Graduate Courses, Summer 2025 – Spring 2026
Summer 2025
For Detailed Course Outlines, click on course numbers.
7999-1
Problem Sessions in Algebra—practice for PhD Qualifying Exam in Algebra
- Time:
- Instructor:
- Prerequisite:
- Text:
7999-2
Problem Sessions in Analysis—practice for PhD Qualifying Exam in Analysis
- Time:
- Instructor:
- Prerequisite:
- Text:
7999-3
Problem Sessions in Topology—practice for PhD Qualifying Exam in Topology
- Time:
- Instructor:
- Prerequisite:
- Text:
7999-4
Problem Sessions in Applied Math—practice for PhD Qualifying Exam in Applied Math
- Time:
- Instructor:
- Prerequisite:
- Text:
7999-n Assorted Individual Reading Classes
- No additional information.
8000-n Assorted Sections of MS-Thesis Research
- No additional information.
9000-n Assorted Sections of Doctoral Dissertation Research
- No additional information.
Fall 2025
For Detailed Course Outlines, click on course numbers. Core courses are listed in bold.
4997-1
Vertically Integrated Research: Combinatorial Topology. Profs. Bibby and Schreve
- Time: 11:30-12:20 MWF
- Instructor: Profs. Bibby and Schreve
- Prerequisite: A first course in linear algebra; experience with topology or graph theory may be useful but not required
- Text: None
- Description: This is a project-based seminar class in geometry and topology. Students work together in small groups to tackle problems in combinatorial topology and geometric combinatorics.
4997-2
Vertically Integrated Research: Stochastic processes and partial differential equations. Prof. Fehrman
- Time: 1:30-2:20 MWF
- Instructor: Prof. Fehrman
- Prerequisite: Calculus (necessary), real analysis (preferred, but not necessary), probability (preferred, but not necessary)
- Text: The course will be based on notes provided by the professor. A good secondary reference is "Probability: Theory and Examples" by Rick Durrett (a free pdf copy is available on the author's website)
- Description: The purpose of this course is to develop the heuristic connections between random processes and partial differential equations on both theoretic and numerical levels. The course will begin with an overview of measure theoretic probability theory, including probability spaces, sigma algebras, filtrations, (Gaussian) random variables, and independence. The well-posedness of ordinary differential equations with Lipschitz continuous coefficients will then be established in order to motivate the study of discrete stochastic differential equations, which are, roughly speaking, ordinary differential equations perturbed by a random noise. Such equations model everything from the diffusion of aerosols in the wind, the advection of passive quantities like dye or energy in a fluid, and the evolution of a stock price. By developing a version of Ito's formula, which is the fundamental theorem of calculus for stochastic processes, we will show that the density of the solution to a stochastic differential equation can be described using the solution to a certain partial differential equation. This connection is known as the Feynman--Kac formula, and it will be the fundamental result of the course. We will then spend a significant amount of time applying this connection to model stochastic processes with connections to biology, physics, finance, and machine learning. In particular, we will derive the Black--Scholes equation, which is the most fundamental partial differential equations used to estimate European-style options in finance; discuss diffusion models in machine learning, which are a state-of-the-art technique for generating synthetic data, including images, in machine learning; and analyze stochastic gradient descent algorithms.
7001
Communicating Mathematics I. Prof. Shipman and Dr. Ledet
- Time: 3:00-4:50 TT
- Instructor: Prof. Shipman and Dr. Ledet
- Prerequisite: Consent of department. This course is required for all first-year 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.
7210
Algebra I. Prof. X. Wang
- Time: 8:30-9:20 MWF
- Instructor: Prof. X. Wang
- 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: A Tour of Analytic Number Theory. Prof. Kopp
- Time: 12:00-1:20 TT
- Instructor: Prof. Kopp
- Prerequisite: Math 7210, Math 7311, Math 7350 (recommended)
- Text: To be determined
- Description: This course will be split into three parts, each visiting a different realm within the vast landscape of analytic number theory. Part 1 covers multiplicative number theory, including Dirichlet series and the proof of the prime number theorem. Part 2 covers additive number theory, including the Hardy–Littlewood circle method as applied to partitions. Part 3 covers topics in computational number theory or other topics determined by student interest. Coursework will include problem sets due regularly and student presentations to be given at the end of the course.
7250
Representation Theory. Prof. Ng
- Time: 9:00-10:20 TT
- Instructor: Prof. Ng
- Prerequisite:
- Text:
- Description: Representations of finite groups, group algebras, character theory, induced representations, Frobenius reciprocity, Lie algebras and their structure theory, classification of semisimple Lie algebras, universal enveloping algebras and the PBW theorem, highest weight representations, Verma modules, and finite-dimensional representations.
7260
Homological Algebra. Prof. Hoffman
- Time: 8:30-9:20 MWF
- Instructor: Prof. Hoffman
- Prerequisite: Familiarity with basic Algebra (groups, rings, modules) and elementary Topology. A prior course in Algebraic Topology is helpful but not required.
- Text: Charles A. Weibel, An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics, Series Number 38). This is available electronically in the LSU Library.
-
Description:
Topics covered include
- Derived Functors, Ext, Tor. Chapters 1 -3 of Weibel
- Spectral Sequences, Chapter 5 of Weibel
- Cohomology of Groups, Chapter 6 of Weibel
- Simplicial Methods, Chapter 8 of Weibel
- The Derived Category, Chapter 10 of Weibel
7311
Real Analysis (a.k.a. Analysis I). Prof. Lipton
- Time: 9:30-10:20 MWF
- Instructor: Prof. Lipton
- Prerequisite: Undergraduate real analysis
- Text:
- 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 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 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. Ólafsson
- Time: 12:30-1:20 MWF
- Instructor: Prof. Ólafsson
- Prerequisite: Math 7311
- Text: Lecture notes and Complex Analysis by Elias Stein and Rami Shakarchi, Princeton Lectures in Analysis II.
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Description: Theory of holomorphic functions of one complex variable; path integrals, power series, singularities, mapping properties, normal families, other topics.
Detailed Description: This is a standard course in complex analysis in one variable. We start out by discussing basic definitions concerning the complex numbers and continuous functions. Then we will cover the following material: Holomorphic functions; power series; path integrals, Cauchy’s integral singularities, sequences of holomorphic functions; singularities and meromorphic functions; Schwarz reflection principle. The Fourier transform in the complex plane. Paley-Wiener theorem. Entire functions including Jensen's formula and infinite products. Conformal mapping including the Riemann mapping theorem. Some special functions including the Gamma function, the Zeta functions and an introduction to the theory of elliptic functions. Other topics might be discussed depending on the time and requests from the students. We will use our own lecture notes which follows closely the book by E. Stein and R. Shakarchi, but some of the material will be taken from other books.
7360
Probability Theory: Prof. Ganguly
- Time: 1:30-2:50 TT
- Instructor: Prof. Ganguly
- Prerequisite: Math 7311
- Text: Probability and Stochastics by Erhan Cinlar
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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.
This is a self-contained introduction to modern probability theory. Starting from the concept of probability measures, it would introduce random variables, and independence. After a study of various modes of convergence, the Kolmogorov strong law of large numbers and results on random series will be established. Weak convergence of probability measures will be discussed in detail, which would lead 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. Stochastic processes such as Brownian motion and martingales will be introduced, and their essential features, studied.
7365
Applied Stochastic Analysis. Prof. Fehrman
- Time: 11:30-12:20 MWF
- Instructor: Prof. Fehrman
- Prerequisite: Calculus (necessary), real analysis (necessary), probability (preferred, but not necessary). The course can be taken concurrently with Math 7360.
- Text: Probability: Theory and Examples by Rick Durrett (a pdf copy is available on the author's website)
- Description: The evolution of many physical systems is fundamentally random. Even relatively simple processes, like random walks generated by a sequence of independent coin flips, can exhibit remarkably interesting behaviors, and such systems form the basis of our understanding of diverse phenomena in physics and biology and several of the computational techniques used in finance and machine learning. This course will focus, in particular, on stochastic processes in discrete time. We will develop the theory of discrete time martingales and Markov processes. Topics will include Doob's decomposition theorem, Doob's Martingale inequalities, the Burkholder-Davis-Gundy inequality, Kolmogorov's equations, generators, stationary measures, and some results in ergodic theory. We will also discuss some stochastic algorithms like Markov Chain Monte Carlo and diffusion models in machine learning. The course is provides the necessary background for Math 7366, which builds on Math 7365 to develop the theory of continuous-time martingales and Markov processes and and the theory of stochastic differential equations.
7380
Seminar in Functional Analysis: Operator Theory. Prof. Shipman
- Time: 2:30-3:20 MWF
- Instructor: Prof. Shipman
- Prerequisite: Real and complex analysis
- Text: Operator Theory by Barry Simon, AMS
- Description: This material is foundational broadly in analysis and its applications. Operator theory is ubiquitous in all flavors of mathematical physics, including PDE, quantum mechanics, and quantum field theory. The topics include compact operators, index theory, and spectral theory, and perhaps orthogonal polynomials and/or Banach algebras.
7390-1
Seminar in Functional Analysis: Iterative Methods for Linear Systems. Prof. Brenner
- Time: 10:30-11:50 TT
- Instructor: Prof. Brenner
- Prerequisite: Math 7710
- Text: Iterative Methods for Solving Linear Systems by Anne Greenbaum (This book can be downloaded for free through the LSU library.)
- Description: Basic Iterative Methods, Krylov Subspace Methods, Preconditioning, Domain Decomposition, Multigrid
7382
Introduction to Applied Mathematics. Prof. Tarfulea
- Time: 10:30-11:50 TT
- Instructor: Prof. Tarfulea
- Prerequisite: Simultaneous enrollment in Math 7311
- Text: Lecture notes by Massatt and Shipman
- 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.
7386
Theory of Partial Differential Equations. Prof. Zhu
- Time: 10:30-11:20 MWF
- Instructor: Prof. Zhu
- Prerequisite: Math 7330
- Text: Partial Differential Equations by L. C. Evans
- Description: Introduction to PDE. 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.
7390-2
Seminar in Analysis: Dynamics of Nonlinear Waves and Tools from Harmonic Analysis. Prof. Bulut
- Time: 12:00-1:20 TT
- Instructor: Prof. Bulut
- Prerequisite: Graduate student in Mathematics or consent of instructor.
- Text: Christopher D. Sogge, Lectures on Nonlinear Wave Equations, 2nd edition.
- Description: In this course, we will give a self-contained introduction to the mathematical analysis of the nonlinear wave equation, and give an overview of a variety of tools from harmonic analysis used in this area. The course will be accessible to graduate students in Mathematics and related areas. Students do not need to have taken a prior course in PDE.
7400
Combinatorial Theory. Prof. Bibby
- Time: 9:30-10:20 MWF
- Instructor: Prof. Bibby
- Prerequisite: Calculus (Math 1552), linear algebra (Math 2085), and abstract algebra (Math 4200)
- Text: Enumerative Combinatorics Volume 1 by Stanley (pdf available through LSU Library here)
- Description: Problems of existence and enumeration in the study of arrangements of elements into sets; combinations and permutations; other topics such as generating functions, recurrence relations, inclusion-exclusion, Polya’s theorem, graphs and digraphs, combinatorial designs, incidence matrices, partially ordered sets, matroids, finite geometries, Latin squares, difference sets, matching theory.
7510
Topology I. Prof. Bălibanu
- Time: 9:00-10:20 TT
- Instructor: Prof. B&alibanu
- 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
- Time: 1:30-2:50 TT
- Instructor: Prof. Cohen
- Prerequisite: MATH 7512
- Text: <\em>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: Sigma-invariants of finitely generated groups. Prof. Dani
- Time: 10:30-11:20 MWF
- Instructor: Prof. Dani
- Prerequisite: MATH 7210 and 7512
- Text: Notes on the Sigma invariants by Ralph Strebel (available on arXiv)
- Description: The Sigma-invariants of a finitely generated group (also called BNSR-invariants) are an important tool in geometric group theory, with connections to many other topics, such as the Thurston norm and fibering of manifolds over S^1, Novikov homology, and resonance varieties. Defined in the 1980s by Bieri-Neuman-Strebel (in dimension 1) and Bieri-Renz (in higher dimensions), they are certain subsets of the character sphere of a group G, which encode a vast amount of information about the finiteness properties of subgroups of G. Recently, they have been successfully used in the study of interesting group theoretic questions such as virtual algebraic fibering and commensurability. Although they are rather hard to compute in general, they are now understood, at least in low dimensions, for various well-known classes of groups, such as generalized Baumslag-Solitar groups as well as many Artin and braid groups. In this course, we will begin with the geometric approach to the 1-dimensional Sigma invariants, following Strebel's "Notes on the Sigma invariants". Following that, we will explore further applications, explicit computations in specific classes of groups, and connections to other topics, to be determined by the interests of the class.
Spring 2026
For Detailed Course Outlines, click on course numbers.
4997-1
Vertically Integrated Research: Calculus of Variations. Dr. Rios and Prof. Wolenski
- Time: TT 1:30-2:50
- Instructors: Dr. Rios and Prof. Wolenski
- Prerequisite:
- Text:
- Description: This course will cover the mathematical theory of the Calculus of Variations (CofV). This is an old subject of fundamental importance in Physics and we shall begin with a detailed treatment of several classical problems. Students will form teams and be assigned to study a particular problem to illustrate how the theory can be applied. In the latter half of the course, we shall see how optimal control arose as a natural extension with a myriad of modern applications.
4997-2
Vertically Integrated Research: Topics in Combinatorics. Profs. Bibby and Z. Wang
- Time: MWF 11:30-12:20
- Instructors: Profs. Bibby and Z. Wang
- Prerequisite: Linear Algebra (Math 2085 or 2090)
- Text: None
- Description: This is a project-based seminar class in combinatorics. Students work together in small groups to tackle problems in topics such as graph theory; matroid theory; order theory; enumerative and algebraic combinatorics; geometric and topological combinatorics. Previous experience in combinatorics is not required.
4997-3
Vertically Integrated Research: TBA. Prof. Drenska
- Time: TT 3:00-4:20
- Instructors: Prof. Drenska
- Prerequisite: 2085 or equivalent, as well as a 4000-level mathematics course with a grade of C or better, or obtain permission of the department
- Text: Gilbert Strang, Linear Algebra and Learning from Data
- Description: This is a project-based course that involves real-world applications of machine learning. The course provides opportunities for students to consolidate their mathematical knowledge, and to obtain a perspective on the meaning and significance of that knowledge. Course work will emphasize communication skills, including reading, writing, and speaking mathematics.
4997-4
Vertically Integrated Research: Arithmetic of subspace packings and quantum designs. Prof. Kopp
- Time: MWF 9:30-10:20
- Instructors: Prof. Kopp
- Prerequisite: Multivariable calculus (Math 2057, Math 2058, or equivalent) and linear algebra (Math 2085, Math 2090, or equivalent) are required. Some experience reading and writing mathematical proofs and some experience writing computer code is highly recommended. Some projects, but not all, will use elementary number theory (Math 4181) and abstract algebra including Galois theory (Math 4200 and Math 4201). As project leaders, PhD students in the course should have Math 7210 and should have some knowledge of algebraic number theory.
- Text: Waldron, An Introduction to Finite Tight Frames
- Description: In this research-focused course, we will use computational tools to search for structures of interest to algebraic number theory, quantum information theory, and coding theory. These structures (called subspace packings, quantum designs, and/or fusion frames) are arrangements of subspaces in a finite-dimensional Hilbert space satisfying highly restrictive geometric constraints or symmetries. The prototypical examples are SIC-POVMs (symmetric, informationally complete positive operator-valued measures), or sets of d^2 complex equiangular lines in d-dimensional Hilbert space, whose conjectural classification ties them to deep number theory (class field theory for real quadratic fields). SIC-POVMs make up a small sliver of a vast space of potentially interesting subspace packings. The discovery of the number-theoretic features of SIC-POVMs was an accident made possibly by massive computer searches, and such searches are needed in other directions. Background on finite frame theory and connections to number theory will be provided through lectures, and students will pursue computational research projects in groups. Half of the in-class time will be spent in a computer lab.
4997-5
Vertically Integrated Research: Quantum cohomology. Prof. Achar
- Time: TT 12:00-1:20
- Instructors: Prof. Achar
- Prerequisite: Math 4153 and Math 4200, or permission of the instructor
- Text: J. Kock and I. Vainsencher, An Invitation to Quantum Cohomology
- Description: Quantum cohomology is a topic with connections to classical problems in enumerative geometry as well as to modern ideas coming from mathematical physics, such as mirror symmetry and Gromov-Witten invariants. This course will be a beginners' introduction to these topics, with a focus on understanding the quantum cohomology of complex projective space.
7002
Communicating Mathematics II. Prof. Shipman and Dr. Ledet
- Time: TT 3:00-4:50
- Instructor: Prof. Shipman and Dr. Ledet
- Prerequisite: Consent of department. This course is required for all first-year 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.
7211
Algebra II. Prof. Long
- Time: MWF 1:30-2:20
- Instructor: Prof. Long
- Prerequisite: Math 7210 or equivalent
- Text: David Dummit and Richard Foote, Abstract Algebra, 3rd Edition, John Wiley and Sons, 2003
- 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. Dr. Saad
- Time: MWF 2:30-3:20
- Instructors: Dr. Saad
- Prerequisite:
- Text:
- Description:
7240
Topics in Algebraic Geometry: Sheaf Theory. Prof. Achar
- Time: TT 10:30-11:50
- Instructors: Prof. Achar
- Prerequisite: Algebra II and Topology II. Prior exposure to algebraic topology and to homological algebra will be helpful
- Text: none
- Description: This course will cover the basics of sheaf theory, including abelian and derived categories of sheaves, (derived) sheaf functors, and major theorems about them, such as the proper base change theorem and the projection formula. Topics for later in the semester may include Poincare-Verdier duality, Borel-Moore homology, and Artin's vanishing theorem. If time permits, I may introduce perverse sheaves and discuss applications in representation theory.
7290
Seminar in Algebra and Number Theory: Category Theory. Prof. Ng
- Time: MWF 12:30-1:20
- Instructor: Prof. Ng
- Prerequisite: Algebra II or Math7250
- Text: Categories for the Working Mathematician (Graduate Texts in Mathematics) 2nd edition
- Description: The course will cover basic notions in category theory which include Hom-functors, Yoneda’s lemma, limits, adjoint functors, and abelian categories. The second part of the course covers monoidal/tensor categories, coherence theorem, graphical calculus, braidings, pivotal structures, ribbon categories, and their applications to representations of Hopf algebras, quantum groups and mapping class groups. If time allowed Vafa’s Theorem and Verlinde’s formula could be covered.
7320
Ordinary Differential Equations. Prof. Neubrander
- Time: MWF 10:30-11:20
- Instructor: Prof. Neubrander
- Prerequisite: The basics of Real Analysis or even just Advanced Calculus.
- Text:
- Description: The course will cover the qualitative theory of Ordinary Differential Equations. This includes the usual existence and uniqueness theorems, linear systems, stability theory, hyperbolic systems (the Grobman-Hartman Theorem), phase portraits, and discrete systems. if time permits, an introduction to Control Theory will be included. The course will be accessible to those who do not aspire to be analysts, but expect at some point to teach an undergraduate course in ODEs.
7330
Functional Analysis (a.k.a. Analysis II). Prof. Vempati
- Time: TT 1:30-2:50
- Instructor: Prof. Vempati
- Prerequisite: Math 7311 or its equivalent
- Text: A course in Functional analysis by John Conway
- Description: Banach spaces and their generalizations; Baire category, Banach-Steinhaus, open mapping, closed graph, and Hahn-Banach theorems; duality in Banach spaces, weak topologies; other topics such as commutative Banach algebras, spectral theory, distributions, and Fourier transforms.
