All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu
All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu
This course will focus primarily on ideals in polynomial rings. A Gröbner basis of such an ideal is a generating set that has certain properties with respect to a partial order on the monomials. An algorithm for finding a Gröbner basis was developed by Buchberger in his dissertation in the early 70's, but was largely ignored until advancements in computer speed and memory made the algorithm practical in many cases. Gröbner bases are useful in solving systems of polynomial equations, and, in some cases, they provide a generalization for the method of Gaussian elimination that is used for systems of linear equations. We will study the basic ideas in the theory, and consider applications to problems in commutative algebra, affine algebraic geometry, and algebraic coding theory. We will also see how to use software such as Mathematica, Macaulay2, and SAGE to compute Gröbner bases, and students will be asked to use such software for some of the exercises.
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.
Combinatorial commutative algebra. This course concerns the interplay between combinatorial geometry of convex bodies with integral vertices and the geometry of algebraic varieties defined by monomials or differences of monomials. This is an active and accessible area of current research, with a surprisingly diverse range of connections, including applications in computational algebra, statistical modeling, singularity theory and elsewhere. This course will cover the the basic theory of affine semigroups, and toric varieties and a selection of topics including free resolutions of binomial ideals, combinatorial descriptions of these objects, and applications.
An algebraic group is an affine variety that is also a group, such that the group multiplication map and the inverse map are morphisms of varieties. Thus, algebraic groups are the algebraic-geometry version of Lie groups. In this course, we will cover the basic structure theory of reductive algebraic groups (Borel subgroups, maximal tori, Weyl groups, Bruhat decomposition), and we will classify the simple algebraic groups. If time permits, we may begin discussing the representation theory of algebraic groups. Note: Algebraic geometry is not a prerequisite for this course; we will cover as much algebraic geometry as we need during the course.
Sentential logic deals with formulae such as "(p&q) --> (not r)," where p,q,r are "sentence symbols," each capable of being assigned the value "true" or "false." We shall prove the compactness theorem for sentential logic. First-order logic deals with formulae such as: "for all x, there exists a y such that y is a prime number and x is less than y." First-order logic (unlike sentential logic) is admirably suited to expressing and analyzing the deductions encountered in mathematics. We shall prove the soundness of first-order logic, and its converse, viz., Goedel's completeness theorem for first-order logic. Finally, we shall (if the teacher and students move quickly enough) prove Goedel's (first) incompleteness theorem, viz., number theory is not recursively axiomatizable.
We will treat measure theory and integration on measure spaces. The examples of the real line and of Euclidean space will be emphasized throughout. Topics will include the Hopf extension theorem, completion of the Borel measure space, Egoroff's theorem, Lusin's theorem, Lebesgue dominated convergence, Fatou's lemma, product measures, Fubini's theorem, absolute continuity, bounded variation, Vitali's covering theorem, Lebesgue differentiation theorems, and the Radon-Nykodim theorem. Applications to L^{p} and its dual, and the Riesz-Markov-Saks-Kakutani theorem will be presented if there is sufficient time. Visit the class website for further information.
Functional Analysis is the language of modern mathematics. The course provides an introduction to the theory of normed spaces and metric spaces, linear operators and linear functionals, Banach and Hilbert spaces, spectral theory, and other important topics. The students will become familiar with such notions as isometry, completeness, orthogonal projections, compactness, duality, and many others.
The course starts right from the concept of a random variable and proceeds to discuss independence of random variables, strong limit theorems, weak convergence, characteristic functions, the central limit theorem, conditional expectation, and basics of the Brownian motion. This is a basic and standard course in modern probability theory and is based on the development and contributions of A. N. Kolmogorov and J. L. Doob.
This is an introduction to finite difference methods for the numerical solution of differential equations. We will derive many fundamental finite difference schemes and develop mathematical tools for their analysis. Topics will include:
We will study the basic theory of stochastic integration with applications to finance. Many concrete examples will be used to motivate the concepts and theorems. We will assume the advanced calculus and elementary probability theory. Basic knowledge of measure theory and Hilbert space will be helpful, but is not absolutely necessary. Below are some items to be covered in this course:
The heat equation is one of the more important partial differential equations. We will discuss its relation to complex analysis and infinite dimensional analysis. The main topic are:
More information is posted at http//www.math.lsu.edu/~olafsson.
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.
Topology I has been revised to cover the fundamentals of basic point-set topology as well as homotopy and the fundamental group. The revised syllabus for Topology I is available at Math 7510 Topics.
A fundamental problem in topology is that of determining whether or not two spaces are topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects (groups, rings, etc.) to a topological space in such a way that topologically equivalent spaces get assigned isomorphic objects. Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. Two spaces with inequivalent invariants cannot be topologically equivalent. The focus of this course will be on homology, cohomology. and homotopy. We will start off approximately where last springs 7512 left off. We will study topics from chapters 3 and 4 of Hatcher's book, including Poincare duality. Geometric examples, including surfaces, projective spaces, lens spaces, etc., will be used to illustrate the techniques. We may also discuss characteristic classes from Milnor's book.
