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On a rotating hollow sphere, there are exactly two points that don't move at all (we might call them the "north and south poles"). The sphere belongs to a large class of important topological spaces with two key features: (1) they have some (perhaps more than one) kind of rotational symmetry, and (2) they have finitely many points that don't move when the space is rotated. "Equivariant cohomology," the subject of this seminar, is a powerful tool for studying the geometry of such spaces. Using equivariant cohomology, we will explore remarkable connections to combinatorics and group theory.
This semester, we will discuss an important result from current research in representation theory called the geometric Satake isomorphism, restricting ourselves to the simplest possible case of the 2 by 2 invertible complex matrices GL2(C). Roughly speaking, this result states that representations of GL2(C) (i.e., group homomorphisms from GL2(C) to GLn(C) for any n) can be understood in terms of the equivariant cohomology of a topological space called the affine Grassmannian. The goal of this course is to make sense of this isomorphism as explicitly as possible and to come up with a new simple proof in this case. (No background in representation theory is assumed.)
Mathematical control theory is one of the most central and fast growing areas of applied mathematics. This course will help prepare students for research at the interface of engineering and applied mathematics. The first part provides a self-contained introduction to the mathematics of control systems, focusing on feedback stabilization and Lyapunov functions. The second part will be a series of lectures by faculty from the LSU College of Engineering about open problems in control. The third part will explore ways of solving the problems. The only prerequisite is a graduate or advanced undergraduate course on the theory of differential equations. Students from engineering or mathematics are encouraged to enroll.
Cluster Algebras are a topic of great interest in current mathematics.
They were defined by Sergey Fomin and Andrei Zelevinsky in 2001 in relation to problems in combinatorics and Lie groups. Only a few years later they started playing a key role in a number of developments in representation theory, topology, combinatorics and algebraic geometry.
The beauty of the subject is that a great deal of it requires almost no prerequisites. Thus undergraduate students who register will be able to understand and lecture on a number of topics.
One of the main goals of the course is to go over applications and relations to various areas of mathematics. Graduate students specializing in representation theory, topology, combinatorics and algebraic geometry will see relations to each of these areas and will be asked to make presentations on their area of expertise.
The colored Jones polynomial is one of the more mysterious objects in knot theory. We will start with various definitions of it and will try to develop some of its properties. The methods will be elementary.
This course provides practical training in the teaching of mathematics at the pre-calculus level, how to write mathematics for publication, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both pre-calculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
This is the first semester of the first year graduate algebra sequence. It will cover the basic notions of group, ring, and module theory. Topics will include symmetric and alternating groups, Cayley's theorem, group actions and the class equation, the Sylow theorems, finitely generated abelian groups, polynomial rings, Euclidean domains, principal ideal domains, unique factorization domains, finitely generated modules over PIDs and applications to linear algebra, field extensions, and finite fields.
Much of commutative algebra was developed (by Zariski, particularly) as a precise language for geometry. This course will present the fundamental concepts of the field with careful attention to the geometric meaning. I plan to treat the following topics: localization, prime ideals and primary decomposition, spectra, the Nullstellensatz, flatness, dimension theory, valuations, integral dependence and Noether normalization (which are treated in chapters 2-13 of the text), as well as a selection of topics from the remaining parts of the text as time permits. Students will be expected to participate actively by giving brief talks on a rotating basis and preparing notes.
The aim is to develop the basics of both subjects to the point where the main theorems established by Wiles, Taylor and others connecting these two can be understood (these theorems led to the proof of Fermat's theorem).
Algebraic number theory is the study of algebraic number fields and the rings of algebraic integers in these fields. An algebraic number field is any field obtained by adjoining to the field of rational numbers a single root of a polynomial with rational coefficients. A background in algebraic number theory is essential to any student interested in algebra, algebraic geometry, arithmetic geometry, and of course number theory.
This will be a beginning course in algebraic number theory. Students taking this course will be be prepared to take the course Class Field Theory offered in spring, 2011.
Lie groups and Lie algebras form a topic of great importance, with connections to many areas of mathematics and physics. This course will cover the basic structure theory of complex algebraic Lie groups and their associated Lie algebras. Topics include: solvable groups and Lie algebras; maximal tori and Cartan subalgebras; Borel subgroups and subalgebras; Weyl groups; root systems; classification of simple groups and Lie algebras by Dynkin diagrams; weights and representations. If time permits, we will also cover the Borel-Weil theorem, which gives a geometric construction of all irreducible representations. (Note: the term "algebraic" means that the proofs will be based on methods from algebraic geometry, rather than differential geometry and analysis. However, no knowledge of algebraic geometry is needed to take this course.)
This course will address the classical theory of real valued functions, measure, and integration.
A standard first year graduate course in complex analysis. Topics include holomorphic functions, covering spaces and the monodromy theorem, winding numbers, residues, Runge's theorem, Riemann mapping theorem, harmonic functions. .
Coverage:
In the first two weeks I will give a brief review of elementary probability
theory and measure theory. Topics to be covered include the following:
1. Kolmogorov's extension theorem
2. Various types of convergence
3. Laws of large numbers
4. Convergence of random series
5. Law of iterated logarithm
6. Characteristic functions
7. Bochner theorem
8. Levy's continuity theorem
9. Levy's equivalence theorem
10. Central limit theorem
11. Stable and infinitely divisible laws
This is an introductory course in the theory of the Radon transform, one of the main objects in modern analysis, integral geometry, and tomography. Topics to be studied include fractional integration and differentiation of functions of one and several variables, Radon transforms in the n-dimensional Euclidean space and on the sphere, related aspects of the harmonic analysis, functional analysis, and function theory.
This is an introductory course of numerical solution of partial differential equations subject to random perturbations. Topics include sampling techniques, such as Monte Carlo methods, and non-sampling techniques, such as polynomial chaos methods, for ordinary differential equations, elliptic and parabolic equations. Algorithm development, numerical analysis and computer implementation issues will be addressed.
This is a basic introduction to stochastic analysis with a view to applications. After a short review of the framework of probability theory and basic tools, we will study the Brownian motion process, including its construction and its sometimes strange features. Then we will proceed to the Ito theory of stochastic integration and familiarize ourselves with the celebrated Ito lemma, a basic tool of stochastic calculus, and stochastic differentials. We will occasionally take time out to look at nearby scenery, such as Gaussian measures in infinite dimensions and infinite-dimensional stochastic analysis. Aside from this we will also look at a few applications. These will include stochastic differential equations in finance and the famous Black-Scholes formula.
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The Fourier transform and Fourier series are basic tools in several parts of mathematics including PDEs. This course covers the basic theory of Fourier series and Fourier integrals. We discuss function spaces like the space of compactly supported functions, rapidly decreasing functions, L^{2}-functions, and discuss the basic theory of distributions. On the way we will introduce the basic concepts of topological vector spaces. The material will cover selected parts from chapter 1 to 4 in the above mentioned lecture notes. The course will be very useful for Math 7380-1 (Singular Integrals) and Math 7390-1 (Harmonic Analysis-II: representations of classical groups) in spring 2011.
The first half of this course covers classical min-max results like Menger theorem, max-flow-min-cut theorem, and Konig theorem. Then we establish the connection between these results and Integer Programming. Under this general framework, we discuss more min-max results concerning packing and covering various combinatorial objects.
This course is a preparation course for the Core I examination in topology. It will cover general (point set) topology, the fundamental group, and covering spaces. We will also introduce simplicial complexes and manifolds, using them often as examples. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online .
This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group in MATH 7510, the homology groups in MATH 7512) 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 7512. 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, we may pursue some of these connections, such as cohomology of groups or the De Rham theorem.
In this course we study smooth closed 4-manifolds. We will describe standard examples and study how to construct exotic 4-manifolds (manifolds that are homeomorphic but not diffeomorphic to a standard example). Along the way we will work with fundamental groups, symplectic 4-manifolds, and Seiberg-Witten invariants. This course should be interesting to students who want an overview of 4-manifold constructions and want to learn how to use advance techniques in topology, geometry, and global analysis to distinguish different smooth 4-manifolds.
On a rotating hollow sphere, there are exactly two points that don't move at all (we might call them the "north and south poles"). The sphere belongs to a large class of important topological spaces with two key features: (1) they have some (perhaps more than one) kind of rotational symmetry, and (2) they have finitely many points that don't move when the space is rotated. "Equivariant cohomology," the subject of this seminar, is a powerful tool for studying the geometry of such spaces. Using equivariant cohomology, we will explore remarkable connections to combinatorics and group theory.
This semester, we will discuss an important result from current research in representation theory called the geometric Satake isomorphism, restricting ourselves to the simplest possible case of the 2 by 2 invertible complex matrices GL2(C). Roughly speaking, this result states that representations of GL2(C) (i.e., group homomorphisms from GL2(C) to GLn(C) for any n) can be understood in terms of the equivariant cohomology of a topological space called the affine Grassmannian. The goal of this course is to make sense of this isomorphism as explicitly as possible and to come up with a new simple proof in this case. (No background in representation theory is assumed.)
This introductory course to recently emerging topic of quantum information theory will introduce students to major recent developments such as quantum cryptography, teleportation, error correction, and quantum computing. Basic concepts of quantum theory such as quantum states, entanglement, measurement, etc. will be incorporated into the course, as well as a few other basic background ideas such as elementary information theory. The mathematical content will center on matrix theory (unitary and Hermitian matrices, positive and completely operators, Gram-Schmidt decomposition, etc.) together with some probabilistic content.
Cluster Algebras is a topic of great interest in current mathematics. They were defined by Sergey Fomin and Andrei Zelevinsky in 2001 in relation to problems in combinatorics and Lie groups. Only a few years later they started playing a key role in a number of developments in representation theory, topology, combinatorics and algebraic geometry. The beauty of the subject is that a great deal of it requires almost no prerequisites. Thus undergraduate students who register will be able to understand and lecture on a number of topics. One of the main goals of the course is to go over applications and relations to various areas of mathematics. Graduate students specializing in representation theory, topology, combinatorics and algebraic geometry will see relations to each of these areas and will be asked to make presentations on their area of expertise.
During the second semester of the course we will be able to fully explore relations with other subjects such as topology, combinatorics, and group theory. There will be more guest lectures by professors in those fields and more student presentations.
For all students who decide to join the class from the second semester, we will arrange for introductory lectures by current students and the instructors.
We will study representations of fundamental groups of knot complements and their combinatorics. Topics will be: The A-polynomial of knots, representations of knot groups into SU(n), combinatorial interpretations of certain knot group representations.
This is the second semester of the first-year graduate algebra sequence. In this course, we will further develop the topics introduced in the first semester (7210). Specific topics include: normal and separable field extensions; Galois theory and applications; solvable groups, normal series, and the Jordan-Hoelder theorem; tensor products and Hom for modules; noetherian rings; the Hilbert Basis Theorem; and algebras over a field.
Matrix computations lies at the heart of most scientific computer codes. In this course, we will study how to perform such computations efficiently and accurately. Topics will include Gaussian elimination, singular value decomposition, eigenvalue solvers and iterative methods for linear systems.
Algebraic geometry has its origin in the study of solutions to systems of polynomial equations. It is of fundamental importance in a wide range of areas of mathematics such as number theory, representation theory, and mathematical physics and also has surprising applications to such fields as statistics, mathematical biology, control theory, and robotics.
Modern algebraic geometry is based on the fundamental notion of a scheme. This course will give an introduction to schemes and their geometry, with particular emphasis on motivating the definitions and constructions and providing many examples. Topics covered will include algebraic varieties, sheaf theory, affine schemes, and projective schemes. .
Class Field Theory is the study of abelian extensions of number fields and more generally of global fields. It describes abelian extensions of a global field K in terms of arithmetic invariants of K such as the ideal class group of K and its generalizations. The most basic example is the description of the abelian extensions of Q as subfields of cyclotomic fields (the Kronecker-Weber Theorem).
One of the goals of this course will be to prove Artin's reciprocity law, which relates abelian extensions to generalized ideal class groups. Artin's reciprocity law can be viewed as a sophisticated generalization of Gauss' quadratic reciprocity law. From the analytic point of view, Artin's reciprocity law states that L-functions associated to one-dimensional representations of abelian Galois groups are identical to Hecke L-series. This is the starting point (i.e. the "easy" case) of a far-reaching theory called the Langlands Program.
This course provides fundamental tools for research in pure and applied mathematics. The topics to be covered are Banach spaces and their generalizations; Baire category, Banach-Steinhaus, open mapping, closed graph, and Hahn-Banach theorems; duality in Banach spaces, weak topologies; Hilbert Spaces, other topics such as commutative Banach algebras, spectral theory, and Gelfand theory.
This is an introductory course to Fourier Analysis whose emphasis is placed on the boundedness of Singular Integral Operators of convolution type. Such a boundedness property plays a fundamental role in various applications in pure and applied analysis. The course also covers such classical topics as Interpolation, Maximal Functions, Fourier Series, and possibly Littlewood-Paley Theory if time permits.
Convex Optimization is an important topic in optimization theory and applications. In this course, we will discuss convex sets, convex functions, convexity and optimization, duality theory and some basic algorithms in convex optimization.
This course provides an introduction to the variational approach to quasi-static fracture in brittle materials. We will first review topics on the linear theory of elasticity and the classical theory of linear fracture mechanics. We will then focus on the derivation of the variational approach to fracture from Griffith criterion and energy conservation. We will review some of the properties of the model, then focus on various numerical approximation methods. Part of the course evaluation will be in the form of student projects, be they numerical or theoretical.
The unitary representation theory of classical groups has a wide range of applications in geometry, quantum mechanics and number theory. In this course, we will focus on the unitary representation theory of SL_{2}(R) and the oscillator representations. We will cover the basic theory of induced representation, spherical transform, Plancherel formula, discrete series and the oscillator representation. We shall also discuss some applications in quantum mechanics and geometry.
This course will cover three related topics in stochastic analysis: (1) stochastic integration, (2) abstract Wiener space theory, and (3) white noise theory. We will study the basic material for these topics and explain connections among them. Recent developments in these areas will also be addressed.
What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
This course will introduce the homology theory of topological spaces. To each space X and nonnegative integer k, there is assigned an abelian group the kth homology group of X. We will learn to calculate these groups, and use them to prove topological results such as the Brouwer Fixed point theorem (in all dimensions) and generalizations of the Jordan curve theorem. Homology theory is important in many parts of modern mathematics. Depending on what was covered in 7510, we may also discuss some other topics related to the fundamental group.
This course gives an introduction to smooth manifolds, which are spaces which locally resemble Euclidean space and have enough structure to support the basic concepts of calculus. We shall cover such fundamental ideas as submanifolds, tangent and cotangent vectors and bundles, smooth mappings and their derivatives, vector fields, differential forms and Stokes's theorem. If time permits, we shall also cover the de Rham Theorem.
Links in a 3-dimensional manifold are 1-dimensional submanifolds. A skein module of a 3-manifold is a module generated by the set of isotopy classes of links in the 3-manifold subject to certain linear relations specified by geometric modifications. Skein modules and related algebra may be used to construct link invariants and 3-manifold invariants. We will study the Jones polynomial and the Witten-Reshetikhin-Tureav invariants of 3-manifolds from the skein theory point of view.
All inquiries about our graduate program are warmly welcomed and answered daily:
grad@math.lsu.edu
Department of Mathematics
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