All inquiries about our graduate program are warmly welcomed and answered daily:
All inquiries about our graduate program are warmly welcomed and answered daily:
This course is something of a hybrid course, with a goal of providing students with at least an elementary knowledge of two important mathematics disciplines: control theory and Lie theory. Hence the course will provide an introduction to basic ideas of control theory and basic ideas of Lie theory (in the context of linear matrix groups), and develop connections between the two areas. Topics to be covered include matrix Lie groups and their Lie algebras and exponential maps, control systems, particularly linear control systems and control systems on matrix groups, notions of controllability, observability, optimaility, and stability, Riccati equations and optimal control on linear systems with quadratic cost, control sets and semigroups, and the interpretation of certain mechanical systems as control systems.
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.
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.
This course is an introduction to "commutative algebra" which - besides being an area in its own right - is the basis for algebraic number theory and algebraic geometry. Topics include: Rings and modules, Spec of a commutative ring, tensor products, Nakayama's lemma, Integral extensions (going-up and going-down), Nullstellensatz, Noetherian and Artinian rings, Discrete valuation rings, Dedekind domains, Dimension theory, regular local rings, Kaehler differentials, smooth and etale extensions, UFD's, etc. I will regularly assign exercises and expect some of them to be turned in.
Lie algebras are essential in many areas of mathematics and theoretical physics. In this course, after covering the definition and basic properties of Lie algebras, we will study the structure theory of semisimple Lie algebras and the classification of simple Lie algebras, and then take a brief look at their representation theory. Throughout the course, we'll keep in mind the example of sl(2), the smallest semisimple Lie algebra. It's small enough (only 3-dimensional) to easily do explicit calculations in, and at the same time interesting enough to give us a good idea of what goes on in higher-dimensional Lie algebras.
We will treat measure theory and integration on measure spaces. The cases of the real line and Euclidean space will be emphasized. 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, the Radon-Nykodim theorem, Vitali's covering theorem, bounded variation, and the Riesz-Markov-Saks-Kakutani theorem if there is sufficient time. Visit the class website for further information: http://www.math.lsu.edu/personal/rich/7311info .
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.
This is a second graduate course in complex analysis, in one and several complex varibles. The first part of the course will cover results in one variable, including the modular function, the Picard theorems, the boundary behavior of conformal maps, the proof of the Bieberbach conjecture and the thoery of Hp spaces. The second part will introduce the basic ideas from the analysis of functions of several complex variables.
This course focusses on the numerical implementation of elliptic solvers on parallel supercomputers, using finite differences and finite element methods. I will briefly review the numerical analysis of these methods, together with the required mathematical tools, if necessary. I will present some peripheral issues (mesh generation, iterative solvers, visualization). I will give some elements of parallel programming, using MPI and PETSc, which will be the basis of the numerical project. Time permitting, I will also discuss domain decomposition methods and their implementation, and related problems. The numerical part of this class will be based on MPI (Message Passing Interface) and PETSc (Portable, Extensible Toolkit for Scientific Computation developed at Argonne National Laboratories) and will require some basic programming skills in C or Fortran. Numerical project and homework problems will include the parallel implementation of finite difference and finite element method.
This course will cover topics in stochastic analysis with a view towards applications in finance. In the first few weeks we will go over the basic framework of probability theory (probability spaces, random variables) and essential tools. We will then study Brownian motion and other stochastic processes, examining the beautiful and surprising relationship between such processed and partial differential equations. A typical question of interest is: how long does it take for a random process to escape from a given region? In this context we will study the Feynman-Kac formula, which arose in physics but has application in finance. On the financial side, we will see a very simple approach to the celebrated Black - Scholes - Merton formula for pricing stock options. We will examine the structure of certain financial products such as bond options, swaps, and credit derivatives. Pricing and risk management of these derivatives will be examined. Credit derivatives form a currently highly active area both in the "real world" of finance and in development of the theory.
Numerical methods for partial differential equations generate large ill-conditioned systems. Since direct methods are too expensive, such systems are usually solved by iterative methods. In this course we will discuss two classes of modern iterative methods for numerical PDEs: multigrid and domain decomposition. The goal is to gain insight through the analysis and implementation of various algorithms. The audience for this course should have some background in finite element methods. Experience in MATLAB is desirable but not required.
This is the first part of a one year course on Harmonic Analysis and Represeentation Theory. It is also an introductory course on Fourier Series and Integrals. Topics include: Periodic functions, Fourier series, classical functions spaces, in particular the space of rapidly decreasing functions, the Fourier transform and its application to differential equations (heat equation and the wave equation), Hermite polynomials and functions, distributions, and the continuous wavelet transform. More advance topics might be introduced if there is time. Further information is available at http://www.math.lsu.edu/~olafsson/sy7390.html
This course will begin with a review of continuous optimization in Euclidean space, and quickly move to the main topic, so-called dynamic optimization. The first half will cover the classical material of the calculus of variations, including topics such as the Euler-Lagrange equation, Weierstrass maximality condition, Erdmann corner conditions, and Jacobi conjugate points. Plenty of examples will be covered. There is a natural transition into optimal control, which will be the focus of the rest of the course from a neo-classical point of view.
The course provides an introduction to the field of homogenization theory as well as a guide to the current research literature useful for understanding the mathematics and physics of complex heterogeneous media. The course is suitable for both mathematicians and engineers. The first part of the course develops the variational tools and asymptotic analysis necessary for characterizing the macroscopic behavior of heterogeneous media. The techniques introduced in this course include the Hashin-Shtrikman variational principles, analytic continuation methods, and the method of two-scale asymptotic expansions. Explicit solutions of field equations inside special microstructures will be developed including the space filling coated spheres construction and confocal coated ellipsoid construction. The second part of the course focuses on very recent developments related to multi-scale inverse problems known as de-homogenization theory. The techniques introduced here include the maximum principle for harmonic functions, applications of the Hölder inequality, and convex duality. These techniques provide a characterization of microscopic field behavior inside complex materials in terms of applied macroscopic fields and the statistics of the microgeometry.
The study of matroids is a branch of discrete mathematics with basic links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics. This course presents more advanced topics in the theory of matroids. In particular, the course will consider problems relating to the representability of matroids over fields especially over small finite fields; constructions for matroids; higher connectivity for matroids; and the structure of certain classes of matroids.
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 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, universal coefficient theorems, cup products, and Poincare duality. Geometric examples, including surfaces, projective spaces, lens spaces, etc., will be used to illustrate the techniques.
The purpose of this course is to introduce students to the basic concepts and results in symplectic geometry/topology and develop Gromov's theory of pseudo-holomorphic curves. The content of the course will be on pseudo-holomorphic curves, including: basic definitions and local properties of pseudo-holomorphic curves, Fredholm theory and transversality of moduli space, Gromov's compactness theorem, pseudo-holomorphic curves in dimension 4, some applications in symplectic topology.
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, including Wedderburn's and Maschke's Theorems.
Roughly speaking we will cover the first two chapters of Hartshorne's book, but we will supplement with other material. For instance we will do some introduction to software systems (Macaulay2 and Singular) for computing things. Thus we will do an introduction to affine and projective varieties, sheaves and schemes.
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 to the cohomology of groups and Galois cohomology. We will begin with basic homological algebra and define the cohomology of a group G with coefficients in a G-module M. We will discuss methods for explicit computation and the classical interpretations of the low-dimensional cohomology groups. The second part will be devoted to the cohomology of profinite groups and Galois cohomology. We will give some standard applications such as the cohomological interpretation of the Brauer group. We will also discuss Galois cohomology with nonabelian coefficients and the theory of torsors, with applications to algebra and algebraic geometry (invariants for quadratic forms, elliptic curves, Severi-Brauer varieties).
Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of Rn which itself is often given by polynomial inequalities. The main objective of the course is to give useful characterizations of such polynomials. It takes as a starting point Hilbert's 17th Problem (which was to prove that if a polynomial f in n variables with real coefficients takes only nonnegative values on Rn, then f can be written as a sum of squares of real rational functions (the converse being obvious)), and explains how E. Artin's solution of that problem (1926) eventually led to the development of "real algebra" towards the end of the 20th century. Topics include ordered fields, real closed fields, semialgebraic sets, elimination of quantifiers, valuation theory, quadratic forms over real fields, the Positivstellensatz and the real Nullstellensatz, the real spectrum of a commutative ring, and Schmuedgen's celebrated theorem (1991) on polynomials that are positive on compact semialgebraic sets in Rn
Systems and control theory is one of the most central and fast growing areas of applied mathematics and engineering. This course provides a basic introduction to the mathematics of finite-dimensional, continuous-time, deterministic control systems at the beginning graduate student level. More advanced topics may also be included depending on the backgrounds of the students. It is intended for PhD students in applied mathematics and engineering graduate students with a basic background in real analysis and nonlinear ordinary differential equations. The course is designed to help students prepare for research at the interface of applied mathematics and control engineering. This is a rigorous proof-oriented systems theory course that goes beyond classical frequency-domain or more applied engineering courses. Emphasis will be placed on controllability and stabilization.
The four main topic in this class are: Topological groups, homogeneous spaces, basic representation theory and representation theory of compact groups. Representations of topological groups are central in several branches of mathematics: In number theory and the study of authomorphic functions and forms, in geometry as a tool to construct important vector bundles and differential operators, and in the study of Riemannian symmetric spaces. Finally, those are important tools in analysis, in particular analysis on some special homogeneous manifolds like the sphere, Grassmanians, the upper half plane and its generalizations. Representations even shows up in branches of applied mathematics as generalizations of the windowed Fourier transform and wavelets. Several examples of those applications in analysis and geometry will be discussed in the class.
The purpose of this course is to introduce students to structures and techniques in analysis that have far-reaching applications. The concepts and techniques will form part of a necessary basis for a wide variety of specializations in analysis, including variational calculus, numerical analysis, mathematical physics, and many areas of partial differential equations. Participants will gain a working understanding of Hilbert space, operators in Hilbert space, and spectral decomposition of self-adjoint operators, as well as generalized solutions to differential equations, variational problems, and integral equations. The medium for presenting the material will be applications to problems from the material sciences. We will not prove the theorems of spectral theory, for this would require a course in itself, but our course will follow a rigorous development with these theorems as a basis. Some of the possible topics are these:
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 uled 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 uled. 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.
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.
We will study knots and 3 dimensional manifolds. We will study the Jones and Alexander polynomials, branched covers, surgery descriptions of 3-manifolds, Quantum invariants of 3-manifolds.
Heegaard Floer homology is a recent development due to Peter Ozsvath and Zoltan Szabo. It assigns a complex to a three-dimensional manifold, whose homology is an invariant of that manifold. The definition involves Heegaard diagrams, symmetric products, and holomorphic disks (all of which I will explain in the class). The definition extends to give invariants of 4-dimensional manifolds and also invariants of knots in 3-manifolds. All are defined using holomorphic disks, though a simple combinatorial definition is possible in some special cases (including knots in the three-sphere). This theory is the focus of a great deal of current research and has led to a lot of progress on longstanding problems in knot theory and three-manifolds. I will describe the invariants for three-manifolds and knots and their main properties. (This will involve some discussion of four-manifolds.) I will also discuss some of the applications.