Hyperbolic geometry has been an important tool for the study of three-manifolds since the groundbreaking work of Thurston in the 1970s. In particular, Thurston's geometrization conjecture, recently proved by Perelman, is a far-reaching generalization of the Poincare conjecture. This course will discuss the geometry of hyperbolic space and the definition of a hyperbolic structure on a manifold. Examples of hyperbolic structures on 2- and 3-manifolds will be given, and we will discuss the idea of hyperbolic Dehn surgery.
A good source (though not a required text) for this material and more is John G. Ratcliffe, Foundations of Hyperbolic Manifolds (2nd ed.), Springer GTM 149. Also of interest: William P. Thurston, Three-Dimensional Geometry and Topology (ed. Silvio Levy), Princeton Mathematical Series 35.
This is an LSU VIR course, or vertically integrated research course, for graduate and undergraduate students. It will be taught as Math. 4999-1, to allow both undergraduate and graduate students to participate for credit. The topic is the foundations of special relativity and the related mathematics, including algebra, analysis, and geometry. Topics to be covered include 4--dimensional space--time, reference frames and coordinates, Lorentz transformations, simultaneity, time dilation and length contraction, causality and prohibition of motion faster than light, and composition of velocities. Geometric ideas will be stressed, and there will be an attempt to make connections with hyperbolic geometry. Students will have the opportunity to discuss selected topics in smaller groups and then present them to the others. The course serves as an introduction to research. The expectation is therefore, that the smaller groups will discuss open problems and work on them as we get more familiar with the topic.
This is a 3-hour research course with graduate credit. Students will read research papers and solve open problems. More details on the topic, Computing the bandwidth of graphs, can be found in Prof. Ding's website.
We will discuss the differential geometry of curves and surfaces in space. The course will also incorporate the use of Mathematica in the computation and visualization. As applications we will offer projects in Computer Graphics and 3-dimensional positioning systems (location detection in space with the help of two cameras).
Project: Each student would complete a project (possibly in groups) and make a presentation on a geometric topic.
On a rotating hollow sphere, there are exactly two points that don't move at all (we might call them the "north and south poles"). The sphere belongs to a large class of important topological spaces with two key features: (1) they have some (perhaps more than one) kind of rotational symmetry, and (2) they have finitely many points that don't move when the space is rotated. "Equivariant cohomology," the subject of this seminar, is a powerful tool for studying the geometry of such spaces. Using equivariant cohomology, we will explore remarkable connections to combinatorics and group theory.
This seminar will investigate the dynamical behavior of complex materials and fluids and the behavior of fields that they host. One of the topics will be the interaction of electromagnetic waves with composite materials consisting of natural and artificial components. This includes the creation and characterization of meta-materials and the phenomena of resonance. The mathematics will involve a wide range of topics in analysis and partial differential equations, including homogenization, scattering theory, special functions, and integral equations.
This course provides practical training in the teaching of calculus, how to write mathematics for publication, how to give a mathematical talk, 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.
This is the second semester of the first-year graduate algebra sequence. In this course, we will further develop the topics introduced in the first semester. Specific topics include: normal and separable field extensions; Galois theory and applications; solvable groups, normal series, and the Jordan-Hoelder theorem; tensor products and Hom for modules; noetherian rings; the Hilbert Basis Theorem; and algebras over a field.
Numerical linear algebra is central to scientific computing. A good understanding of the basic algorithms (derivation, applicability, efficient implementation) in numerical linear algebra is indispensable to anyone who wants to do research involving large scale computation. The following topics will be covered in this course: LU factorization, Cholesky factorization, QR factorization, Singular Values, Condition Numbers, Backward Stability, Rayleigh Quotient Iteration, QR Algorithm, Jacobi Iteration, Gauss-Seidel Iteration, SOR Iteration, Steepest Descent, Conjugate Gradient.
This is an introductory course. About 2/3 of the course will be devoted to algebraic tools in number theory. These include Dedekind domains, module theory and ideal class groups, discrete valuation rings, p-adic numbers, completions. The remaining 1/3 will include analytic tools such as zeta functions and applications to classical results such as the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.
Geometric methods play a central role in modern representation theory. Indeed, geometry often reveals unexpected connections between disparate phenomena. For example, the irreducible representations of the Weyl group of SL(n,C) are parametrized by partitions of n. Nilpotent conjugacy classes in the Lie algebra of SL(n,C) are also parametrized by partitions of n. Is this just a combinatorial coincidence? No--it is possible to define a natural bijection between these two sets via a geometric construction, and this bijection (the Springer correspondence) has been of fundamental importance in representation theory.
In this course, we will discuss several examples of geometric methods in representation theory. In particular, we will study flag varieties and the nilpotent cone and how they can be used to study representations (of Weyl groups, simple algebraic groups, Hecke algebras, etc) from a geometric perspective. Time permitting, we will also examine the building of a simple algebraic group.
In this course we will develop the mathematical theory of finite element methods, one of the main comuptational tools in science and engineering. The following topics will be covered: Hilbert spaces, variational formulations of elliptic boundary value problems, Ritz-Galerkin methods, constructions of finite element spaces, interpolation error estimates, discretization error estimates, nonconforming methods, mixed methods and adaptive methods.
