LSU College of Science
LSU
Mathematics

Graduate Course Outlines, Summer 2019-Spring 2020

Contact


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

Summer 2019

  • 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 2019

  • MATH 4997-1: Vertically Integrated Research: Buildings
  • 12:00-1:20 TTh
  • Instructor:
  • Profs. Achar and Sage
  • Prerequisites: Math 4200 and 4153
  • Text: Buildings—Theory and Applications by P. Abramenko and K. Brown
  • A building is a certain type of simplicial complex introduced by Jacques Tits to provide a geometric framework for studying simple algebraic groups. They have since turned out to have important applications to group theory, representation theory, geometry, and discrete mathematics. The simplest examples of buildings are (not necessarily finite) graphs with very strong regularity properties. For example, if F is a field, then the spherical building associated to SL_3(F) (the 3x3 matrices with entries in F and determinant one) is a bipartite graph with a vertex for each line and for each plane in F^3; two vertices are connected precisely when they correspond to a line contained in a plane. More generally, the building associated to SL_n(F) for n>3 is an (n-1)-dimensional simplicial complex with simplices corresponding to “flags”—chains of subspaces in F^n. In this course, we will provide an introduction to the theory of buildings, with a particular focus on concrete examples. We will also discuss various applications.
  • MATH 4997-2: Vertically Integrated Research: Invariants in Low-Dimensional Topology
  • 10:30-11:50 TTh
  • Instructor:
  • Profs. Vela-Vick and Wong
  • Prerequisites: None
  • Text: None. Suggested references will be provided during the course.
  • In recent years, research has found many interesting interactions between low-dimensional topology and contact and symplectic geometry. This has resulted in the discovery of invariants that can help us distinguish different knots and links, and explore the geometry of 3- and 4-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
  • 12:30-1:20 MWF
  • Instructor: Prof. Ng.
  • Prerequisites: Math 4200 or equivalent
  • Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
  • This is the first semester of the first year graduate algebra sequence, and covers the material required for the Comprehensive Exam in Algebra. Topics will include group actions and Sylow Theorems, finitely generated abelian groups; rings and modules: PIDs, UFDs, finitely generated modules over a PID, applications to Jordan canonical form.
  • MATH 7230: Class Field Theory.
  • 10:30-11:50 TTh
  • Instructor: Prof. Mahlburg.
  • Prerequisites: Algebraic Number Theory is essential. A good understanding of Galois Theory at the level of Graduate Algebra II is recommended.
  • Text: Class Field Theory, by Nancy Childress, Springer Universitext, 2009. Freely available as an E-book through LSU Libraries.
  • Class Field Theory is the study of abelian extensions of number fields (which may be global or local). These extensions are described in terms of arithmetic invariants such as the ideal and ray class groups. One of the main results is Artin's Reciprocity Law, which generalizes quadratic reciprocity, and can be viewed analytically as a first case of Langlands Program.
  • MATH 7260: Homological Algebra.
  • 9:00-10:20 TTh
  • Instructor: Prof. Achar.
  • Prerequisites:
  • Text:
  • MATH 7290: Modular Tensor Categories and Quantum Invariants.
  • 10:30-11:20 MWF
  • Instructor: Profs. Ng and Wang
  • Prerequisites:
  • Text:
  • This course introduces modular tensor categories and the Reshetikhin-Turaev (RT) 3-dimensional topological quantum field theories (TQFTs). We will start with the theory of modular tensor categories and their graphical calculus. We will then give an axiomatic definition of TQFTs and discuss some examples arise from finite groups and quantum groups.
  • MATH 7311: Real Analysis I.
  • 10:30-11:20 MWF
  • Instructor: Prof. Sundar.
  • Prerequisites: Math 4032 or 4035 or equivalent.
  • Text: Real Analysis, Modern Techniques and Their Applications, by G. B. Folland.
  • This is an standard course on measure theory and integration. After introducing measures, we discuss measurable functions, and develop integration of real and complex valued functions. An important example would be the Lebesgue integral on the line and n-dimensional Euclidean space. Next, we will present the connection between the Lebesgue integral and the Riemann integral, and study functions of bounded variation, absolute continuity, and Lebesgue differentiation theorems. Important topics such as Lebesgue convergence theorems, product measures, Fubini's theorem and the Radon-Nikodym derivative will be covered. We will give a short introduction of Banach and Hilbert spaces and then apply it to Lp spaces. The Riesz-Markov-Kakutani theorem will be discussed if there is sufficient time. A sampler of very good books on analysis and measure theory are:
    A. Friedman: Foundations of Modern Analysis
    L. Richardson: Measure and Integral: An Introduction to Real Analysis
    H. L. Royden: Real Analysis
    R. L. Wheeden and A. Zygmund: Measure and Integral: An Introduction to Real Analysis
  • MATH 7325: Finite Element Methods.
  • 12:00-1:20 TTh
  • Instructor: Prof. Walker.
  • Prerequisites: Analysis (MATH 4031, 4032), Linear Algebra (MATH 7710)
  • Texts: (required): The Mathematical Theory of Finite Element Methods, 3rd Edition (2008) by S. Brenner, R. Scott; Instructor will also use course notes.
    Reference Texts (not required): Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition (2007) by D. Braess; The Finite Element Method for Elliptic Problems, (Classics in Applied Mathematics, 2002) by Philippe G. Ciarlet.
  • The finite element method (FEM) is one of the most successful computational tools in dealing with partial differential equations (PDE) arising in science and engineering (solid and fluid mechanics, electromagnetism, mathematical physics, etc.). The formulation of the FEM, its properties, stability, and convergence will be discussed. Implementation of the FEM will also be addressed mainly via Python programming projects.

    Additional topics may be introduced depending on the interests of the class.

    Prerequisites: Some basic functional analysis and PDE theory (MATH 4340, or MATH 7386 or equivalent) will be reviewed. Prior exposure to numerical analysis (MATH 4065 or 4066) and Python would be useful but not mandatory.

  • MATH 7350: Complex Analysis.
  • 1:30-2:50 TTh
  • Instructor: Prof. Sage.
  • Prerequisites: Math 7311 or its equivalent.
  • Text: Narasimhan, R. and Nievergelt, Y., Complex Analysis in One Variable, second edition, Birkhauser, Boston, 2001.
  • This course is an introduction to complex analysis at the graduate level. Topics include holomorphic functions, covering spaces and the monodromy theorem, winding numbers, residues, Runge's theorem, the Mittag-Leffler and Weierstrass theorems on construction of functions with specified zeros or poles, the Riemann mapping theorem, and an introduction to Riemann surfaces.
  • MATH 7360: Probability Theory.
  • 1030-11:50 TTh
  • Instructor: Prof. Ganguly.
  • Prerequisites: Familiarity with Measure theory (Math 7311) is desirable but not required. However a good mathematical maturity and a solid understanding of undergraduate analysis at the level of Math 4027-4032 is important.
  • Text:
  • The course is a self-contained 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. Well-known 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 continuous-time 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.
  • MATH 7380: Spherical Harmonics and Integral Geometry.
  • 8:30-9:20 MWF
  • Instructor: Prof. Rubin.
  • Prerequisites: Differential calculus in n-dimensional space (MATH 4035), Lebesgue integration (MATH 7311).
  • Text: Prof. Rubin will provide his notes following the book:
    B. Rubin: Introduction to Radon transforms: With elements of fractional calculus and harmonic analysis (Encyclopedia of Mathematics and its Applications), Cambridge University Press (2015).
  • Syllabus: The focus of this course/seminar is introduction to the Fourier analysis on the unit sphere in the n-dimensional real Euclidean space and related problems of integral geometry. This topic plays an important role in PDE, harmonic analysis, group representations, mathematical physics, geometry, and many other areas of mathematics and applications.
  • MATH 7384: Topics in Material Science: Optical Metamaterials and Plasmonic Composites.
  • 2:30-3:20 MWF
  • Instructor: Prof. Lipton.
  • Prerequisites: Any one of Math 2065, 3355, 3903, 4031, or 4038, or their equivalent.
  • Text:
  • In this course we provide an introduction to theory behind the design of meta-materials for the control of light. Here we develop basic ideas and intuition as well as introduce the mathematics and physics of spectral theory necessary for the rational design of meta-materials. The course provides a self contained introduction as well as a guide to the current research literature useful for understanding the mathematics and physics of wave propagation inside complex heterogeneous media. The course begins with an introduction to Bloch waves in crystals and provides an introduction to local plasmon resonance phenomena inside crystals made from nano-metallic particles. We then show how to apply these techniques to construct media with exotic properties. We provide the mathematical underpinnings for characterizing the interaction between surface plasmon spectra and Mie resonances and its effect on wave propagation. This understanding is used for design of structured media supporting backward optical and infrared waves and behavior associated with an effective negative index of refraction. Potential applications include drug delivery, optical communication, and holography.
  • MATH 7386: Partial Differential Equations.
  • 12:00-1:20 MW
  • Instructor: Prof. Bulut.
  • Prerequisites: Math 7311 (Real Analysis I) or equivalent, or consent of department.
  • Text: Primary Text: Partial Differential Equations by Lawrence C. Evans
    Additional optional references:
    Elliptic partial differential equations by Qing Han and Fanghua Lin
    Semilinear Schrödinger Equations by Thierry Cazenave.
  • This course provides an introduction to the theory of partial differential equations. Topics to be covered include:
    1. Introduction to elliptic, parabolic, and hyperbolic partial differential equations
    2. Examples: Laplace's equation, the heat equation, and the wave equation
    3. Introduction to Sobolev spaces, weak derivatives, existence and uniqueness of solutions.
    4. Elliptic equations: existence, regularity and the maximum principle.
    5. Selected additional topics (chosen according to time and class interest).
    Additional topics may include:
    a) Nonlinear dispersive PDE with an emphasis on the nonlinear Schrodinger equation.
    b) Introduction to Calculus of Variations: Euler-Lagrange equations, existence of minimizers, eigenvalues of self-adjoint elliptic operators.
  • MATH 7390-1: Topics in Numerical Analysis.
  • 130-2:20 MWF
  • Instructor: Prof. Wan.
  • Prerequisites: Numerical Linear Algebra: Math 4064, Probability Math 3355, or equivalent.
  • Text: Lecture notes from the instructor.
  • This is an introductory course on some topics in uncertainty quantification and machine learning that are closely related to scientific computing. We will introduce some techniques in deep learning that are widely used and still in active development such as deep nueral networks, Bayesian regression, generative modeling, etc. We will pay attention to the relation of these techniques to classical theories or problems such as approximation theory, inverse problems, model reduction, etc. We will also introduce the implementation of these techniques using Python and some open libraries such as Tensorflow, scikit-learn, etc.
  • MATH 7390-2: Introduction to Control Theory.
  • 9:00-10:20 TTh
  • Instructor: Prof. Malisoff.
  • Prerequisites: MATHs 4027 and 4035, or their equivalents or permission from instructor.
  • Text:
  • This course provides an introduction to basic ideas and results from control theory, which is an interdisciplinary field that provides the foundation for much current research in multiple branches of engineering.
    Course topics will be chosen from among the following: review of differential equations, linear control systems, open and closed loop control, Lyapunov functions, feedback control of nonlinear systems, robustness to uncertainty, adaptive control, and control systems with delays.
  • 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.
  • 11:30-12:20 MWF
  • Instructor: Prof. Cohen.
  • Prerequisites: Math 7512
  • Text: Algebraic Topology by Allen Hatcher (this can be obtained free online)
  • This is a continuation of Math 7512. We will discuss cup products in cohomology, Poincare and Lefshetz duality, and beginning homotopy theory.
  • MATH 7590: Trisections on smooth 4-manifolds.
  • 1:30-2:50 TTh
  • Instructor: Prof. Baldridge.
  • Prerequisites: Differential Geometry (7550) and Topology II (7512)
  • Text: 4-Manifolds and Kirby Calculus by R. Gompf and A. Stipsicz, and papers by Peter Lambert-Cole on trisections
  • In this course, we study trisections on smooth closed 4-manifolds with the goal of understanding the minimal genus of all embedded smooth surfaces representing a given homology class (The Thom Theorem). A trisection of a 4-manifold is analogous to a Heegaard decompositions of a 3-manifold. It is a cutting-edge tool that has already shown great promise in understanding and proving some of the hardest problems in 4-manifold theory. This course will benefit students who want an overview of 4-manifold constructions and want to learn how to use advanced techniques in topology and geometry to study smooth manifolds.