7366
Stochastic Analysis: Stochastic Differential Equations. Prof. Fehrman
- Time: 12:00-1:20
- Instructor: Prof. Fehrman
- Prerequisite: Math 7311, and Math 7360 or its equivalent
- Text: Primary text: Notes provided by the professor. Secondary text: Continuous Martingales and Brownian Motion by Daniel Revuz and Mark Yor
-
Description: Stochastic differential equations (SDEs) model systems that are sensitive to the microscopic fluctuations of their surroundings. A physical example is the diffusion of a perfume, where the spreading out of the individual perfume molecules is caused by their microscopic and essentially random collisions with molecules in the air. Similarly, in finance, the price of a stock is also modeled as a random process determined by the small-scale decisions of millions of individual investors. This stands in contrast to ordinary differential equations (ODEs), which model systems that are sensitive to only macroscopic properties of their surrounds. It is the difference between a leaf floating down a calmly flowing stream, which is modeled using an ODE, versus a diffusing drop of dye in the stream, which is modeled using a SDE.
The first aim of the course is to define the noise that drives a stochastic differential equation. The most important example is Brownian motion, which we will construct as a scaling limit of the simple random walk. Brownian motion is a martingale in continuous time, and this type of martingale will define the randomness entering into a SDE. We will develop the full theory of martingales in continuous time, including stochastic integration, Itô's formula with applications, local times, the Meyer--Tanaka formula, and the Girsanov theorem.
The second aim of the course is to prove the well-posedness of stochastic differential equations, and to establish techniques for modeling such equations with Python (no prior programming experience required). We will see that SDEs are well-posed in regularity regimes for which the corresponding ODE is not, including with coefficients that are not Lipschitz continuous, like the square root. We will also prove the Feynam--Kac formula, which connects the solution of a SDE to a second-order linear parabolic partial differential equation.
The final section of the course will introduce the theory of Markov chains in continuous time, where our primary examples will be solutions of SDEs and interacting particle systems. We will cover topics including the generator of a Markov process, the Hille--Yosida theorem, and the well-posedness of the martingale problem.
7375
Wavelets. Prof. Huang
- Time: MWF 8:30-9:20
- Instructor: Prof. Huang
- Prerequisite: Math 7311
- Text: Introduction to Fourier Analysis and Wavelets, by M. A. Pinsky, Graduate Studies in Mathematics, Volume 102, AMS
- Description: This course is a basic introduction to Fourier analysis and wavelets. Topics to be covered include Fourier series, Fourier transform, interpolation theorems, the windowed/short time Fourier transform, multiresolution analysis, and the construction of various wavelets.
7380
Seminar in Functional Analysis: Topics in Harmonic Analysis. Prof. Nguyen
- Time: MWF 11:30-12:20
- Instructor: Prof. Nguyen
- Prerequisite: Math 7311
-
Text: Loukas Grafakos, Classical Fourier Analysis, Third Edition, GTM 249, Springer, New York, 2014. xviii+638 pp. ISBN: 978-1-4939-1193-6.
Recommended References: 1. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press,Princeton, 1970. 2. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993. 3. Javier Duoandikoetxea, Fourier Analysis, Graduate Studies in Math., Vol. 29. Translated and revised by David Cruz-Uribe, SFO, 2000.
- Description: This is an introductory course that covers basic topics of Harmonic Analysis. Topics include Fourier transform, Lorentz spaces, interpolation theorems, the Hardy-Littlewood maximal function, and Calderon-Zygmund singular integral operators. Moreover, the Littlewood-Paley theory will also be discussed if time permits.
7390
Seminar in Analysis: Introduction to Radon transforms. Prof. Rubin
- Time: 12:00-1:20 TT
- Instructor: Prof. Rubin
- Prerequisite: Math 7311, 7330, or equivalent
-
Text: B. Rubin, Introduction to Radon transforms (with elements of fractional calculus and harmonic analysis), Encyclopedia of Mathematics and Its Applications, 160, Cambridge University Press, 2015
(Notes will be provided by the instructor).
Additional Reference: B. Rubin, Fractional integrals, potentials, and Radon transforms (2nd edition), Chapman and Hall/CRC, 2024.
- Description: This is an introductory course-seminar in the theory of the Radon transform, one of the main objects in modern analysis, integral geometry, and tomography. Topics include fractional integration and differentiation of functions of one and several variables, Radon transforms in the n-dimensional Euclidean space and on the sphere, related aspects of harmonic analysis, functional analysis, and function theory.
7390
Seminar in Analysis: Nonlinear Optimization Theory an Algorithms. Prof. Zhang
- Time: MWF 1:30-2:20
- Instructor: Prof. Zhang
- Prerequisite:
- Text: Lecture Notes
- Description: This class will cover classical nonlinear optimization theory and algorithms. Tentative topics include but not limited to Line search methods, Newton and quasi-Newton methods Conjugate gradient methods, KKT optimality conditions, Penalty methods, Trust region methods, Sequential quadratic programming, Structured nonsmooth optimization.
7410
Graph Theory. Prof. Z. Wang
- Time: TT 1:30-2:50
- Instructor: Prof. Z. Wang
- Prerequisite: None
- Text: Graph Theory by Reinhard Diestel, Fifth Edition, Springer, 2016
- Description: The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, Eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, surface embeddings, Turan theorems, Ramsey theorems, regularity lemma, and graph minors.
7490
Seminar in Combinatorics, Graph Theory, and Discrete Structures: Matroid Theory. Prof. Oxley
- Time: MWF 8:30-9:20
- 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. Baldridge
- Time: TT 10:30-11:50
- Instructor: Prof. Baldridge
- Prerequisite: Math 7510
- Text: Algebraic Topology by Hatcher
- Description: This course covers the basics of homology and cohomology theory. Topics discussed may 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. Baldridge
- Time: TT 9:00-10:20
- Instructor: Prof. Baldridge
- Prerequisite: Math 7210 and 7510; or equivalent.
- Text: Topology and Geometry by Glen Bredon
- Description: Manifolds, vector fields, vector bundles, transversality, deRham cohomology, metrics, other topics.
7590-1
Seminar in Geometry and Algebraic Topology: Rational Homotopy Theory. Prof. Bibby
- Time: MWF 9:30-10:20
- Instructor: Prof. Bibby
- Prerequisite: Strongly recommend some familiarity with homotopy theory (as in Math 7520 Algebraic Topology) and differential forms (as in Math 7550 Differential Geometry)
- Text: Rational Homotopy Theory and Differential Forms by Griffiths & Morgan (pdf available through LSU Library here)
- Description: The course will start with some basics of homotopy theory, fibrations, Postnikov towers, and de Rham cohomology. The goal is then to study the rational homotopy type of a simply-connected space X, which is the homotopy type of its localization (or rationalization). The homotopy and homology groups of the rationalization are rationalizations of those for X, killing all torsion. The rational homotopy type of X has the advantage of being more computable than the (ordinary) homotopy type of X, thanks to the algebraic models (using differential graded algebras or Lie algebras) from Sullivan and Quillen. The story is more complicated when X is not simply connected, when one needs to make sense of how to "rationalize" the (possibly non-abelian) fundamental group.
7590-2
Seminar in Geometry and Algebraic Topology: Fibering of groups and manifolds. Prof. Schreve
- Time: TT 9:00-10:20
- Instructor: Prof. Schreve
- Prerequisite: Topology II
- Text: The class will be based on a number of papers.
- Description: The goal of this class is to give a fairly complete proof of a recent theorem of Kielak connecting algebraic fibering of groups to the vanishing of L^2-homology. We will go through the basics of group cohomology, classical constructions of embedding group rings into division rings, and the connection of Bieri-Neumann-Strebel invariants to Novikov homology. Given time, we will see some other recent applications of this theory to the study of 3-manifolds and free-by-cyclic groups.
7710
Advanced Numerical Linear Algebra. Prof. Wan
- Time: TT 12:00-1:20
- Instructor: Prof. Wan
- Prerequisite: Linear Algebra and Advanced Calculus and Programming Experience
- Text: TBA
- Description: In this course, we will focus on how to perform matrix computations efficiently and accurately. Topics will include Gaussian elimination, singular value decomposition, eigenvalue solvers, and iterative methods for linear systems. In addition to these classical topics, we will also pay attention to randomized algorithms. Both theoretical analysis and numerical experiments will be discussed. A programming language is required, and Python is recommended.
7999-n Assorted Individual Reading Classes
- No additional information.
8000-n Assorted Sections of MS-Thesis Research
- No additional information.
9000-n Assorted Sections of Doctoral Dissertation Research
- No additional information.