities.
  • 9:00-10:20 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 7510: Topology I
  • 9:30-10:20 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.
  • 12:30-1:20 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.
  • 1:30-2:50
  • 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 3-manifolds, 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 4997-1: Vertically Integrated Research:
  • Instructor: Profs. Achar and Sage
  • Prerequisites:
  • Text:
  • MATH 4997-2: Vertically Integrated Research:
  • Instructor: Profs. Vela-Vick and Wong
  • Prerequisites:
  • Text:
  • MATH 7002: Communicating Mathematics II
  • 3:00-4:50 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 one-semester-hour'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
    1. A First Course in Modular Forms (Graduate Texts in Mathematics, Vol. 228), Diamond & Shurman
    2. A Classical Introduction to Modern Number Theory, Ireland &Rosen, (Graduate Texts in Mathematics) (v. 84)
    3. An Introduction to the Langlands Program, Bernstein& Gelbart (eds)
  • The focus is on L-functions. These are at the center of much research today. Themes: Euler products; analytic continuation and functional equations; special values; relations among different kinds of L-functions. Briefly there are two broad classes of L-functions: those that arise from automorphic forms and those that arise from geometry. There are conjectural relations between the L-functions of the first kind with those of the second kind. We start with the classical Dirichlet L-functions and move on to the L-functions defined by modular forms. We give an exposition of Tate's thesis which gives a unified account of the L-functions that arise from abelian characters of Galois groups. We also give an introduction to L-functions 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 7384-1: Topics in Material Science:
  • Instructor: Prof. Shipman.
  • Prerequisites:
  • Text:
  • MATH 7384-2: Topics in Material Science:
  • Instructor: Prof. Lipton.
  • Prerequisites:
  • Text:
  • MATH 7390: Topics Course (TBA)
  • Instructor: Prof. Zhang.
  • 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 7590-1: Geometric Topology: Symplectic Geometry.
  • Instructor: Prof. Vela-Vick.
  • Prerequisites:
  • Text:
  • MATH 7590-2: Bounded Cohomology
  • Instructor: Prof. Dani
  • Prerequisites:
  • Text:
  • MATH 7710: Advanced Numerical Linear Algebra.
  • Instructor: Prof. Walker.
  • Prerequisites:
  • Text: