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# Course Descriptions Summer 2002 - Spring 2003

## Contact

## Summer 2002

- MATH 7350: Complex Variables
- Instructor: Jorge Morales
- Prerequisite: undergraduate (real) analysis such as MATH 4032 (Advanced Calculus II)
- Text: "Functions of One Complex Variable" by John B. Conway, Springer-Verlag Graduate Texts in Mathematics.

We will cover at least the following topics:

- Power series and analytic functions
- Complex integration
- Cauchy's Theorem
- Goursat's Theorem
- Classification of singularities
- Residues
- The maximum principle
- Analytic continuation

Time permitting, we will discuss some applications to analytic number theory.

## Fall 2002

- MATH 7200: Geometric and Abstract Algebra
- Instructor: Charles N. Delzell Tel. 578-1602; Lockett 346.
- Prerequisite: undergraduate abstract algebra.
- Text: Thomas W. Hungerford's `Algebra,' Grad. Texts in Math., Vol. 73, Springer, 1st ed. 1974; corr. 8th printing 1996.

Topics: groups (including the structure of finitely generated Abelian groups), rings, modules (including the structure of finitely generated modules over a principal ideal domain), and linear algebra (including the Jordan canonical form and the rational canonical form of a matrix or linear transformation). For a more detailed list of topics, see the department's syllabus for the core-1 comprehensive exam in algebra. That syllabus also includes a list of 150 sample problems. This course should prepare students for that exam. There will be approximately six homework assignments, a mid-term exam, and a final exam.

- MATH 7211: Commutative Rings
- Instructor: Augusto Nobile, Lockett 342, Tel: 578-1604
- Prerequisite: Math 7210.
- Text: "Introduction to Commutative Algebra", by M. F. Atiyah and I.G. Macdonald

This course will be a standard introduction to the theory of Commutative Rings. This material plays a fundamental role in the study of Algebraic Geometry, and an important one in Algebraic Number Theory, other types of Geometry, Algebraic Topology, Mathematical Logic, etc. We'll try to cover a variety of topics, to make the course useful to people interested in various disciplines. For instance, we expect to discuss (at least in introductory form) integral extensions, completions, valuations, etc. Our basic reference will be the book "Introduction to Commutative Algebra", by M. F. Atiyah and I. G. Macdonald, although probably we won't follow it very closely. As in this reference, our approach will be "traditional", i.e. we won't emphasize computational or constructive methods (although we hope to say something about this important approach, that has attracted renewed attention in recent times.)

- MATH 7280-1: Toric Varieties
- Instructor: James J. Madden
- Prerequisite: 7200 Geometric and Abstract Algebra and 7210 Algebra I. Recommended but not necessary: 7211 Topics in Commutative Algebra.
- Text: Combinatorial Convexity and Algebraic Geometry by Guenter Ewald (Springer); Toric Varieties by William Fulton (recommended but not required).

An algebraic variety is a subset of complex n-space that is defined by a system of polynomial equations. Algebraic geometry studies the properties of such sets that are preserved by polynomial (or rational) mappings. Toric varieties are a special kind of algebraic variety that are defined by particularly simple equations---systems of differences between monomials, e.g. , z x3-w2 y2. Because of the special form of the defining equations, there is a beautiful correspondence between the geometry of toric varieties and the geometry of lattice polytopes (the latter being in a loose and informal sense the "logarithms" of the varieties). The study of toric varieties, therefore, has a strongly combinatorial flavor. This course will be run as a seminar in which students will do a substantial amount of speaking. In my experience, students find this approach to instruction both challenging and fun.

- MATH 7280-2: Axiomatic Set Theory
- Instructor: Charles N. Delzell Tel. 578-1602; Lockett 346.
- Prerequisite: Minimal acquaintance with `naive' set theory (as commonly taught in courses on real variables or point-set topology), plus mathematical maturity.
- Text: Paul J. Cohen's "Set Theory and the Continuum Hypothesis," W.A. Benjamin, Inc., 1966. Unfortunately, Cohen's text is out of print; used copies of it are now collector's items selling for $135 or for other prices at ZShops or Amazon. However, the publisher will allow us to make our own copies of the book if students pay the publisher a royalty of $11.65; I plan to have our campus bookstore copy the book for us as a course-packet, which should be reasonably priced. While there are many excellent texts on set theory available, Cohen's book seems to be at just the right level for this course; it has the merit of "getting right to the point," with a minimum of formalities.

We will study:

- The Zermelo-Fraenkel axioms for set theory (denoted by ZF)
- The axiom of choice (AC)
- The continuum hypothesis (CH, conjectured by Georg Cantor in 1878)
- Kurt Gödel's celebrated proof (1938) that neither AC nor CH can be disproved from ZF.

Other topics to be covered along the way:

- ordinal numbers, cardinal numbers, and the development of the natural numbers, the integers, the rational numbers, and the real numbers from the empty set.

There will be approximately six homework assignments, a mid-term exam, and a final exam.

- MATH 7311: Analysis I
- Instructor: Padmanaban Sundar, Lockett 316
- Prerequisite: Advanced Calculus, such as Math 4031 and/or 4032.
- Text: `Foundations of Modern Analysis' by Avner Friedman.

This is a core graduate course in Real Analysis in which students learn about Lebesgue measure and integration, Lp spaces, Banach and Hilbert spaces, and operators on such spaces. The course will cultivate a rigorous and modern understanding of differentiation, integration and other limit operations. The role played by concepts such as compactness, functions of bounded variation, and completions of spaces will be discussed. Students will learn several important theorems such as the Hahn-Banach theorem, closed graph theorem, uniform boundedness principle, and implicit function theorem. There will be several homework assignments as well as a mid-term and a final examination. This course should prepare the students for the core-1 qualifying exam in Analysis. If you have further questions about the course please contact Dr. Sundar in his office or by e-mail.

- MATH 7330: Functional Analysis
- Instructor: Ambar Sengupta
- Prerequisite: Measure and Integration [Math 7312], Topological Spaces, Linear Algebra
- Text: Rudin's "Functional Analysis" will be a useful reference

Our objective will be to study the spectral theorem for operators on Hilbert spaces. On the way, we shall study the Gelfand theory of commutative Banach algebras. Click here for either the Course Web Page; or Dr. Sengupta's web page.

- MATH 7360: Probability Theory, 2:40-3:30 MWF
- Instructor: George Cochran, 308 Lockett, 578-1614
- Prerequisite: A good understanding of fundamental analysis concepts and techniques of proof, as covered in the course Math 7311. Prior knowledge of probability will not be assumed, although it would be helpful in understanding the motivation behind the theory. Prior knowledge of measure theory will also not be assumed, although it would be helpful in learning the definitions and methods of the theory.
- Text: Kai Lai Chung, A Course in Probability Theory, 2nd ed Revised, (available in paperback), 419 pp, Academic Press. Online prices: Approx. $60 new, $45 used for paperback version.

This is the standard measure-theory-based course on the foundations of probability. Topics: Extension of measure, Random objects and their distributions, Moments, Laws of Large Numbers and Central Limit Theorems, Conditional expectation and martingales, Basic properties of Brownian motion.

- MATH 7380-1: Stochastic Analysis & Applications to Mathematical Finance
- Instructor: H-H Kuo
- Prerequisite: Math 4031 Advanced Calculus and Math 4055 Introduction to Probability
- Text: No textbook is required. The lectures will be based on Dr. Kuo's book manuscript.

This course consists of two parts: the mathematics of stochastic integration and some applications to mathematical finance. We will present motivations and explanations (without technical details of the mathematical theory) of the mathematical concepts such as Brownian motion, stochastic integrals, martingales, Ito's lemma, Markov processes, and stochastic differential equations. For applications we will cover topics from continuous-time finance theory such as trading strategies, arbitrage pricing, valuation of derivative securities, the Black-Scholes analysis, hedging portfolio, and option pricing. Visit Dr. Kuo's web page.

- MATH 7380-2: Calculus of Variations and Optimal Control Theory T,Th 1:30-3:00, 132 Lockett
- Instructor: Peter R. Wolenski Lockett 326; Tel. 578--1606
- Prerequisite: A good advanced calculus background.
- Text: There will be no formal textbook.

- Examples, History, and Problem Formulation: (A) The Brachistochrone and minimum distance problems;
- Necessary Conditions: (A) Euler--Lagrange equation and Erdmann corner conditions; (B) The Weierstrass excess function and the Legendre condition; (C) Unified conditions, canonical equations; (D) The Jacobi condition and conjugate points.v
- Sufficient conditions: (A) Jacobi's strengthening of the Legendre condition; (B) Field theory and embedding theorems; (C) The verification procedure and the Hamilton--Jacobi equation.
- Hamilton--Jacobi Theory: (A) Principle of least action; (B) Noether's theorem; (C) Generalized solution concepts, viscosity solutions.
- Optimal Control: (A) The Pontryagin maximum principle; (B) Existence theory.

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

- MATH 7490: Discrete Optimization 9.30-10.30 MWF
- Instructor: Manoj Chari, 256 Lockett Hall, Tel.: 578-1677
- Prerequisite: Linear algebra and Vector spaces (MATH 4153 or equivalent) or permission of instructor.
- Text: Introduction to Linear Optimization, D. Bertsimas and J. Tsitsiklis, Athena Scientific Co. 1997. (Reference: The Theory of Linear and Integer Programming, A. Schrijver, Wiley 1986.)

Linear and integer programming are some of the most useful techniques in applied (discrete) mathematics. A large number of industrial optimization problems involving scheduling, facility location, product mixing, resource allocations, inventory planning etc. are modeled and solved using these techniques. Linear programming ideas have also proved useful in both theoretical and algorithmic aspects of combinatorics, mathematical economics and computer science. There is elegant mathematics underlying these techniques that combines geometric, combinatorial and algorithmic ideas. In this course, we will develop the mathematical theory of linear programming and discuss some of its applications. Evaluation will be primarily based on regular homework problems, a midterm exam and a final project. Syllabus: Formulation of linear programming problems and applications, Mathematical preliminaries, Geometry of linear programming, Facial structure of polyhedra, Fourier-Motzkin elimination, Duality theory in linear programming with applications, The Simplex method and its variations, Network simplex and Dual simplex algorithms, The ellipsoid algorithm and selected interior point methods, Integrality of polyhedra and total unimodularity with applications to networks and combinatorics.

- MATH 7510-2: Topology I, Tuesday and Thursday, 1:40-3:00 PM
- Instructor: Robert Perlis, Lockett Hall 212; Tel: 225 578-1673
- Prerequisite: A one semester advanced calculus course along the lines of our Math 4031, in which the student was required to prove theorems.
- Text: Topology Notes, by Prof. J. Lawson. At the start of the Fall semester, these 41-page notes will be available for purchase at cost at the Kinko's near campus.

Topology, sometimes called `rubber sheet mathematics', deals with properties of objects that do not change by stretching or bending (tearing, however, is not allowed). Modern topology grew out of a combination of geometry and classical analysis with the (very successful) attempt to generalize the notion of continuity for functions on euclidean space to functions on more general spaces. The focus of this course will be on becoming acquainted with these spaces and with the major theorems concerning them. We will follow notes written by my LSU colleague J. Lawson. The notes contain the relevant definitions, examples, and statements of lemmas and theorems. The theorems have been carefully broken down into manageable small steps, and the participants in the course will supply the proofs of these steps. So this is a course intended to develop and sharpen your theorem-proving skills. Every student will make many presentations at the board during the semester.

- MATH 7520: Algebraic Topology
- Instructor: Daniel Cohen, Tel.: (225) 578-1576
- Prerequisite: MATH 7200 and 7510, or equivalent The exposure to algebraic topology provided by MATH 7512 would be useful, but not absolutely essential.
- Text: J. R. Munkres, Elements of Algebraic Topology, required

A fundamental problem in topology is that of determining, for two spaces, whether or not they are topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects (groups, rings, etc.) to a topological space in such a way that topologically equivalent spaces get assigned isomorphic objects. The fundamental group introduced in MATH 7512 is one example. Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. Two spaces with inequivalent invariants cannot be topologically equivalent.

The focus of this course will be on homology theory (which complements the study of algebraic topology begun in MATH 7512). To a topological space, we will associate a sequence of abelian groups, called the homology groups. These homology groups are often more accessible than the fundamental group, so sometimes provide an easier means for distinguishing between topological spaces. We will concretely study simplicial and singular homology, the homology of CW-complexes, and related topic such as homology with coefficients, Mayer-Vietoris sequences, degrees of maps, and Euler characteristics.

Geometric examples, including surfaces, projective spaces, lens spaces, etc., will be used to illustrate the techniques. We will also discuss a number of applications, including Brouwer and Lefschetz Fixed Point Theorems, and the Jordan Curve Theorem. A continuation of this course will be offered in Spring 2003. Then we will study cohomology (dual to homology), and duality on (compact) manifolds.

- MATH 7590: Knot Theory
- Instructor: R. A. Litherland
- Prerequisite: Math 7520 (Algebraic Topology)
- Text: W. B. Raymond Lickorish, An Introduction to Knot Theory, Springer, 1997 (required)

This course is an introduction to the theory of knots and links in three-dimensional space. (Formally, a knot is a subspace of R3 homeomorphic to a circle. To make a model, tie a knot in a piece of nylon cord and then fuse the ends together by applying a match. To make a link, repeat with more pieces of cord entangling themselves and the previous pieces.) We will use three broad classes of techniques. First, there is the method of studying intersections of surfaces in a 3--manifold. This is used to prove results on the factorization of knots, as well as more recent results on the geometry of alternating links. Second, there is the application of algebraic topology to knot theory: the Alexander polynomial, the fundamental group, and signatures, with applications to branched coverings and slice knots. Finally, there is the recent application of combinatorial methods to produce the Jones and other polynomial invariants of knots, and the quantum invariants of 3--manifolds introduced by Witten. I do not assume any previous acquaintance with piecewise-linear or differential topology. Some basic results on regular neighborhoods and general position will have to be taken on faith, but in three dimensions these are easy to grasp intuitively. I expect to cover Chapters 1-13 of the text (except for Chapter 7, which deals with material from 7512), and perhaps some of Chapters 14-16 if time allows.

## Spring 2003

- MATH 7280: Linear Representations of Finite Groups
- Instructor: Prof. Jorge Morales.
- Prerequisite: Required: A graduate-level linear algebra course (such as Math 7200). Recommended: Math 7210 or similar.
- Text: We will follow for the most part J.-P. Serre's book "Linear Representations of Finite Groups" (Graduate Texts in Mathematics Vol 42, Springer-Verlag 1996).

At least half of the course will be devoted to the classical theory of linear representations of finite groups over the complex numbers. In the second half, we will discuss rationality questions and representations of general semisimple finite-dimensional algebras. Time permitting, we will give an introduction to representations of finite groups over fields of positive characteristic.

The main topics of this course are:

- Representations of finite groups over C. Characters. Representation rings. Generalization to compact groups.
- Induced representations. Frobenius reciprocity. Artin's Theorem.
- Semisimple finite-dimensional algebras. The group ring. Rationality questions (descent). Real representations.
- (Time permitting) Representations of finite groups in positive characteristic. Introduction to Brauer theory.

- MATH 7290: Algebraic Number theory
- Instructor: Prof. R. Perlis, Lockett 212, Tel.: 578-1673
- Prerequisites: Familiarity with basic concepts of algebra (groups, rings, fields); Some familiarity with finite fields, including the fact that the multiplicative group of a finite field is always cyclic; Knowing that every euclidean domain is a principal ideal domain, and every principal ideal domain is a unique factorization domain. If you knew this once but forgot it, and if you promise to review before the course starts, that should be sufficient. No special knowledge of number theory will be assumed. Everyone interested is welcome to participate.
- Text: Algebraic number theory, by A. Frohlich & M. J. Taylor, Cambridge studies in advanced mathematics 27

Algebraic number theory is very beautiful, very modern, and very ancient, all at the same time. This course will try to give a sense of the beauty and breadth of the field.

- MATH 7312: Measure and Integration, Lockett 111, Tuesdays and Thursdays 9:10am-10:30am
- Instructor: Prof. Ambar Sengupta
- Text: Rudin's Real and Complex Analysis will be a useful reference. Extensive notes will be posted.

This course will build the basic apparatus of measure theory and integration. Topics will include abstract measure and integration, convergence theorems for integration, Fubini's theorem, the Radon-Nikodym theorem, the Riesz-Markov representation theorem, and duals of certain function spaces. Time permitting we will do some functional analysis, with a look at software from Hilbert spaces, Banach spaces, and topological vector spaces, and hardware from Sobolev inequalities.

- MATH 7320: Ordinary Differential Equations; T, Th 1:30-3:00
- Instructor: Prof. Peter Wolenski, Lockett 326
- Text: Ordinary Differential Equations by Wolfgang Walter that is translated from the German by Thompson. It is published by Springer in the Graduate Texts in Mathematics series, Vol. 182, and is currently on sale for $39.50. I will use additional lecture notes and standard mathematical software to help in the calculations and for visualization.

This course is an introduction into the theory and applications of ordinary differential equations. The topics covered are standard, and include existence and uniqueness of solutions, dependence on initial conditions, linear theory, stability theory, and aspects of dynamical system theory. We shall also attempt to incorporate some applications in modern science and engineering through mathematical modeling and computer experiments. Finally, if time permits, we will give an introduction to optimal control theory.

- MATH 7380-1: Operator Theory - Representation Theory
- Instructor: Prof. R. Fabec.
- Prerequisite: Real Analysis - I (Math 7311) and Measure Theory (Math 7312).
- Text: Fundamentals of Infinite Dimensional Representation Theory by R. Fabec.

This is a course which will develop the important relation between operator algebras, their representations, and the representation theory of locally compact groups. It will start with the classical results of Gelfand on commutative C* algebras, develop the representation theory of C* and W* algebras, and introduce the group C* algebra and W* algebra. Direct integral decompositions of algebras and the corresponding decomposition of representations will be presented. Next we will look at inducing group actions and representations. Using Mackey's orbit method, the unitary duals of some groups will be obtained. Then we discuss the abstract Fourier transform, and if time permits, work out some concrete examples.

- MATH 7390-1: Applied Harmonic Analysis and Wavelets
- M-W-F 10:40 - 11:30 AM, 237 Lockett
- Instructor: Gestur Olafsson Lockett 322.
- Prerequisite: MATH 7311 Analysis I
- Text: C. Blatter: Wavelets: A Primer. A.K. Peters

Harmonic analysis deals with the problem of decomposing functions into simpler functions. What ``simpler functions'' means depends on the situation, and which properties we are looking for. The classical situation is the Fourier analysis which decomposes arbitrary functions simultaneously into waves and into eigenfunctions of differential operators with constant coefficients. The object of Fourier series is to expand arbitrary periodic square integrable function in terms of exponential functions, or cos and sin. The cos and sin have exact localization in the frequency, i.e., the Fourier variable, but have no precise localization in space. This makes Fourier series not always the right tool for signal analysis, where both good localization in the frequency and space parameter is required. In the last 20 years a new theory has emerged that deals exactly with this problem. The wavelet theory tries to construct an orthonormal basis that has localization properties both in space and frequency. It uses both translation and dilation to zoom into given part of the spatial variable. Wavelet theory lies on the boundary between

- Mathematics, in particular harmonic analysis
- Signal processing,
- Image processing
- Scientific calculation

In this course we will give an overview of the wavelet theory and its applications. This course will deal with the following topics,

- Introduction to Fourier series and integrals.
- Hilbert spaces, orthonormal basis and Riesz basis. Continuous operators between Hilbert spaces.
- Multiresolution analysis
- Haar wavelets and Daubechies Wavelets.
- Construction of wavelets in one dimension.
- Signal compression and denoising.

For more information see www.math.lsu.edu/~olafsson/teaching.html.

- MATH 7390-2: Advanced Topics in Probability Theory
- Instructor: H.-H. Kuo.
- References:

- Kuo, H.-H.: Gaussian Measures in Banach Spaces, Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975
- Kuo, H.-H.: Stochastic Integration. (In preparation)
- Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996

This course will cover the following topics:

- Brownian motion
- Wiener-Ito decomposition theorem
- Measures on infinite dimensional spaces
- Wiener space
- Abstract Wiener spaces
- Nuclear space
- Stochastic integrals
- The Ito formula
- Mathematical finance applications
- Theory of generalized functions on Rn
- White noise theory

- MATH 7512: Topology II
- Instructor: Patrick Gilmer.
- Prerequisite: Math 7510 (Topology 1) , Math 7200 (Geometric and Abstract Algebra)
- Text: Topology, 2/E (second edition) by James Munkres, (required)

To each topological space, we will associate a group, called the fundamental group of the space. We study the basic properties of the fundamental group. We will give such applications of this group as the Brouwer fixed point theorem, the fundamental theorem of calculus, and the Jordan curve theorem. We will learn to calculate this group, and relate it to the theory of covering spaces. Time permitting we will study surfaces and their classification.

- MATH 7590-2: Cohomology Theory Tues. & Thurs. 12:10 - 1:30 PM, 381 Lockett
- Instructor: Dan Cohen.
- Prerequisite: MATH 7520 Algebraic Topology
- Text: Elements of Algebraic Topology, by J. R. Munkres, Perseus Publishing,1984.

This course continues the study of algebraic topology begun in MATH 7512 and MATH 7520. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group in MATH 7512, the homology groups in MATH 7520) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., isomorphic groups). Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces: two spaces with inequivalent invariants cannot be topologically equivalent. The focus of this course will be on cohomology theory, dual to the homology theory developed in MATH 7520. One reason to pursue cohomology theory is that the cohomology of a space may be given a natural ring structure. This additional algebraic structure provides another topological invariant. In developing this structure, we will study several products relating homology and cohomology. These considerations will be used to study the topology of manifolds, yielding a number of duality theorems also relating homology and cohomology, with a variety of applications. In addition to its importance within topology, cohomology theory also provides connections between topology and other subjects, including algebra and geometry. Depending on the interests of the audience, and of course time, we may pursue some of these connections, such as the de Rham theorem or cohomology of groups.