Spring 2020

  • MATH 4997-1: Vertically Integrated Research:
  • Instructor:
  • Prerequisites:
  • Text:
  • MATH 4997-2: Vertically Integrated Research:
  • Instructor:
  • 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. Hoffman.
  • Prerequisites:
  • Text:
  • MATH 7230: Introduction to Elliptic Curves and Modular Forms
  • Instructor: Prof. Tu.
  • Prerequisites:
  • Text:
  • MATH 7280: D-modules
  • Instructor: Prof. Sage.
  • Prerequisites:
  • Text:
  • MATH 7290: Group Schemes
  • Instructor: Prof. Casper.
  • Prerequisites:
  • Text:
  • MATH 7320: Ordinary Differential Equations.
  • Instructor: Prof. Malisoff
  • Prerequisites:
  • Text:
  • MATH 7330: Functional Analysis.
  • Instructor: Prof. He.
  • Prerequisites:
  • Text:
  • MATH 7366: Stochastic Analysis.
  • Instructor: Prof. Kuo.
  • Prerequisites:
  • Text:
  • MATH 7370: Lie Groups and Representations
  • Instructor: Prof. Olafsson.
  • Prerequisites:
  • Text:
  • MATH 7375: Wavelets:
  • Instructor: Prof. Nguyen.
  • Prerequisites:
  • Text:
  • MATH 7384: Topics in Material Science:
  • Instructor: Prof. Li.
  • Prerequisites:
  • Text:
  • MATH 7490: Combinatorial Optimization
  • Instructor: Prof. Ding.
  • Prerequisites:
  • Text:
  • MATH 7550: Differential Geometry.
  • Instructor: Prof. Zeitlin.
  • Prerequisites:
  • Text:
  • MATH 7590-1: Geometric Topology: Hyperbolic Geometry.
  • Instructor: Prof. Zimmer.
  • Prerequisites:
  • Text:
  • MATH 7590-2: Modular categories and Reshetikhin-Turaev TQFTs
  • Instructor: Prof. Wang
  • Prerequisites:
  • Text:
  • MATH 7710: Advanced Numerical Linear Algebra.
  • Instructor: Prof. Zhang.
  • Prerequisites:
  • Text: