All inquiries about our graduate program are warmly welcomed and answered daily:
All inquiries about our graduate program are warmly welcomed and answered daily:
We will cover at least the following topics:
Time permitting, we will discuss some applications to analytic number theory.
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.
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.)
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.
We will study:
Other topics to be covered along the way:
There will be approximately six homework assignments, a mid-term exam, and a final exam.
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.
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.
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.
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.
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.
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.
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.
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.
Visit Dr. Cohen's webpage for this course.
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.
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:
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.
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.
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.
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.
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 reqiuered. 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
In this corse we will give an overview of the wavelet theory and its applications. This course will deal with the following topics,
For more information see www.math.lsu.edu/~olafsson/teaching.html.
This course will cover the following topics:
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.
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.