LSU
Mathematics

# Course Descriptions Summer 2003 - Spring 2004

## Summer 2003

• MATH 7380: Introduction to Hilbert and Banach Space
• Instructor: Prof. Sundar
• Prerequisite: Math 7311 or Permission of Instructor
• Text: Class Lecture Notes

The first half of this course will cover basic Hilbert space theory. Topics include Bessel's inequality, orthonormal systems, Riesz representation theorem, Lax-Milgram Theorem, projections, operator (self-adjoint, normal, unitary, and compact) theory, elements of spectral theory, eigenvalue problems, and Fourier transforms. The second half of the course will highlight applications of the theory, and will be adjusted to the interests of the students. Topics could include integral and differential equations, partial differential equations, wavelets, and the calculus of variations.

## Fall 2003

• MATH 7200: Geometric and Abstract Algebra
• Instructor: Prof. Hurrelbrink.
• Prerequisite: Math 2085 Linear Algebra or equivalent
• Text: Hungerford: Algebra, Springer-Verlag Graduate Texts in Mathematics 73

Math 7200 is the basic course in Algebra. It prepares the students for many mathematical fields. We will cover selected topics from Chapters I through V in Hungerford's book.

• MATH 7211: Commutative Rings
• Prerequisite: Math 7200 and 7210.
• Text: Introduction to Commutative Algebra and Algebraic Geometry by E. Kunz, Birkhauser (1997), ISBN: 0-8176-3065-1

This course is an introduction to commutative algebra, the algebra which was motivated by problems in algebraic geometry and number theory. Thus, this course will serve as a foundational course for further study in either of these two disciplines. The goal of the text being used is to present many of the standard topics of commutative algebra as they are needed in order to study a specific interesting problem, that of representing an algebraic variety (the zero set of finitely many polynomials) as the intersection of the smallest number of  hypersurfaces (zero set of a single polynomial) possible. Some of the specific topics to be studied include: Hilbert basis theorem, Hilbert's nullstellensatz, prime spectrum of a ring, dimension theory of rings and varieties, rings and modules of fractions, and the sheaf of regular functions on an algebraic variety, the local-global principle in commutative algebra, Krull's principal ideal theorem, and applications to dimension, regular local rings.

• MATH 7280: Algebraic Geometry
• Prerequisite: Math 7210
• Texts: (1) Klaus Hulek, Elementary Algebraic Geometry, AMS2003. Also (2) Kenji Ueno, Algebraic Geometry I: From Algebraic Varieties to Schemes, AMS 1999.

This course is an introduction to some of the basic concepts of algebraic geometry, with an emphasis on the geometry. Topics include:

• affine and projective varieties, smooth and singular points,dimension
• algebraic curves, (including: intersection multiplicity and Bezout’s theorem, resolution of singularities, genus, divisors and linear systems)
• special additional topics according to interest e.g.: a)sheaves and schemes, b) toric varieties, c) semi algebraic sets and real geometry.
• MATH 7290: Lie Algebras and Lie Groups
• Instructor: Prof. Hoffman.
• Prerequisite: Math 7200 and 7510
• Text: Lie Groups: Beyond an introduction, by Anthony Knapp Birkhaeuser Progress in Mathematics vol 140

Topics:

• Lie algebras, definitions.
• Lie groups, definition and examples.
• Lie theory - the relation between 1 and 2 above.
• Structure of complex semi-simple Lie algebras.
• Introduction to representation theory.

We will illustrate the general theory with examples principally drawn from the so-called classical groups. This course has minimal prerequisites - basically groups, rings, modules, fields, and elementary topology. However, this course will go deep fast, so that familiarity with such topics as manifolds, representations, covering spaces, etc., while not required, is desirable. At any rate, what we can cover will depend in part on the preparation and interests of the class.

• MATH 7311: Real Analysis I
• Instructor: Prof. Davidson
• Prerequisite: Math 4032 or the equivalent.
• Text: Real Analysis (3rd Edition) by H. L. Royden

This is our introductory course in real analysis. The core of the course is Part One of Royden's book. Topics include Lebesgue measure and integration on the real line, convergence theorems, functions of bounded variation and absolute continuity, differentiation, and the classical Banach spaces. As time permits we may consider abstract Banach spaces and/or abstract measure and integration. This course continues as Math 7312 in the spring semester.

• MATH 7350: Complex Variables
• Instructor: Prof. Neubrander
• Prerequisite: MATH 7311: Real Analysis I
• Text: Lecture notes; available in PDF format on a web site specifically maintained for this course.

This course is an introduction to some of the basic concepts and applications of complex analysis. The course is largely self-contained; i.e., many results and techniques from real analysis will be re-introduced and re-proved using the fundamental ideas and language of modern analysis. The following topics will be covered:Chapter I: Complex Numbers, Analytic Functions, The Complex Exponential, Cauchy-Riemann Theorem, Contour Integrals. Chapter II: Antiderivatives, Cauchy's Theorem, Cauchy's Integral Formula, Cauchy's Theorem for Chains, Principles of Linear Analysis, Cauchy's Theorem for Vector-Valued Analytic Functions, Power Series, Maximum Principle, Laurent's Series and Isolated Singularities, Residue Calculus. Chapter III: Dunford Functional Calculus, Systems of Linear ODEs, Laplace Transforms, Semigroups, Stability and Convergence for Numerical Approximations, Tauberian Theorems, Prime Number Theorem, Asymptotic Analysis and Formal Power Series.

• MATH 7360: Probability Theory
• Instructor: Prof. Sengupta, Office: 324 Lockett; 578-1607. Office hours by appointment. Homepage .
• Prerequisite: Math 7312 (measure and integration), or permission of the department.
• Text: Kiyosi Ito's Introduction to Probability Theory.

The course will build the basic apparatus of probability theory. We will study the fundamental results concerning probability measures and random variables. Topics will include limit theorems and elements of stochastic processes.

• MATH 7380-1: Stochastic Analysis & Applications to Mathematical Finance
• Instructor: Prof. Sundar
• Prerequisite: Math 4031 and Math 3355
• Text: Based on lecture notes given in class

The aim of the course is to provide a mathematically precise account of the arbitrage theory of financial derivatives. A self-contained treatment of stochastic differential equations and the Ito calculus will be presented, and will include the Feynman-Kac formula, and the Kolmogorov equations. Risk neutral valuation formulas and martingale measures will be introduced through Feynman-Kac representations. The course will cover pricing and hedging problems in complete as well as incomplete markets. Barrier options, options on dividend-paying assets, as well as currency markets will be studied.Interest rate theory will include short rate models and the Heath-Jarrow-Morton approach to forward rate models. A self-contained treatment of stochastic optimal control theory will be used to study optimal consumption/investment problems. The course will be geared towards concrete computations involving stochastic differential equations.

• MATH 7380-2: Mathematical Topics in Systems Theory
• Instructor: Prof. Malisoff
• Prerequisites: MATH 7311 (Real Analysis I), MATH 7312 (Measure and Integration), and MATH 7320 (Ordinary Differential Equations), or permission from the instructor.
• Text: Sontag, E.D., Mathematical Control Theory, Deterministic Finite-Dimensional Systems, Second Edition, Texts in Applied Mathematics Vol. 6, Springer-Verlag, New York, 1998 (ISBN: 0-387-98489-5). Additional material as appropriate from author's web site.

This course provides a basic introduction to the mathematics of finite-dimensional, continuous-time, deterministic control systems at the beginning graduate student level. The course is intended for PhD students in applied mathematics, and for engineering graduate students with a background in graduate real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for interdisciplinary research at the interface of applied mathematics and control engineering. This will be a rigorous, proof-oriented systems theory that goes beyond classical frequency-domain or more applied engineering courses. Emphasis will be placed on controllability and stabilization. Additional information about this course is available at http://www.math.lsu.edu/~malisoff/m7380-2.pdf .

• MATH 7390-1: Harmonic Analysis - I
• Instructor: Prof. Olafsson
• Prerequisite: Math 7311 (Real Analysis I) or equivalent.
• Text: Lecture notes by R. Fabec and G. Olafsson

The subject proper of harmonic analysis is to decompose functions into simpler functions. In the theory of differential equations that means to write an arbitrary functions as a sum or integral of eigenfunctions. If we have a symmetry group acting on the system, then we would like to write an arbitrary function as a sum of functions that transforms in a simple and controllable way under the symmetry group. The simplest example is the use of polar coordinates and radial functions for rotation symmetric equations. The course is an introduction into the basic theory of classical Fourier analysis. The main topics will be:

• Basic properties of Fourier series;
• Convergence of Fourier series;
• Function spaces on Rn, in particular the space of rapidly decreasing functions, Lp-spaces, and the space of compactly supported functions;
• Basic properties of the Fourier transform on Rn, inversion formula for rapidly decreasing functions;
• The L2-theory for the Fourier transform, in particular the Plancherel theorem;
• We include some application, which will be chosen depending on time available and the interests of the students. Possible topics include: distribution theory and Fourier transforms of distributions, differential equations, Sobolev spaces, the Fock-space and Hermite polynomials, introduction to harmonic analysis on the Heisenberg group, the short time Fourier transform, and the continuous wavelet transform.

The class will be continued by R. Fabec in the spring 2004. We can view Rn as a set or as a manifold. But we can also view Rn as an abelian group. In that sense Rn is a part of abelian harmonic analysis. The simplest examples of  non-abelian harmonic analysis are harmonic analysis on the (ax+b)-group, which is the basic tool in the wavelet theory, the Heisenberg group, classical linear Lie groups.The second part discusses the basic properties of topological groups and homogeneous spaces. Then representations of topological groups will be discussed and several examples worked out.

• MATH 7390-2: Applied Analysis in the Materials Sciences
• Instructor: Prof. Lipton
• Prerequisite: Math 7311 (Real Analysis I) or equivalent.
• Text: Partial Differential Equations, by L. C. Evans, and Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn.

Motivated by applications in physics and engineering it is desirable to identify heterogeneous materials with optimal properties.Mathematically this is a problem of distributed parameter optimal control. Here the state equation is a partial differential equation and the control variable is the Lebesgue measurable coefficient in the principal part of the differential operator. The preliminary part of the course provides a self contained introduction to the theory of Sobolev Spaces and weak solutions of elliptic partial differential equations. For this part we will follow chapters 5 and 6 of the book Partial Differential Equations, by L. C. Evans. The primary part of the course will develop the Calculus of Variations as it applies to the optimal design of heterogeneous media. For this part we will follow chapters 3, 6, and 7 of the book Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn.

• MATH 7490: Discrete Optimization
• Instructor: Prof. Ding
• Prerequisite: Linear Algebra
• Text: useful but not required): Combinatorial Optimization, by W. Cook, W. Cunningham, W. Pulleyblank, and A. Schrijver.

In this course, students will be motivated by real world problems. They will learn how to solve fundamental problems indiscrete optimization. They will also be introduced to advanced techniques that deal with computationally hard problems. A more detailed course description can be found at http://www.math.lsu.edu/~ding/sy7490.html

• MATH 7510: Topology I
• Instructor: Prof. Perlis
• Prerequisite: advanced calculus, such as Math 4031
• Text: We will use notes of Prof. Jimmie Lawson--no need to buy a textbook!

In the early 1900's, great effort was made to carry over the powerful notion of continuity of functions on subsets of Euclidean space Rn to functions on more general sets. This generalization is called point-set topology and is the subject of this course. Prof. Lawson's notes contain definitions, exercises, and statements of theorems. Your job as a student is to learn the definitions and to supply proofs of the exercises and theorems. Class will be highly interactive. Only rarely will I lecture; most of the time students will present their solutions on the board, followed by a discussion. Active class participation is required! The material itself covers standard point-set topology. Some of the topics we will discuss are: topological space, open set, basis, metric space, product topology, closure, partial order, net, Hausdorff space, homeomorphism, relative topology, quotient maps, connectedness, and compactness.

• MATH 7590: Geometric Topology
• Instructor: Prof. Cohen,Tel.: (225) 578-1576
• Prerequisite: MATH 7512 Topology II
• References: Much of the subject matter may be found in Moran"The Mathematical Theory of Knots and Braids." Other potential sources include Birman, Hansen, Orlik-Terao, and Prasolov-Sossinsky.

The course will be a broad survey of selected current research areas in topology within our Department, such as Braid Theory, Arrangements, 3-manifolds, Complements of Knots and Links, and Mapping Class Groups of 2-manifolds. Concepts from geometric topology/group theory will be explored in the contexts of these subjects. These include aspects of infinite group theory such as the lower central series, word and conjugacy problems, Fox calculus, etc.. Other (related) topics include braid group representations and classical knot and link invariants (e.g., Alexander polynomials). The course will also provide an introduction to configuration spaces and selected generalizations, and connections to subjects such as combinatorics and the theory of reflection groups.

• MATH 7690: Riemannian Geometry
• Instructor: Prof. Lawson
• Prerequisite: Math 7550 (Differential Geometry) or equivalent.
• Text: An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby

A Riemannian manifold is a differentiable manifold equipped with a special kind of metric called a Riemannian metric. The presence of a Riemannian metric on a differentiable manifold allows the introduction and treatment of many classical geometric notions: distance, geodesics (an appropriate generalization of lines), angles, etc. Indeed an important class of Riemannian manifolds, the class of symmetric manifolds, includes as special cases all the classical geometries: Euclidean, non-Euclidean, projective, etc. Thus one might say that Riemannian geometry provides a modern, elegant, comprehensive view of classical geometry, although its theory goes much further. This course provides an introduction into the basic theory of Riemannian manifolds and their associated geometry. The main topics include:

• The geometry of space curves
• Covariant differentiation
• Geodesic curves and their properties
• The Riemannian connection and the curvature tensor
• Symmetric Riemannian manifolds
• Gaussian and mean curvature
• Manifolds of constant curvature

## Spring 2004

• MATH 7210: Algebra - I
• Instructor: Prof. Perlis.
• Prerequisite: Math 7200 and a good undergraduate theorem-proving abstract algebra course.
• Text: Hungerford, Algebra.

This is primarily a course on Field Theory and Galois Theory. It is expected that the students are familiar with the theory of finite groups, including the Sylow theorems. We will begin with Chapter V in Hungerford's book. Galois theory has its roots in the attempt to generalize the familiar quadratic formula for finding the complex solutions of a polynomial equation ax2 + bx + c = 0 of degree two to polynomials of higher degree. In 1798 Ruffini published an incomplete proof of the assertion that in general there is no algebraic formula (involving only finitely-many sums, products, quotients, and nth roots) to solve a polynomial equation of degree five or greater. In 1826, at the age of 24, Abel published a proof of Ruffini's statement. By today's standards, Abel's proof is a bit fuzzy. Galois theory gives an elegant way to analyze the problem, and is so powerful that the central ideas if not the details have carried over to algebraic topology, geometry, and even parts of differential equations. If you have questions, please contact R. Perlis using the link above under Instructor.

• MATH 7280: Elliptic Curves and Modular Forms
• Instructor: Prof. Verrill.
• Prerequisite: (1) Undergraduate complex analysis (you should be comfortable with meromorphic and analytic functions; (2) Groups, Galois groups (definitions and examples).
• Text: Introduction to the arithmetic theory of automorphic forms, G. Shimura. (Strongly recommended, but not required; used as reference material; we will cover roughly the first 4 chapters; some topics not covered in the text will be presented as handouts.)

This course is primarily an introduction to the theory of modular forms, with applications to elliptic curves. A modular form is an analytic function on the upper half complex plane having many symmetries, and an expansion of the form f(z) = a0 + a1 q + a2 q2 + a3 q3 + .... where q=exp(2pi iz). An elliptic curve is (usually) given by an equation y2 = x3 + ax + b (and has only an historical connection to ellipses). The modularity of elliptic curves is the surprising fact that for every elliptic curve (with a, b rationals), there is a corresponding modular form such that the number of points on the curve, mod p, for all but finitely many primes p, is p minus the coefficient ap of the modular form. Some of the topics we will cover are: Congruence subgroups, fundamental domains for subgroups of PSL2(Z), Hecke operators, the group law of an elliptic curves, p-adic numbers, and Galois representations associated with elliptic curves. A goal is to be able to sketch an outline of the ingredients that go into how the modularity of elliptic curves is used to prove Fermat's last theorem. We will also discuss computational problems, such as how to compute Hecke eigen modular forms, and fundamental domains. These will be illustrated with the Magma computer algebra package. Questions? Please contact me at the email address linked to my name above, and visit the web-site.

• MATH 7290: Linear Algebraic Groups
• Instructor: Prof. Morales.
• Prerequisite: Algebra I and II (MATH 7210, 7211) or equivalent
• Text: Class lecture notes. *Reference books (strongly recommended, but not required): (1) "Linear Algebraic Groups" by James E. Humphreys, Springer-Verlag 1999. (2) "Linear Algebraic Groups" by T. A. Springer, Birkhauser 1998.

Linear (or affine) algebraic groups are affine algebraic varieties endowed with a group structure compatible with the structure of variety. Examples of such objects are the classical groups GLn, SLn, SOn and more generally the groups of automorphisms of vector spaces equipped with a tensor. Algebraic groups are the algebraic counterpart of Lie groups. When the ground field is the field of complex numbers C, the two theories have a large intersection. The goal of this course is to give an introductory overview of this beautiful theory. Background material from algebraic geometry will be developed as we go.

Outline:

1. Some Algebraic Geometry
2. Linear Algebraic groups
3. Derivations, differentials, Lie algebras
4. Homogeneous Spaces
5. Semi-simple and Unipotent Elements
6. Solvable Groups
7. Borel Subgroups
8. Reductive Groups
9. Semi-simple Groups
• MATH 7312: Measure and integration
• Instructor: Prof. Sundar
• Prerequisite: Math 7311
• Text: Real and Complex Analysis by Walter Rudin

The course is a sequel to Real Analysis I, and introduces measure theory, elements of Hilbert and Banach spaces, and analysis of functions of several variables. The syllabus is as follows:

1. Chapter I: Abstract measures, integration, convergence theorems, Lp-spaces and inequalities.
2. Chapter II: Hilbert and Banach spaces, consequences of Baire's theorem, Hahn-Banach theorem and applications.
3. Chapter III: Signed measures, Radon-Nikodym theorem, decomposition theorems, duality.
4. Chapter IV: Product measures, Fubini and Tonelli theorems.
5. Chapter V: Differentiation, implicit and inverse function theorems, change of variables in integrals.
6. Chapter VI: Sobolev spaces, Embedding theorems.
• MATH 7320: Ordinary Differential Equations
• Prerequisite: Either Math 7311 or else both Math 4031 and Math 4032.
• Text: W.Walter, Ordinary Differential Equations, Springer, 1998.

This is a first graduate course in ordinary differential equations. The following topics will be covered. Existence and uniqueness results for first order equations and systems, and for higher order equations. Linear equations and systems. Complex linear systems. Boundary value and eigenvalue problems. Stability.

• MATH 7380-1: Partial Differential Equations
• Instructor: Prof. Shipman.
• Prerequisite: A first course in graduate analysis, such as Math 7311; elementary exposure to ordinary differential equations.
• Text: Partial Differential Equations, by L. C. Evans (AMS Graduate Studies in Mathematics, Vol. 19)

This course is an introduction to mathematical theories of partial differential equations and will be complemented by physical motivations and applications. The course will be mostly self-contained, presuming a solid foundation in basic analysis and exposure to basic (ordinary) differential equations.

1. The basic linear equations will be covered in depth as well as methods of solution such as transforms and fundamental solutions. Basic nonlinear theory will include the method of characteristics, conservation laws, and existence and uniqueness theory. Other major topics include the calculus of variations and elliptic PDEs.
2. Some special topics will be studied in some detail, such as nonlinear waves, nonlinear spectral transforms, asymptotic methods, or the mathematics of electrodynamics.
3. An attempt will be made to give a broad survey of topics in PDEs by discussing briefly a sampling of topics. These may include evolution equations, operator and semigroup methods, nonlinear hyperbolic equations, boundary-integral methods, energy methods, numerical methods, integrable systems, elliptic regularity theory, or pseudo-differential operators.
• MATH 7380-2: Integral and Functional Equations
• Instructor: Prof. Antipov.
• Prerequisite: MATH 4036 Complex variables
• Text: Lecture notes and F.D.Gakhov, Boundary value problems.

Aims: To introduce students to singular integral and integro-differential equations and functional-difference equations which can be tackled using methods for boundary-value problems of the theory of analytic functions.

Objectives: Students should know a range of analytical methods for solving model problems in applied mathematics including fracture mechanics and diffraction theory.
Content: The Cauchy integral, the Riemann-Hilbert problem on an axis and an open curve, the Wiener-Hopf method, the Carleman problem, matrix factorization, elements of elliptic functions and elliptic surfaces, model problems of fracture, elasticity and diffraction of electromagnetic and acoustic waves.

• MATH 7380-3: Numerical Methods & Applications
• Instructor: Prof. Bourdin.
• Prerequisite: Math 7311 or equivalent; Basic knowledge of harmonic analysis, PDE, or numerical analysis is a plus, but not a requirement. Basic programming skills in C, fortran, pascal , or other "low" level language.
• Text: Lecture notes, online resources provided during the class.

This is an introduction class to the numerical analysis and implementation of Partial Differential Equations. Its focus is mainly on the Finite Element Method for elliptic problems. In a first part, we will briefly survey some theoretical results for PDEs (variational formulation, Galerkin method...). Then, we will discuss multi-dimensional interpolation, differentiation and integration, which will lead us to the core of the finite element method. Time permitting, we will either cover the basic of domain decomposition and parallel processing, and/or iterative methods for solving large sparse linear systems of equations. Numerical projects will represent a major part of the work (and of the grade). It is my intention to make this class suited to students with a theoretical background, willing to learn about advanced numerics, as well as to those already familiar with numerics, and interested in a more theoretical approach. Mathematics, computer science and engineering students are welcome here!

• MATH 7390-1:  Harmonic Analysis - II: The Classical Groups
• Instructor: Prof. Fabec
• Prerequisite: Math 7311 and 7312.
• Text: will be provided on-line.

This is the second in a sequence of courses on harmonic analysis. The sequence starts with classical Fourier series and the Fourier integral, extends these objects to distributions and explores their uses in solving differential equations. Next the Fourier integral is used to study functions and function spaces on the Heisenberg group. This group is fundamentally important for both its importance in quantum mechanics where its representation theory encodes the Heisenberg uncertainty principle and as an example of the difficulties arising when one attempts to do harmonic analysis on groups which are no longer commutative. Here one must deal with infinite dimensional representations and from this one can be familiarized with this theory. We then discuss compact groups. These can be handled with a very nice theory. Tools developed here can be used to study functions on spheres or other compact homogeneous spaces in terms of spherical functions. These functions behave well with respect to representations and invariant differential operators on these spaces. In dealing with this part of the subject, we present the notion of a Gelfand pair. The first semester course begins with an overview of Fourier series of periodic functions on the line; then goes on to Fourier analysis on Euclidean spaces. Material covered includes the Fourier transform, inversion formula, Plancherel theorem, the Paley-Wiener theorem, and extensions to distributions and algebras. The role of the Fourier transform in analysis of partial differential equations with constant coefficients is discussed.The Heisenberg group H is introduced and the intimate connection between classical Fourier analysis and harmonic analysis on H is developed. Along with group and analytic structure, the representation theory of H will be presented. This includes the Stone-von Neumann theorem, connections to quantum mechanics, the Plancherel theorem, the Schrödinger model, the equivalent Fock-Bargmann representation, twisted convolution, the strong Stone-von Neumann theorem and transforms of Schwartz functions as operators with smooth kernels. The large automorphism group of H is used to introduce semidirect products, compact Lie groups and semi-simple Lie groups. Similarities between harmonic analysis for these groups and H along with new subtleties introduced by the further non-commutativity of these groups will be discussed. The groups introduced this way may include SO(n), SU(n), SL(2,R), the Poincaré group, the Euclidean motion group and the symplectic group. The division between first and second semester material is fluid. We will review materials from topics covered in Course I needed for the second semester.

• MATH 7390-3: Advanced Topics in Probability Theory
• Instructor: Prof. Kuo .
• Prerequisite:
• References:
1. Kuo, H.-H.: Introductory Stochastic Integration. (In preparation)
2. Oksendal, B.: Stochastic Differential Equations. 5th edition,Springer, 2000
3. Kuo, H.-H.: Gaussian Measures in Banach Spaces, Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975
4. Kuo, H.-H.: White Noise Distribution Theory, CRC Press,1996

This course contains two parts: (1) basic theory of stochastic integration with applications to mathematical finance (which appeals to finance students and non-probability math graduate students), (2) advanced research topics for Ph.D. students (which I will outline in an overview for further independent study). Below are some items to be covered in this course:

1. Brownian motion
2. Construction of Brownian motion
3. Wiener integrals
4. Ito's integrals
5. Stochastic integrals for martingales
6. The Ito formula
7. Girsanov theorem
8. Stochastic differential equations
9. Arbitrage and option pricing
10. Black-Scholes analysis
11. Wiener-Ito decomposition theorem
12. White noise theory
• MATH 7520: Algebraic Topology
• Instructor: Prof. Litherland.
• Prerequisite: Math 7512 or the equivalent.
• Text: Marvin J. Greenberg and John R.Harper, Algebraic Toplogy, A First Course, Perseus Books, 1981.

The basic idea in algebraic topology is to associate to a topological space an algebraic object (e.g. an integer, a polynomial, a group, or a module over some ring), in such a way that homeomorphic spaces get assigned equivalent objects (e.g. equal integers, or isomorphic groups). The object assigned to a space is an invariant of the space, and provides a tool for distinguishing between topological spaces: if two spaces have inequivalent invariants, they are not homeomorphic. One such invariant of a space X, the fundamental group, is covered in Math 7512. In Math 7520 we deal with a family of invariants, the homology groups Hn(X) (n a non-negative integer). Very roughly, these groups count the numbers of holes of different dimensions in X. We should cover Part II of the text, which deals with the definitions, methods of calculation, and some applications of these invariants.

• MATH 7550: Differential Geometry
• Instructor: Prof. Fabec.
• Prerequisite: MATH 7200 and 7510 at least a senior-level undergraduate analysis course.
• Text: Foundations of Differential Manifolds and Lie Groups by Frank Warner, and posted class notes.

The course title is Differential Geometry and Topology, but the content is a basic introduction to manifolds. When one studies analysis on real n-space one is studying local phenomena; when one studies analysis on manifolds one is studying global phenomena. This subject is called global analysis. It is particularly useful for differential topology, differential equations, differential geometry, harmonic analysis, and mathematical physics. The prerequisites are advanced calculus, point set topology, basic group theory and linear algebra. The topics to be covered will be:

• Manifolds
• Vector Bundles
• Frobenius' Theorem
• Lie Derivative
• Differential Forms
• Integration

Other books of interest are:

1. Differential Geometry volume 1 by Michael Spivak This book is an excellent book and series to read. It gives long detailed mathematical and historical explanations, but does not make a particularly good textbook.
2. Differential and Riemannian Manifolds} by Serge Lange This book covers the infinite dimensional theory. Lange writes, "It is possible to lay down at no extra cost the foundations ... for manifolds modelled on Banach or Hilbert spaces."
3. Calculus on Manifolds by Michael Spivak This book was written as an advanced undergraduate textbook. It has few prerequisites, however, it uses less sophisticated ideas and it gives less insight.
• MATH 7590: Finite Type Invariants
• Instructor: Prof. Dasbach.
• Prerequisite: Some knowledge of Topology or Differential Geometry.
• Text: Sergei Chmutov, Sergei Duzhin: "Introduction to Vassiliev Knot Invariants" (preliminary version of a book, available online) and handouts.

Low dimensional topology in general and knot theory in particular is an exciting subpart of topology. E.g. applications are in the understanding of the structure of our space or in the understanding of human DNA. This course is concerned with invariants on knots that were first introduced by Vassiliev in the early 90s. The course is largely self-contained. We will give a review of the necessary background from topology and other fields. We will start with a quick trip through knot theory. We will go on to topics like Jones polynomials and Quantum invariants before we introduce Vassiliev's ideas and the combinatorics that grew out of it. We will also spend some time introducing algorithms used in computer programs like KNOTSCAPE that effectively deal with knots.

• MATH 7380: Introduction to Hilbert and Banach Space
• Instructor: Prof. Sundar
• Prerequisite: Math 7311 or Permission of Instructor
• Text: Class Lecture Notes

The first half of this course will cover basic Hilbert space theory. Topics include Bessel's inequality, orthonormal systems, Riesz representation theorem, Lax-Milgram Theorem, projections, operator (self-adjoint, normal, unitary, and compact) theory, elements of spectral theory, eigenvalue problems, and Fourier transforms. The second half of the course will highlight applications of the theory, and will be adjusted to the interests of the students. Topics could include integral and differential equations, partial differential equations, wavelets, and the calculus of variations.

## Fall 2003

• MATH 7200: Geometric and Abstract Algebra
• Instructor: Prof. Hurrelbrink.
• Prerequisite: Math 2085 Linear Algebra or equivalent
• Text: Hungerford: Algebra, Springer-Verlag Graduate Texts in Mathematics 73

Math 7200 is the basic course in Algebra. It prepares the students for many mathematical fields. We will cover selected topics from Chapters I through V in Hungerford's book.

• MATH 7211: Commutative Rings
• Prerequisite: Math 7200 and 7210.
• Text: Introduction to Commutative Algebra and Algebraic Geometry by E. Kunz, Birkhauser (1997), ISBN: 0-8176-3065-1

This course is an introduction to commutative algebra, the algebra which was motivated by problems in algebraic geometry and number theory. Thus, this course will serve as a foundational course for further study in either of these two disciplines. The goal of the text being used is to present many of the standard topics of commutative algebra as
they are needed in order to study a specific interesting problem, that of representing an algebraic variety (the zero set of finitely many polynomials) as the intersection of the smallest number of  hypersurfaces (zero set of a single polynomial) possible. Some of the specific topics to be studied include: Hilbert basis theorem, Hilbert's nullstellensatz, prime spectrum of a ring, dimension theory of rings and varieties, rings and modules of fractions, and the sheaf of regular functions on an algebraic variety, the local-global principle in commutative algebra, Krull's principal ideal theorem, and applications to dimension, regular local rings.

• MATH 7280: Algebraic Geometry
• Prerequisite: Math 7210
• Texts: (1) Klaus Hulek, Elementary Algebraic Geometry, AMS2003. Also (2) Kenji Ueno, Algebraic Geometry I: From Algebraic Varieties to Schemes, AMS 1999.

This course is an introduction to some of the basic concepts of algebraic geometry, with an emphasis on the geometry. Topics include:

• affine and projective varieties, smooth and singular points,dimension
• algebraic curves, (including: intersection multiplicity and Bezout’s theorem, resolution of singularities, genus, divisors and linear systems)
• special additional topics according to interest e.g.: a)sheaves and schemes, b) toric varieties, c) semi algebraic sets and real geometry.
• MATH 7290: Lie Algebras and Lie Groups
• Instructor: Prof. Hoffman.
• Prerequisite: Math 7200 and 7510
• Text: Lie Groups: Beyond an introduction, by Anthony Knapp Birkhaeuser Progress in Mathematics vol 140

Topics:

• Lie algebras, definitions.
• Lie groups, definition and examples.
• Lie theory - the relation between 1 and 2 above.
• Structure of complex semi-simple Lie algebras.
• Introduction to representation theory.

We will illustrate the general theory with examples principally drawn from the so-called classical groups. This course has minimal prerequisites - basically groups, rings, modules, fields, and elementary topology. However, this course will go deep fast, so that familiarity with such topics as manifolds, representations, covering spaces, etc., while not required, is desirable. At any rate, what we can cover will depend in part on the preparation and interests of the class.

• MATH 7311: Real Analysis I
• Instructor: Prof. Davidson
• Prerequisite: Math 4032 or the equivalent.
• Text: Real Analysis (3rd Edition) by H. L. Royden

This is our introductory course in real analysis. The core of the course is Part One of Royden's book. Topics include Lebesgue measure and integration on the real line, convergence theorems, functions of bounded variation and absolute continuity, differentiation, and the classical Banach spaces. As time permits we may consider abstract Banach spaces and/or abstract measure and integration. This course continues as Math 7312 in the spring semester.

• MATH 7350: Complex Variables
• Instructor: Prof. Neubrander
• Prerequisite: MATH 7311: Real Analysis I
• Text: Lecture notes; available in PDF format on a web site specifically maintained for this course.

This course is an introduction to some of the basic concepts and applications of complex analysis. The course is largely self-contained; i.e., many results and techniques from real analysis will be re-introduced and re-proved using the fundamental ideas and language of modern analysis. The following topics will be covered:Chapter I: Complex Numbers, Analytic Functions, The Complex Exponential, Cauchy-Riemann Theorem, Contour Integrals. Chapter II: Antiderivatives, Cauchy's Theorem, Cauchy's Integral Formula, Cauchy's Theorem for Chains, Principles of Linear Analysis, Cauchy's Theorem for Vector-Valued Analytic Functions, Power Series, Maximum Principle, Laurent's Series and Isolated Singularities, Residue Calculus. Chapter III: Dunford Functional Calculus, Systems of Linear ODEs, Laplace Transforms, Semigroups, Stability and Convergence for Numerical Approximations, Tauberian Theorems, Prime Number Theorem, Asymptotic Analysis and Formal Power Series.

• MATH 7360: Probability Theory
• Instructor: Prof. Sengupta, Office: 324 Lockett; 578-1607. Office hours by appointment. Homepage .
• Prerequisite: Math 7312 (measure and integration), or permission of the department.
• Text: Kiyosi Ito's Introduction to Probability Theory.

The course will build the basic apparatus of probability theory. We will study the fundamental results concerning probability measures and random variables. Topics will include limit theorems and elements of stochastic processes.

• MATH 7380-1: Stochastic Analysis & Applications to Mathematical Finance
• Instructor: Prof. Sundar
• Prerequisite: Math 4031 and Math 3355
• Text: Based on lecture notes given in class

The aim of the course is to provide a mathematically precise account of the arbitrage theory of financial derivatives. A self-contained treatment of stochastic differential equations and the Ito calculus will be presented, and will include the Feynman-Kac formula, and the Kolmogorov equations. Risk neutral valuation formulas and martingale measures will be introduced through Feynman-Kac representations. The course will cover pricing and hedging problems in complete as well as incomplete markets. Barrier options, options on dividend-paying assets, as well as currency markets will be studied.Interest rate theory will include short rate models and the Heath-Jarrow-Morton approach to forward rate models. A self-contained treatment of stochastic optimal control theory will be used to study optimal consumption/investment problems. The course will be geared towards concrete computations involving stochastic differential equations.

• MATH 7380-2: Mathematical Topics in Systems Theory
• Instructor: Prof. Malisoff
• Prerequisites: MATH 7311 (Real Analysis I), MATH 7312 (Measure and Integration), and MATH 7320 (Ordinary Differential Equations), or permission from the instructor.
• Text: Sontag, E.D., Mathematical Control Theory, Deterministic Finite-Dimensional Systems, Second Edition, Texts in Applied Mathematics Vol. 6, Springer-Verlag, New York, 1998 (ISBN: 0-387-98489-5). Additional material as appropriate from author's web site.

This course provides a basic introduction to the mathematics of finite-dimensional, continuous-time, deterministic control systems at the beginning graduate student level. The course is intended for PhD students in applied mathematics, and for engineering graduate students with a background in graduate real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for interdisciplinary research at the interface of applied mathematics and control engineering. This will be a rigorous, proof-oriented systems theory that goes beyond classical frequency-domain or more applied engineering courses. Emphasis will be placed on controllability and stabilization. Additional information about this course is available at http://www.math.lsu.edu/~malisoff/m7380-2.pdf .

• MATH 7390-1: Harmonic Analysis - I
• Instructor: Prof. Olafsson
• Prerequisite: Math 7311 (Real Analysis I) or equivalent.
• Text: Lecture notes by R. Fabec and G. Olafsson

The subject proper of harmonic analysis is to decompose functions into simpler functions. In the theory of differential equations that means to write an arbitrary functions as a sum or integral of eigenfunctions. If we have a symmetry group acting on the system, then we would like to write an arbitrary function as a sum of functions that transforms in a simple and controllable way under the symmetry group. The simplest example is the use of polar coordinates and radial functions for rotation symmetric equations. The course is an introduction into the basic theory of classical Fourier analysis. The main topics will be:

• Basic properties of Fourier series;
• Convergence of Fourier series;
• Function spaces on Rn, in particular the space of rapidly decreasing functions, Lp-spaces, and the space of compactly supported functions;
• Basic properties of the Fourier transform on Rn, inversion formula for rapidly decreasing functions;
• The L2-theory for the Fourier transform, in particular the Plancherel theorem;
• We include some application, which will be chosen depending on time available and the interests of the students. Possible topics include: distribution theory and Fourier transforms of distributions, differential equations, Sobolev spaces, the Fock-space and Hermite polynomials, introduction to harmonic analysis on the Heisenberg group, the short time Fourier transform, and the continuous wavelet transform.

The class will be continued by R. Fabec in the spring 2004. We can view Rn as a set or as a manifold. But we can also view Rn as an abelian group. In that sense Rn is a part of abelian harmonic analysis. The simplest examples of  non-abelian harmonic analysis are harmonic analysis on the (ax+b)-group, which is the basic tool in the wavelet theory, the Heisenberg group, classical linear Lie groups.The second part discusses the basic properties of topological groups and homogeneous spaces. Then representations of topological groups will be discussed and several examples worked out.

• MATH 7390-2: Applied Analysis in the Materials Sciences
• Instructor: Prof. Lipton
• Prerequisite: Math 7311 (Real Analysis I) or equivalent.
• Text: Partial Differential Equations, by L. C. Evans, and Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn.

Motivated by applications in physics and engineering it is desirable to identify heterogeneous materials with optimal properties.Mathematically this is a problem of distributed parameter optimal control. Here the state equation is a partial differential equation and the control variable is the Lebesgue measurable coefficient in the principal part of the differential operator. The preliminary part of the course provides a self contained introduction to the theory of Sobolev Spaces and weak solutions of elliptic partial differential equations. For this part we will follow chapters 5 and 6 of the book Partial Differential Equations, by L. C. Evans. The primary part of the course will develop the Calculus of Variations as it applies to the optimal design of heterogeneous media. For this part we will follow chapters 3, 6, and 7 of the book Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn.

• MATH 7490: Discrete Optimization
• Instructor: Prof. Ding
• Prerequisite: Linear Algebra
• Text: useful but not required): Combinatorial Optimization, by W. Cook, W. Cunningham, W. Pulleyblank, and A. Schrijver.

In this course, students will be motivated by real world problems. They will learn how to solve fundamental problems indiscrete optimization. They will also be introduced to advanced techniques that deal with computationally hard problems. A more detailed course description can be found at http://www.math.lsu.edu/~ding/sy7490.html

• MATH 7510: Topology I
• Instructor: Prof. Perlis
• Prerequisite: advanced calculus, such as Math 4031
• Text: We will use notes of Prof. Jimmie Lawson--no need to buy a textbook!

In the early 1900's, great effort was made to carry over the powerful notion of continuity of functions on subsets of Euclidean space Rn to functions on more general sets. This generalization is called point-set topology and is the subject of this course. Prof. Lawson's notes contain definitions, exercises, and statements of theorems. Your job as a student is to learn the definitions and to supply proofs of the exercises and theorems. Class will be highly interactive. Only rarely will I lecture; most of the time students will present their solutions on the board, followed by a discussion. Active class participation is required! The material itself covers standard point-set topology. Some of the topics we will discuss are: topological space, open set, basis, metric space, product topology, closure, partial order, net, Hausdorff space, homeomorphism, relative topology, quotient maps, connectedness, and compactness.

• MATH 7590: Geometric Topology
• Instructor: Prof. Cohen,Tel.: (225) 578-1576
• Prerequisite: MATH 7512 Topology II
• References: Much of the subject matter may be found in Moran"The Mathematical Theory of Knots and Braids." Other potential sources include Birman, Hansen, Orlik-Terao, and Prasolov-Sossinsky.

The course will be a broad survey of selected current research areas in topology within our Department, such as Braid Theory, Arrangements, 3-manifolds, Complements of Knots and Links, and Mapping Class Groups of 2-manifolds. Concepts from geometric topology/group theory will be explored in the contexts of these subjects. These include aspects of infinite group theory such as the lower central series, word and conjugacy problems, Fox calculus, etc.. Other (related) topics include braid group representations and classical knot and link invariants (e.g., Alexander polynomials). The course will also provide an introduction to configuration spaces and selected generalizations, and connections to subjects such as combinatorics and the theory of reflection groups.

• MATH 7690: Riemannian Geometry
• Instructor: Prof. Lawson
• Prerequisite: Math 7550 (Differential Geometry) or equivalent.
• Text: An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby

A Riemannian manifold is a differentiable manifold equipped with a special kind of metric called a Riemannian metric. The presence of a Riemannian metric on a differentiable manifold allows the introduction and treatment of many classical geometric notions: distance, geodesics (an appropriate generalization of lines), angles, etc. Indeed an important class of Riemannian manifolds, the class of symmetric manifolds, includes as special cases all the classical geometries: Euclidean, non-Euclidean, projective, etc. Thus one might say that Riemannian geometry provides a modern, elegant, comprehensive view of classical geometry, although its theory goes much further. This course provides an introduction into the basic theory of Riemannian manifolds and their associated geometry. The main topics include:

• The geometry of space curves
• Covariant differentiation
• Geodesic curves and their properties
• The Riemannian connection and the curvature tensor
• Symmetric Riemannian manifolds
• Gaussian and mean curvature
• Manifolds of constant curvature

## Spring 2004

• MATH 7210: Algebra - I
• Instructor: Prof. Perlis.
• Prerequisite: Math 7200 and a good undergraduate theorem-proving abstract algebra course.
• Text: Hungerford, Algebra.

This is primarily a course on Field Theory and Galois Theory. It is expected that the students are familiar with the theory of finite groups, including the Sylow theorems. We will begin with Chapter V in Hungerford's book. Galois theory has its roots in the attempt to generalize the familiar quadratic formula for finding the complex solutions of a polynomial equation ax2 + bx + c = 0 of degree two to polynomials of higher degree. In 1798 Ruffini published an incomplete proof of the assertion that in general there is no algebraic formula (involving only finitely-many sums, products, quotients, and nth roots) to solve a polynomial equation of degree five or greater. In 1826, at the age of 24, Abel published a proof of Ruffini's statement. By today's standards, Abel's proof is a bit fuzzy. Galois theory gives an elegant way to analyze the problem, and is so powerful that the central ideas if not the details have carried over to algebraic topology, geometry, and even parts of differential equations. If you have questions, please contact R. Perlis using the link above under Instructor.

• MATH 7280: Elliptic Curves and Modular Forms
• Instructor: Prof. Verrill.
• Prerequisite: (1) Undergraduate complex analysis (you should be comfortable with meromorphic and analytic functions; (2) Groups, Galois groups (definitions and examples).
• Text: Introduction to the arithmetic theory of automorphic forms, G. Shimura. (Strongly recommended, but not required; used as reference material; we will cover roughly the first 4 chapters; some topics not covered in the text will be presented as handouts.)

This course is primarily an introduction to the theory of modular forms, with applications to elliptic curves. A modular form is an analytic function on the upper half complex plane having many symmetries, and an expansion of the form f(z) = a0 + a1 q + a2 q2 + a3 q3 + .... where q=exp(2pi iz). An elliptic curve is (usually) given by an equation y2 = x3 + ax + b (and has only an historical connection to ellipses). The modularity of elliptic curves is the surprising fact that for every elliptic curve (with a, b rationals), there is a corresponding modular form such that the number of points on the curve, mod p, for all but finitely many primes p, is p minus the coefficient ap of the modular form. Some of the topics we will cover are: Congruence subgroups, fundamental domains for subgroups of PSL2(Z), Hecke operators, the group law of an elliptic curves, p-adic numbers, and Galois representations associated with elliptic curves. A goal is to be able to sketch an outline of the ingredients that go into how the modularity of elliptic curves is used to prove Fermat's last theorem. We will also discuss computational problems, such as how to compute Hecke eigen modular forms, and fundamental domains. These will be illustrated with the Magma computer algebra package. Questions? Please contact me at the email address linked to my name above, and visit the web-site.

• MATH 7290: Linear Algebraic Groups
• Instructor: Prof. Morales.
• Prerequisite: Algebra I and II (MATH 7210, 7211) or equivalent
• Text: Class lecture notes. *Reference books (strongly recommended, but not required): (1) "Linear Algebraic Groups" by James E. Humphreys, Springer-Verlag 1999. (2) "Linear Algebraic Groups" by T. A. Springer, Birkhauser 1998.

Linear (or affine) algebraic groups are affine algebraic varieties endowed with a group structure compatible with the structure of variety. Examples of such objects are the classical groups GLn, SLn, SOn and more generally the groups of automorphisms of vector spaces equipped with a tensor. Algebraic groups are the algebraic counterpart of Lie groups. When the ground field is the field of complex numbers C, the two theories have a large intersection. The goal of this course is to give an introductory overview of this beautiful theory. Background material from algebraic geometry will be developed as we go.

Outline:

1. Some Algebraic Geometry
2. Linear Algebraic groups
3. Derivations, differentials, Lie algebras
4. Homogeneous Spaces
5. Semi-simple and Unipotent Elements
6. Solvable Groups
7. Borel Subgroups
8. Reductive Groups
9. Semi-simple Groups
• MATH 7312: Measure and integration
• Instructor: Prof. Sundar
• Prerequisite: Math 7311
• Text: Real and Complex Analysis by Walter Rudin

The course is a sequel to Real Analysis I, and introduces measure theory, elements of Hilbert and Banach spaces, and analysis of functions of several variables. The syllabus is as follows:

1. Chapter I: Abstract measures, integration, convergence theorems, Lp-spaces and inequalities.
2. Chapter II: Hilbert and Banach spaces, consequences of Baire's theorem, Hahn-Banach theorem and applications.
3. Chapter III: Signed measures, Radon-Nikodym theorem, decomposition theorems, duality.
4. Chapter IV: Product measures, Fubini and Tonelli theorems.
5. Chapter V: Differentiation, implicit and inverse function theorems, change of variables in integrals.
6. Chapter VI: Sobolev spaces, Embedding theorems.
• MATH 7320: Ordinary Differential Equations
• Prerequisite: Either Math 7311 or else both Math 4031 and Math 4032.
• Text: W.Walter, Ordinary Differential Equations, Springer, 1998.

This is a first graduate course in ordinary differential equations. The following topics will be covered. Existence and uniqueness results for first order equations and systems, and for higher order equations. Linear equations and systems. Complex linear systems. Boundary value and eigenvalue problems. Stability.

• MATH 7380-1: Partial Differential Equations
• Instructor: Prof. Shipman.
• Prerequisite: A first course in graduate analysis, such as Math 7311; elementary exposure to ordinary differential equations.
• Text: Partial Differential Equations, by L. C. Evans (AMS Graduate Studies in Mathematics, Vol. 19)

This course is an introduction to mathematical theories of partial differential equations and will be complemented by physical motivations and applications. The course will be mostly self-contained, presuming a solid foundation in basic analysis and exposure to basic (ordinary) differential equations.

1. The basic linear equations will be covered in depth as well as methods of solution such as transforms and fundamental solutions. Basic nonlinear theory will include the method of characteristics, conservation laws, and existence and uniqueness theory. Other major topics include the calculus of variations and elliptic PDEs.
2. Some special topics will be studied in some detail, such as nonlinear waves, nonlinear spectral transforms, asymptotic methods, or the mathematics of electrodynamics.
3. An attempt will be made to give a broad survey of topics in PDEs by discussing briefly a sampling of topics. These may include evolution equations, operator and semigroup methods, nonlinear hyperbolic equations, boundary-integral methods, energy methods, numerical methods, integrable systems, elliptic regularity theory, or pseudo-differential operators.
• MATH 7380-2: Integral and Functional Equations
• Instructor: Prof. Antipov.
• Prerequisite: MATH 4036 Complex variables
• Text: Lecture notes and F.D.Gakhov, Boundary value problems.

Aims: To introduce students to singular integral and integro-differential equations and functional-difference equations which can be tackled using methods for boundary-value problems of the theory of analytic functions.

Objectives: Students should know a range of analytical methods for solving model problems in applied mathematics including fracture mechanics and diffraction theory.
Content: The Cauchy integral, the Riemann-Hilbert problem on an axis and an open curve, the Wiener-Hopf method, the Carleman problem, matrix factorization, elements of elliptic functions and elliptic surfaces, model problems of fracture, elasticity and diffraction of electromagnetic and acoustic waves.

• MATH 7380-3: Numerical Methods & Applications
• Instructor: Prof. Bourdin.
• Prerequisite: Math 7311 or equivalent; Basic knowledge of harmonic analysis, PDE, or numerical analysis is a plus, but not a requirement. Basic programming skills in C, fortran, pascal , or other "low" level language.
• Text: Lecture notes, online resources provided during the class.

This is an introduction class to the numerical analysis and implementation of Partial Differential Equations. Its focus is mainly on the Finite Element Method for elliptic problems. In a first part, we will briefly survey some theoretical results for PDEs (variational formulation, Galerkin method...). Then, we will discuss multi-dimensional interpolation, differentiation and integration, which will lead us to the core of the finite element method. Time permitting, we will either cover the basic of domain decomposition and parallel processing, and/or iterative methods for solving large sparse linear systems of equations. Numerical projects will represent a major part of the work (and of the grade). It is my intention to make this class suited to students with a theoretical background, willing to learn about advanced numerics, as well as to those already familiar with numerics, and interested in a more theoretical approach. Mathematics, computer science and engineering students are welcome here!

• MATH 7390-1:  Harmonic Analysis - II: The Classical Groups
• Instructor: Prof. Fabec
• Prerequisite: Math 7311 and 7312.
• Text: will be provided on-line.

This is the second in a sequence of courses on harmonic analysis. The sequence starts with classical Fourier series and the Fourier integral, extends these objects to distributions and explores their uses in solving differential equations. Next the Fourier integral is used to study functions and function spaces on the Heisenberg group. This group is fundamentally important for both its importance in quantum mechanics where its representation theory encodes the Heisenberg uncertainty principle and as an example of the difficulties arising when one attempts to do harmonic analysis on groups which are no longer commutative. Here one must deal with infinite dimensional representations and from this one can be familiarized with this theory. We then discuss compact groups. These can be handled with a very nice theory. Tools developed here can be used to study functions on spheres or other compact homogeneous spaces in terms of spherical functions. These functions behave well with respect to representations and invariant differential operators on these spaces. In dealing with this part of the subject, we present the notion of a Gelfand pair. The first semester course begins with an overview of Fourier series of periodic functions on the line; then goes on to Fourier analysis on Euclidean spaces. Material covered includes the Fourier transform, inversion formula, Plancherel theorem, the Paley-Wiener theorem, and extensions to distributions and algebras. The role of the Fourier transform in analysis of partial differential equations with constant coefficients is discussed.The Heisenberg group H is introduced and the intimate connection between classical Fourier analysis and harmonic analysis on H is developed. Along with group and analytic structure, the representation theory of H will be presented. This includes the Stone-von Neumann theorem, connections to quantum mechanics, the Plancherel theorem, the Schrödinger model, the equivalent Fock-Bargmann representation, twisted convolution, the strong Stone-von Neumann theorem and transforms of Schwartz functions as operators with smooth kernels. The large automorphism group of H is used to introduce semidirect products, compact Lie groups and semi-simple Lie groups. Similarities between harmonic analysis for these groups and H along with new subtleties introduced by the further non-commutativity of these groups will be discussed. The groups introduced this way may include SO(n), SU(n), SL(2,R), the Poincaré group, the Euclidean motion group and the symplectic group. The division between first and second semester material is fluid. We will review materials from topics covered in Course I needed for the second semester.

• MATH 7390-3: Advanced Topics in Probability Theory
• Instructor: Prof. Kuo .
• Prerequisite:
• References:
1. Kuo, H.-H.: Introductory Stochastic Integration. (In preparation)
2. Oksendal, B.: Stochastic Differential Equations. 5th edition,Springer, 2000
3. Kuo, H.-H.: Gaussian Measures in Banach Spaces, Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975
4. Kuo, H.-H.: White Noise Distribution Theory, CRC Press,1996

This course contains two parts: (1) basic theory of stochastic integration with applications to mathematical finance (which appeals to finance students and non-probability math graduate students), (2) advanced research topics for Ph.D. students (which I will outline in an overview for further independent study). Below are some items to be covered in this course:

1. Brownian motion
2. Construction of Brownian motion
3. Wiener integrals
4. Ito's integrals
5. Stochastic integrals for martingales
6. The Ito formula
7. Girsanov theorem
8. Stochastic differential equations
9. Arbitrage and option pricing
10. Black-Scholes analysis
11. Wiener-Ito decomposition theorem
12. White noise theory
• MATH 7520: Algebraic Topology
• Instructor: Prof. Litherland.
• Prerequisite: Math 7512 or the equivalent.
• Text: Marvin J. Greenberg and John R.Harper, Algebraic Toplogy, A First Course, Perseus Books, 1981.

The basic idea in algebraic topology is to associate to a topological space an algebraic object (e.g. an integer, a polynomial, a group, or a module over some ring), in such a way that homeomorphic spaces get assigned equivalent objects (e.g. equal integers, or isomorphic groups). The object assigned to a space is an invariant of the space, and provides a tool for distinguishing between topological spaces: if two spaces have inequivalent invariants, they are not homeomorphic. One such invariant of a space X, the fundamental group, is covered in Math 7512. In Math 7520 we deal with a family of invariants, the homology groups Hn(X) (n a non-negative integer). Very roughly, these groups count the numbers of holes of different dimensions in X. We should cover Part II of the text, which deals with the definitions, methods of calculation, and some applications of these invariants.

• MATH 7550: Differential Geometry
• Instructor: Prof. Fabec.
• Prerequisite: MATH 7200 and 7510 at least a senior-level undergraduate analysis course.
• Text: Foundations of Differential Manifolds and Lie Groups by Frank Warner, and posted class notes.

The course title is Differential Geometry and Topology, but the content is a basic introduction to manifolds. When one studies analysis on real n-space one is studying local phenomena; when one studies analysis on manifolds one is studying global phenomena. This subject is called global analysis. It is particularly useful for differential topology, differential equations, differential geometry, harmonic analysis, and mathematical physics. The prerequisites are advanced calculus, point set topology, basic group theory and linear algebra. The topics to be covered will be:

• Manifolds
• Vector Bundles
• Frobenius' Theorem
• Lie Derivative
• Differential Forms
• Integration

Other books of interest are:

1. Differential Geometry volume 1 by Michael Spivak This book is an excellent book and series to read. It gives long detailed mathematical and historical explanations, but does not make a particularly good textbook.
2. Differential and Riemannian Manifolds} by Serge Lange This book covers the infinite dimensional theory. Lange writes, "It is possible to lay down at no extra cost the foundations ... for manifolds modelled on Banach or Hilbert spaces."
3. Calculus on Manifolds by Michael Spivak This book was written as an advanced undergraduate textbook. It has few prerequisites, however, it uses less sophisticated ideas and it gives less insight.
• MATH 7590: Finite Type Invariants
• Instructor: Prof. Dasbach.
• Prerequisite: Some knowledge of Topology or Differential Geometry.
• Text: Sergei Chmutov, Sergei Duzhin: "Introduction to Vassiliev Knot Invariants" (preliminary version of a book, available online) and handouts.

Low dimensional topology in general and knot theory in particular is an exciting subpart of topology. E.g. applications are in the understanding of the structure of our space or in the understanding of human DNA. This course is concerned with invariants on knots that were first introduced by Vassiliev in the early 90s. The course is largely self-contained. We will give a review of the necessary background from topology and other fields. We will start with a quick trip through knot theory. We will go on to topics like Jones polynomials and Quantum invariants before we introduce Vassiliev's ideas and the combinatorics that grew out of it. We will also spend some time introducing algorithms used in computer programs like KNOTSCAPE that effectively deal with knots.

The first half of this course will cover basic Hilbert space theory. Topics include Bessel's inequality, orthonormal systems, Riesz representation theorem, Lax-Milgram Theorem, projections, operator (self-adjoint, normal, unitary, and compact) theory, elements of spectral theory, eigenvalue problems, and Fourier transforms. The second half of the course will highlight applications of the theory, and will be adjusted to the interests of the students. Topics could include integral and differential equations, partial differential equations, wavelets, and the calculus of variations.

## Fall 2003

• MATH 7200: Geometric and Abstract Algebra
• Instructor: Prof. Hurrelbrink.
• Prerequisite: Math 2085 Linear Algebra or equivalent
• Text: Hungerford: Algebra, Springer-Verlag Graduate Texts in Mathematics 73

Math 7200 is the basic course in Algebra. It prepares the students for many mathematical fields. We will cover selected topics from Chapters I through V in Hungerford's book.

• MATH 7211: Commutative Rings
• Prerequisite: Math 7200 and 7210.
• Text: Introduction to Commutative Algebra and Algebraic Geometry by E. Kunz, Birkhauser (1997), ISBN: 0-8176-3065-1

This course is an introduction to commutative algebra, the algebra which was motivated by problems in algebraic geometry and number theory. Thus, this course will serve as a foundational course for further study in either of these two disciplines. The goal of the text being used is to present many of the standard topics of commutative algebra as
they are needed in order to study a specific interesting problem, that of representing an algebraic variety (the zero set of finitely many polynomials) as the intersection of the smallest number of  hypersurfaces (zero set of a single polynomial) possible. Some of the specific topics to be studied include: Hilbert basis theorem, Hilbert's nullstellensatz, prime spectrum of a ring, dimension theory of rings and varieties, rings and modules of fractions, and the sheaf of regular functions on an algebraic variety, the local-global principle in commutative algebra, Krull's principal ideal theorem, and applications to dimension, regular local rings.

• MATH 7280: Algebraic Geometry
• Prerequisite: Math 7210
• Texts: (1) Klaus Hulek, Elementary Algebraic Geometry, AMS2003. Also (2) Kenji Ueno, Algebraic Geometry I: From Algebraic Varieties to Schemes, AMS 1999.

This course is an introduction to some of the basic concepts of algebraic geometry, with an emphasis on the geometry. Topics include:

• affine and projective varieties, smooth and singular points,dimension
• algebraic curves, (including: intersection multiplicity and Bezout’s theorem, resolution of singularities, genus, divisors and linear systems)
• special additional topics according to interest e.g.: a)sheaves and schemes, b) toric varieties, c) semi algebraic sets and real geometry.
• MATH 7290: Lie Algebras and Lie Groups
• Instructor: Prof. Hoffman.
• Prerequisite: Math 7200 and 7510
• Text: Lie Groups: Beyond an introduction, by Anthony Knapp Birkhaeuser Progress in Mathematics vol 140

Topics:

• Lie algebras, definitions.
• Lie groups, definition and examples.
• Lie theory - the relation between 1 and 2 above.
• Structure of complex semi-simple Lie algebras.
• Introduction to representation theory.

We will illustrate the general theory with examples principally drawn from the so-called classical groups. This course has minimal prerequisites - basically groups, rings, modules, fields, and elementary topology. However, this course will go deep fast, so that familiarity with such topics as manifolds, representations, covering spaces, etc., while not required, is desirable. At any rate, what we can cover will depend in part on the preparation and interests of the class.

• MATH 7311: Real Analysis I
• Instructor: Prof. Davidson
• Prerequisite: Math 4032 or the equivalent.
• Text: Real Analysis (3rd Edition) by H. L. Royden

This is our introductory course in real analysis. The core of the course is Part One of Royden's book. Topics include Lebesgue measure and integration on the real line, convergence theorems, functions of bounded variation and absolute continuity, differentiation, and the classical Banach spaces. As time permits we may consider abstract Banach spaces and/or abstract measure and integration. This course continues as Math 7312 in the spring semester.

• MATH 7350: Complex Variables
• Instructor: Prof. Neubrander
• Prerequisite: MATH 7311: Real Analysis I
• Text: Lecture notes; available in PDF format on a web site specifically maintained for this course.

This course is an introduction to some of the basic concepts and applications of complex analysis. The course is largely self-contained; i.e., many results and techniques from real analysis will be re-introduced and re-proved using the fundamental ideas and language of modern analysis. The following topics will be covered:Chapter I: Complex Numbers, Analytic Functions, The Complex Exponential, Cauchy-Riemann Theorem, Contour Integrals. Chapter II: Antiderivatives, Cauchy's Theorem, Cauchy's Integral Formula, Cauchy's Theorem for Chains, Principles of Linear Analysis, Cauchy's Theorem for Vector-Valued Analytic Functions, Power Series, Maximum Principle, Laurent's Series and Isolated Singularities, Residue Calculus. Chapter III: Dunford Functional Calculus, Systems of Linear ODEs, Laplace Transforms, Semigroups, Stability and Convergence for Numerical Approximations, Tauberian Theorems, Prime Number Theorem, Asymptotic Analysis and Formal Power Series.

• MATH 7360: Probability Theory
• Instructor: Prof. Sengupta, Office: 324 Lockett; 578-1607. Office hours by appointment. Homepage .
• Prerequisite: Math 7312 (measure and integration), or permission of the department.
• Text: Kiyosi Ito's Introduction to Probability Theory.

The course will build the basic apparatus of probability theory. We will study the fundamental results concerning probability measures and random variables. Topics will include limit theorems and elements of stochastic processes.

• MATH 7380-1: Stochastic Analysis & Applications to Mathematical Finance
• Instructor: Prof. Sundar
• Prerequisite: Math 4031 and Math 3355
• Text: Based on lecture notes given in class

The aim of the course is to provide a mathematically precise account of the arbitrage theory of financial derivatives. A self-contained treatment of stochastic differential equations and the Ito calculus will be presented, and will include the Feynman-Kac formula, and the Kolmogorov equations. Risk neutral valuation formulas and martingale measures will be introduced through Feynman-Kac representations. The course will cover pricing and hedging problems in complete as well as incomplete markets. Barrier options, options on dividend-paying assets, as well as currency markets will be studied.Interest rate theory will include short rate models and the Heath-Jarrow-Morton approach to forward rate models. A self-contained treatment of stochastic optimal control theory will be used to study optimal consumption/investment problems. The course will be geared towards concrete computations involving stochastic differential equations.

• MATH 7380-2: Mathematical Topics in Systems Theory
• Instructor: Prof. Malisoff
• Prerequisites: MATH 7311 (Real Analysis I), MATH 7312 (Measure and Integration), and MATH 7320 (Ordinary Differential Equations), or permission from the instructor.
• Text: Sontag, E.D., Mathematical Control Theory, Deterministic Finite-Dimensional Systems, Second Edition, Texts in Applied Mathematics Vol. 6, Springer-Verlag, New York, 1998 (ISBN: 0-387-98489-5). Additional material as appropriate from author's web site.

This course provides a basic introduction to the mathematics of finite-dimensional, continuous-time, deterministic control systems at the beginning graduate student level. The course is intended for PhD students in applied mathematics, and for engineering graduate students with a background in graduate real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for interdisciplinary research at the interface of applied mathematics and control engineering. This will be a rigorous, proof-oriented systems theory that goes beyond classical frequency-domain or more applied engineering courses. Emphasis will be placed on controllability and stabilization. Additional information about this course is available at http://www.math.lsu.edu/~malisoff/m7380-2.pdf .

• MATH 7390-1: Harmonic Analysis - I
• Instructor: Prof. Olafsson
• Prerequisite: Math 7311 (Real Analysis I) or equivalent.
• Text: Lecture notes by R. Fabec and G. Olafsson

The subject proper of harmonic analysis is to decompose functions into simpler functions. In the theory of differential equations that means to write an arbitrary functions as a sum or integral of eigenfunctions. If we have a symmetry group acting on the system, then we would like to write an arbitrary function as a sum of functions that transforms in a simple and controllable way under the symmetry group. The simplest example is the use of polar coordinates and radial functions for rotation symmetric equations. The course is an introduction into the basic theory of classical Fourier analysis. The main topics will be:

• Basic properties of Fourier series;
• Convergence of Fourier series;
• Function spaces on Rn, in particular the space of rapidly decreasing functions, Lp-spaces, and the space of compactly supported functions;
• Basic properties of the Fourier transform on Rn, inversion formula for rapidly decreasing functions;
• The L2-theory for the Fourier transform, in particular the Plancherel theorem;
• We include some application, which will be chosen depending on time available and the interests of the students. Possible topics include: distribution theory and Fourier transforms of distributions, differential equations, Sobolev spaces, the Fock-space and Hermite polynomials, introduction to harmonic analysis on the Heisenberg group, the short time Fourier transform, and the continuous wavelet transform.

The class will be continued by R. Fabec in the spring 2004. We can view Rn as a set or as a manifold. But we can also view Rn as an abelian group. In that sense Rn is a part of abelian harmonic analysis. The simplest examples of  non-abelian harmonic analysis are harmonic analysis on the (ax+b)-group, which is the basic tool in the wavelet theory, the Heisenberg group, classical linear Lie groups.The second part discusses the basic properties of topological groups and homogeneous spaces. Then representations of topological groups will be discussed and several examples worked out.

• MATH 7390-2: Applied Analysis in the Materials Sciences
• Instructor: Prof. Lipton
• Prerequisite: Math 7311 (Real Analysis I) or equivalent.
• Text: Partial Differential Equations, by L. C. Evans, and Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn.

Motivated by applications in physics and engineering it is desirable to identify heterogeneous materials with optimal properties.Mathematically this is a problem of distributed parameter optimal control. Here the state equation is a partial differential equation and the control variable is the Lebesgue measurable coefficient in the principal part of the differential operator. The preliminary part of the course provides a self contained introduction to the theory of Sobolev Spaces and weak solutions of elliptic partial differential equations. For this part we will follow chapters 5 and 6 of the book Partial Differential Equations, by L. C. Evans. The primary part of the course will develop the Calculus of Variations as it applies to the optimal design of heterogeneous media. For this part we will follow chapters 3, 6, and 7 of the book Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn.

• MATH 7490: Discrete Optimization
• Instructor: Prof. Ding
• Prerequisite: Linear Algebra
• Text: useful but not required): Combinatorial Optimization, by W. Cook, W. Cunningham, W. Pulleyblank, and A. Schrijver.

In this course, students will be motivated by real world problems. They will learn how to solve fundamental problems indiscrete optimization. They will also be introduced to advanced techniques that deal with computationally hard problems. A more detailed course description can be found at http://www.math.lsu.edu/~ding/sy7490.html

• MATH 7510: Topology I
• Instructor: Prof. Perlis
• Prerequisite: advanced calculus, such as Math 4031
• Text: We will use notes of Prof. Jimmie Lawson--no need to buy a textbook!

In the early 1900's, great effort was made to carry over the powerful notion of continuity of functions on subsets of Euclidean space Rn to functions on more general sets. This generalization is called point-set topology and is the subject of this course. Prof. Lawson's notes contain definitions, exercises, and statements of theorems. Your job as a student is to learn the definitions and to supply proofs of the exercises and theorems. Class will be highly interactive. Only rarely will I lecture; most of the time students will present their solutions on the board, followed by a discussion. Active class participation is required! The material itself covers standard point-set topology. Some of the topics we will discuss are: topological space, open set, basis, metric space, product topology, closure, partial order, net, Hausdorff space, homeomorphism, relative topology, quotient maps, connectedness, and compactness.

• MATH 7590: Geometric Topology
• Instructor: Prof. Cohen,Tel.: (225) 578-1576
• Prerequisite: MATH 7512 Topology II
• References: Much of the subject matter may be found in Moran"The Mathematical Theory of Knots and Braids." Other potential sources include Birman, Hansen, Orlik-Terao, and Prasolov-Sossinsky.

The course will be a broad survey of selected current research areas in topology within our Department, such as Braid Theory, Arrangements, 3-manifolds, Complements of Knots and Links, and Mapping Class Groups of 2-manifolds. Concepts from geometric topology/group theory will be explored in the contexts of these subjects. These include aspects of infinite group theory such as the lower central series, word and conjugacy problems, Fox calculus, etc.. Other (related) topics include braid group representations and classical knot and link invariants (e.g., Alexander polynomials). The course will also provide an introduction to configuration spaces and selected generalizations, and connections to subjects such as combinatorics and the theory of reflection groups.

• MATH 7690: Riemannian Geometry
• Instructor: Prof. Lawson
• Prerequisite: Math 7550 (Differential Geometry) or equivalent.
• Text: An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby

A Riemannian manifold is a differentiable manifold equipped with a special kind of metric called a Riemannian metric. The presence of a Riemannian metric on a differentiable manifold allows the introduction and treatment of many classical geometric notions: distance, geodesics (an appropriate generalization of lines), angles, etc. Indeed an important class of Riemannian manifolds, the class of symmetric manifolds, includes as special cases all the classical geometries: Euclidean, non-Euclidean, projective, etc. Thus one might say that Riemannian geometry provides a modern, elegant, comprehensive view of classical geometry, although its theory goes much further. This course provides an introduction into the basic theory of Riemannian manifolds and their associated geometry. The main topics include:

• The geometry of space curves
• Covariant differentiation
• Geodesic curves and their properties
• The Riemannian connection and the curvature tensor
• Symmetric Riemannian manifolds
• Gaussian and mean curvature
• Manifolds of constant curvature

## Spring 2004

• MATH 7210: Algebra - I
• Instructor: Prof. Perlis.
• Prerequisite: Math 7200 and a good undergraduate theorem-proving abstract algebra course.
• Text: Hungerford, Algebra.

This is primarily a course on Field Theory and Galois Theory. It is expected that the students are familiar with the theory of finite groups, including the Sylow theorems. We will begin with Chapter V in Hungerford's book. Galois theory has its roots in the attempt to generalize the familiar quadratic formula for finding the complex solutions of a polynomial equation ax2 + bx + c = 0 of degree two to polynomials of higher degree. In 1798 Ruffini published an incomplete proof of the assertion that in general there is no algebraic formula (involving only finitely-many sums, products, quotients, and nth roots) to solve a polynomial equation of degree five or greater. In 1826, at the age of 24, Abel published a proof of Ruffini's statement. By today's standards, Abel's proof is a bit fuzzy. Galois theory gives an elegant way to analyze the problem, and is so powerful that the central ideas if not the details have carried over to algebraic topology, geometry, and even parts of differential equations. If you have questions, please contact R. Perlis using the link above under Instructor.

• MATH 7280: Elliptic Curves and Modular Forms
• Instructor: Prof. Verrill.
• Prerequisite: (1) Undergraduate complex analysis (you should be comfortable with meromorphic and analytic functions; (2) Groups, Galois groups (definitions and examples).
• Text: Introduction to the arithmetic theory of automorphic forms, G. Shimura. (Strongly recommended, but not required; used as reference material; we will cover roughly the first 4 chapters; some topics not covered in the text will be presented as handouts.)

This course is primarily an introduction to the theory of modular forms, with applications to elliptic curves. A modular form is an analytic function on the upper half complex plane having many symmetries, and an expansion of the form f(z) = a0 + a1 q + a2 q2 + a3 q3 + .... where q=exp(2pi iz). An elliptic curve is (usually) given by an equation y2 = x3 + ax + b (and has only an historical connection to ellipses). The modularity of elliptic curves is the surprising fact that for every elliptic curve (with a, b rationals), there is a corresponding modular form such that the number of points on the curve, mod p, for all but finitely many primes p, is p minus the coefficient ap of the modular form. Some of the topics we will cover are: Congruence subgroups, fundamental domains for subgroups of PSL2(Z), Hecke operators, the group law of an elliptic curves, p-adic numbers, and Galois representations associated with elliptic curves. A goal is to be able to sketch an outline of the ingredients that go into how the modularity of elliptic curves is used to prove Fermat's last theorem. We will also discuss computational problems, such as how to compute Hecke eigen modular forms, and fundamental domains. These will be illustrated with the Magma computer algebra package. Questions? Please contact me at the email address linked to my name above, and visit the web-site.

• MATH 7290: Linear Algebraic Groups
• Instructor: Prof. Morales.
• Prerequisite: Algebra I and II (MATH 7210, 7211) or equivalent
• Text: Class lecture notes. *Reference books (strongly recommended, but not required): (1) "Linear Algebraic Groups" by James E. Humphreys, Springer-Verlag 1999. (2) "Linear Algebraic Groups" by T. A. Springer, Birkhauser 1998.

Linear (or affine) algebraic groups are affine algebraic varieties endowed with a group structure compatible with the structure of variety. Examples of such objects are the classical groups GLn, SLn, SOn and more generally the groups of automorphisms of vector spaces equipped with a tensor. Algebraic groups are the algebraic counterpart of Lie groups. When the ground field is the field of complex numbers C, the two theories have a large intersection. The goal of this course is to give an introductory overview of this beautiful theory. Background material from algebraic geometry will be developed as we go.

Outline:

1. Some Algebraic Geometry
2. Linear Algebraic groups
3. Derivations, differentials, Lie algebras
4. Homogeneous Spaces
5. Semi-simple and Unipotent Elements
6. Solvable Groups
7. Borel Subgroups
8. Reductive Groups
9. Semi-simple Groups
• MATH 7312: Measure and integration
• Instructor: Prof. Sundar
• Prerequisite: Math 7311
• Text: Real and Complex Analysis by Walter Rudin

The course is a sequel to Real Analysis I, and introduces measure theory, elements of Hilbert and Banach spaces, and analysis of functions of several variables. The syllabus is as follows:

1. Chapter I: Abstract measures, integration, convergence theorems, Lp-spaces and inequalities.
2. Chapter II: Hilbert and Banach spaces, consequences of Baire's theorem, Hahn-Banach theorem and applications.
3. Chapter III: Signed measures, Radon-Nikodym theorem, decomposition theorems, duality.
4. Chapter IV: Product measures, Fubini and Tonelli theorems.
5. Chapter V: Differentiation, implicit and inverse function theorems, change of variables in integrals.
6. Chapter VI: Sobolev spaces, Embedding theorems.
• MATH 7320: Ordinary Differential Equations
• Prerequisite: Either Math 7311 or else both Math 4031 and Math 4032.
• Text: W.Walter, Ordinary Differential Equations, Springer, 1998.

This is a first graduate course in ordinary differential equations. The following topics will be covered. Existence and uniqueness results for first order equations and systems, and for higher order equations. Linear equations and systems. Complex linear systems. Boundary value and eigenvalue problems. Stability.

• MATH 7380-1: Partial Differential Equations
• Instructor: Prof. Shipman.
• Prerequisite: A first course in graduate analysis, such as Math 7311; elementary exposure to ordinary differential equations.
• Text: Partial Differential Equations, by L. C. Evans (AMS Graduate Studies in Mathematics, Vol. 19)

This course is an introduction to mathematical theories of partial differential equations and will be complemented by physical motivations and applications. The course will be mostly self-contained, presuming a solid foundation in basic analysis and exposure to basic (ordinary) differential equations.

1. The basic linear equations will be covered in depth as well as methods of solution such as transforms and fundamental solutions. Basic nonlinear theory will include the method of characteristics, conservation laws, and existence and uniqueness theory. Other major topics include the calculus of variations and elliptic PDEs.
2. Some special topics will be studied in some detail, such as nonlinear waves, nonlinear spectral transforms, asymptotic methods, or the mathematics of electrodynamics.
3. An attempt will be made to give a broad survey of topics in PDEs by discussing briefly a sampling of topics. These may include evolution equations, operator and semigroup methods, nonlinear hyperbolic equations, boundary-integral methods, energy methods, numerical methods, integrable systems, elliptic regularity theory, or pseudo-differential operators.
• MATH 7380-2: Integral and Functional Equations
• Instructor: Prof. Antipov.
• Prerequisite: MATH 4036 Complex variables
• Text: Lecture notes and F.D.Gakhov, Boundary value problems.

Aims: To introduce students to singular integral and integro-differential equations and functional-difference equations which can be tackled using methods for boundary-value problems of the theory of analytic functions.

Objectives: Students should know a range of analytical methods for solving model problems in applied mathematics including fracture mechanics and diffraction theory.
Content: The Cauchy integral, the Riemann-Hilbert problem on an axis and an open curve, the Wiener-Hopf method, the Carleman problem, matrix factorization, elements of elliptic functions and elliptic surfaces, model problems of fracture, elasticity and diffraction of electromagnetic and acoustic waves.

• MATH 7380-3: Numerical Methods & Applications
• Instructor: Prof. Bourdin.
• Prerequisite: Math 7311 or equivalent; Basic knowledge of harmonic analysis, PDE, or numerical analysis is a plus, but not a requirement. Basic programming skills in C, fortran, pascal , or other "low" level language.
• Text: Lecture notes, online resources provided during the class.

This is an introduction class to the numerical analysis and implementation of Partial Differential Equations. Its focus is mainly on the Finite Element Method for elliptic problems. In a first part, we will briefly survey some theoretical results for PDEs (variational formulation, Galerkin method...). Then, we will discuss multi-dimensional interpolation, differentiation and integration, which will lead us to the core of the finite element method. Time permitting, we will either cover the basic of domain decomposition and parallel processing, and/or iterative methods for solving large sparse linear systems of equations. Numerical projects will represent a major part of the work (and of the grade). It is my intention to make this class suited to students with a theoretical background, willing to learn about advanced numerics, as well as to those already familiar with numerics, and interested in a more theoretical approach. Mathematics, computer science and engineering students are welcome here!

• MATH 7390-1:  Harmonic Analysis - II: The Classical Groups
• Instructor: Prof. Fabec
• Prerequisite: Math 7311 and 7312.
• Text: will be provided on-line.

This is the second in a sequence of courses on harmonic analysis. The sequence starts with classical Fourier series and the Fourier integral, extends these objects to distributions and explores their uses in solving differential equations. Next the Fourier integral is used to study functions and function spaces on the Heisenberg group. This group is fundamentally important for both its importance in quantum mechanics where its representation theory encodes the Heisenberg uncertainty principle and as an example of the difficulties arising when one attempts to do harmonic analysis on groups which are no longer commutative. Here one must deal with infinite dimensional representations and from this one can be familiarized with this theory. We then discuss compact groups. These can be handled with a very nice theory. Tools developed here can be used to study functions on spheres or other compact homogeneous spaces in terms of spherical functions. These functions behave well with respect to representations and invariant differential operators on these spaces. In dealing with this part of the subject, we present the notion of a Gelfand pair. The first semester course begins with an overview of Fourier series of periodic functions on the line; then goes on to Fourier analysis on Euclidean spaces. Material covered includes the Fourier transform, inversion formula, Plancherel theorem, the Paley-Wiener theorem, and extensions to distributions and algebras. The role of the Fourier transform in analysis of partial differential equations with constant coefficients is discussed.The Heisenberg group H is introduced and the intimate connection between classical Fourier analysis and harmonic analysis on H is developed. Along with group and analytic structure, the representation theory of H will be presented. This includes the Stone-von Neumann theorem, connections to quantum mechanics, the Plancherel theorem, the Schrödinger model, the equivalent Fock-Bargmann representation, twisted convolution, the strong Stone-von Neumann theorem and transforms of Schwartz functions as operators with smooth kernels. The large automorphism group of H is used to introduce semidirect products, compact Lie groups and semi-simple Lie groups. Similarities between harmonic analysis for these groups and H along with new subtleties introduced by the further non-commutativity of these groups will be discussed. The groups introduced this way may include SO(n), SU(n), SL(2,R), the Poincaré group, the Euclidean motion group and the symplectic group. The division between first and second semester material is fluid. We will review materials from topics covered in Course I needed for the second semester.

• MATH 7390-3: Advanced Topics in Probability Theory
• Instructor: Prof. Kuo .
• Prerequisite:
• References:
1. Kuo, H.-H.: Introductory Stochastic Integration. (In preparation)
2. Oksendal, B.: Stochastic Differential Equations. 5th edition,Springer, 2000
3. Kuo, H.-H.: Gaussian Measures in Banach Spaces, Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975
4. Kuo, H.-H.: White Noise Distribution Theory, CRC Press,1996

This course contains two parts: (1) basic theory of stochastic integration with applications to mathematical finance (which appeals to finance students and non-probability math graduate students), (2) advanced research topics for Ph.D. students (which I will outline in an overview for further independent study). Below are some items to be covered in this course:

1. Brownian motion
2. Construction of Brownian motion
3. Wiener integrals
4. Ito's integrals
5. Stochastic integrals for martingales
6. The Ito formula
7. Girsanov theorem
8. Stochastic differential equations
9. Arbitrage and option pricing
10. Black-Scholes analysis
11. Wiener-Ito decomposition theorem
12. White noise theory
• MATH 7520: Algebraic Topology
• Instructor: Prof. Litherland.
• Prerequisite: Math 7512 or the equivalent.
• Text: Marvin J. Greenberg and John R.Harper, Algebraic Toplogy, A First Course, Perseus Books, 1981.

The basic idea in algebraic topology is to associate to a topological space an algebraic object (e.g. an integer, a polynomial, a group, or a module over some ring), in such a way that homeomorphic spaces get assigned equivalent objects (e.g. equal integers, or isomorphic groups). The object assigned to a space is an invariant of the space, and provides a tool for distinguishing between topological spaces: if two spaces have inequivalent invariants, they are not homeomorphic. One such invariant of a space X, the fundamental group, is covered in Math 7512. In Math 7520 we deal with a family of invariants, the homology groups Hn(X) (n a non-negative integer). Very roughly, these groups count the numbers of holes of different dimensions in X. We should cover Part II of the text, which deals with the definitions, methods of calculation, and some applications of these invariants.

• MATH 7550: Differential Geometry
• Instructor: Prof. Fabec.
• Prerequisite: MATH 7200 and 7510 at least a senior-level undergraduate analysis course.
• Text: Foundations of Differential Manifolds and Lie Groups by Frank Warner, and posted class notes.

The course title is Differential Geometry and Topology, but the content is a basic introduction to manifolds. When one studies analysis on real n-space one is studying local phenomena; when one studies analysis on manifolds one is studying global phenomena. This subject is called global analysis. It is particularly useful for differential topology, differential equations, differential geometry, harmonic analysis, and mathematical physics. The prerequisites are advanced calculus, point set topology, basic group theory and linear algebra. The topics to be covered will be:

• Manifolds
• Vector Bundles
• Frobenius' Theorem
• Lie Derivative
• Differential Forms
• Integration

Other books of interest are:

1. Differential Geometry volume 1 by Michael Spivak This book is an excellent book and series to read. It gives long detailed mathematical and historical explanations, but does not make a particularly good textbook.
2. Differential and Riemannian Manifolds} by Serge Lange This book covers the infinite dimensional theory. Lange writes, "It is possible to lay down at no extra cost the foundations ... for manifolds modelled on Banach or Hilbert spaces."
3. Calculus on Manifolds by Michael Spivak This book was written as an advanced undergraduate textbook. It has few prerequisites, however, it uses less sophisticated ideas and it gives less insight.
• MATH 7590: Finite Type Invariants
• Instructor: Prof. Dasbach.
• Prerequisite: Some knowledge of Topology or Differential Geometry.
• Text: Sergei Chmutov, Sergei Duzhin: "Introduction to Vassiliev Knot Invariants" (preliminary version of a book, available online) and handouts.

Low dimensional topology in general and knot theory in particular is an exciting subpart of topology. E.g. applications are in the understanding of the structure of our space or in the understanding of human DNA. This course is concerned with invariants on knots that were first introduced by Vassiliev in the early 90s. The course is largely self-contained. We will give a review of the necessary background from topology and other fields. We will start with a quick trip through knot theory. We will go on to topics like Jones polynomials and Quantum invariants before we introduce Vassiliev's ideas and the combinatorics that grew out of it. We will also spend some time introducing algorithms used in computer programs like KNOTSCAPE that effectively deal with knots.