This is a standard first year graduate course in complex analysis. Topics include holomorphic functions, covering spaces and the monodromy theorem, winding numbers, residues, Runge's theorem, Riemann mapping theorem, harmonic functions.
This is an introductory course to the representation theory of the real linear groups. The language of representation theory is widely used in mathematical physics, number theory, Harmonic analysis and geometry. We will go over the basic structure theory, discussing Haar measure, Lie subgroups, Lie algebras, exponential maps, universal enveloping algebras, and integration formulas. We will then introduce the representation theory of linear groups. We will further discuss three basic types of groups: nilpotent groups, compact groups, and semisimple groups, and their representations.
The whole field of computational complex analysis has grown rapidly during the past decade. Partly, this is due to discovering new methods in the theory of conformal mappings, integral equations and boundary-value problems of analytic functions which have helped to solve long-standing problems in physics and engineering sciences. This course focuses on the theory and computational aspects of conformal mappings for multiply connected regions and the Riemann-Hilbert problem of the theory of automorphic functions and on Riemann surfaces of algebraic functions. We shall discuss elements of the theory of Schottky groups, abelian differentials, the Schottky-Klein prime function, and the Riemann-Roch theorem. Applications: inverse problems in material sciences and fluid mechanics, diffraction of electromagnetic waves, and fracture mechanics.
Positive matrices play a similar role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines,including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. The course will introduce several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. Specific topics include positive and completely positive maps, monotonicity, operator norms, operator inequalities, and matrix means, including very recent developments. The course text has numerous exercises integrated into the text itself and at the end of each chapter. This makes it suitable for teaching in the style of a "working seminar'' with heavy student involvement and participation accompanied by suitable professorial explanation, guidance, coaching, and filling in of background.
The focus of the course is introduction to the Fourier analysis on the unit sphere (Spherical Harmonics) in the n-dimensional real Euclidean space and its application to integral (or, more generally, pseudo-differential) operators commuting with rotations, in particular, to Minkowski-Funk transform and some other transformations of integral geometry. Operators of that type arise in PDE, harmonic analysis, group representations, mathematical physics, geometry, and many other areas of mathematics and applications.
This course contains three parts:
(1) a brief review of the Ito theory of stochastic integration with applications to finance;
(2) infinite dimensional analysis; and
(3) white noise theory. I will explain the basic knowledge in these three related areas and give a comprehensive outline leading to the current research in stochastic analysis.
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, and graph minors. There are many books on graph theory. Prof. Oporowski recommends the references listed above, but they are not absolutely necessaary. He will present the lectures with his own notes, which will be available for download, and which should make good study material. However, the amount of detail in the lecture notes is less than that of either of the mentioned books. If your interest in the subject is anything more than superficial, you would be well advised to get at least one of those books - especially the first one listed. Grades for the course will be based 60% on homework and 40% on two exams (midterm and final). Decisions in borderline cases will be made on the basis of class participation. There will be the total of over twenty problems given as homework, and the two lowest problem scores will be dropped. If you have any questions about the course, do not hesitate to contact Prof. Oporowski.
A fundamental problem in topology is that of determining whether or not two spaces are topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects (groups, rings, etc.) to a topological space in such a way that topologically equivalent spaces get assigned isomorphic objects. Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. Two spaces with inequivalent invariants cannot be topologically equivalent.
The focus of this course will be on homology and cohomology. To a topological space, we will associate sequences of abelian groups, the homology and cohomology groups. Topics (from chapters 2 and 3 of Hatcher's book) include simplicial, singular, and cellular homology, Mayer-Vietoris sequences, and universal coefficient theorems. Geometric examples, including surfaces and projective spaces, will be used to illustrate the techniques. Discussion of cohomology theory, including products and duality, will (presumably) continue in MATH 7520 Algebraic Topology.
This course gives an introduction to smooth manifolds, which are spaces that locally resemble Euclidean space and have enough structure to support the basic concepts of calculus. We shall cover the core material in chapters 1, 2 and 4, including submanifolds, tangent vectors and bundles, smooth mappings and their derivatives, the implicit and inverse function theorems, vector fields, differential forms and Stokes's theorem. If time permits, we shall continue with chapter 5, on sheaves, cohomology and the de Rham Theorem.
Geometric Topology was conceived to be a broad survey of selected current research areas in topology within our Department: braid groups, arrangements, 3-manifolds, in particular complements of knots/links, and mapping class groups of 2-manifolds. This course will start with an introduction to braid groups (with eventual sideways views of the relationship of active research interests in the department: arrangements/configuration spaces and mapping class groups). We will explore the relationship of braids to knots and links, their homology via the technical tool of the free differential calculus and their representations: Burau, Gassner and Lawrence-Krammer. Faithfulness of the Lawrence-Krammer representation will be shown by the methods of Krammer and Bigelow. We will conclude with a survey of related open problems.
We will discuss TQFTs from an axiomatic point of view, as well as several important examples. Then the course will concentrate on one example related to the Jones polynomial of knots. We will also discuss applications to low-dimensonal topology. We will follow a skein theory approach.