Graduate Course Outlines, Summer 2010-Spring 2011

Summer 2010

  • MATH 4999-1 and 4999-2: Problem Labs in Real Analysis-1 and Topology-1 respectively---practice for PhD Qualifying Exams in Analysis and in Topology.
  • Instructor: Prof. Cygan.
  • Prerequisite: Math 7311 (for 4999-1) and Math 7510 (for 4999-2).
  • Text: Online Test Banks.
  • MATH 4999-3: Problem Lab in Algebra-1---practice for PhD Qualifying Exam in Algebra.
  • Instructor: Prof. Perlis.
  • Prerequisite: Math 7210.
  • Text: Online Test Bank.
  • MATH 7999-1 and 7999-2:  Vertically Integrated Research and Mentoring.
  • Instructor: Prof. Davidson.
  • Prerequisite: .
  • Text: .

 

Fall 2010

  • MATH 4997-1: Vertically Integrated Research: Equivariant cohomology---Algebra and the shape of space.
  • Instructor: Profs. Sage and Achar.
  • Prerequisites: For graduate students: 7210 and 7510. For undergraduates: 4200 and 2085, or permission of the instructor.
  • References: .

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.)

  • MATH 4997-2 Vertically Integrated Research: Lyapunov Functions, Stabilization, and Engineering Applications.
  • Instructor: Prof. Malisoff with Prof. de Queiroz, and Prof. Wolenski .
  • Prerequisite: For graduate students: 7320, 7386, or permission of the instructor. For undergraduates: 4027, 4340, or permission of the instructor.
  • Text: Notes and recommended references provided by the instructors.
  • References: M.S. de Queiroz, D.M. Dawson, S. Nagarkatti, and F. Zhang, Lyapunov-Based Control of Mechanical Systems. Control Engineering Series, Birkhauser, Cambridge, MA, 2000. ISBN: 0-8176-4086-X
  • M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions. Communications and Control Engineering Series, Springer-Verlag London Ltd., London, UK, 2009. ISBN: 978-1-84882-534-5

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.

  • MATH 4997-3:  Vertically Integrated Research: Cluster Algebras.
  • Instructor: Prof. Yakimov and Dr. Muller.
  • Prerequisite: Permission of the instructor
  • References:

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.

  • MATH 4997-4:  Vertically Integrated Research: .
  • Instructor: Prof. Dasbach. .
  • Prerequisite: Undergraduate Topology or permission of instructor.
  • References: .

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.

  • MATH 7001: Communicating Mathematics I
  • Instructor: Prof. Oporowski.
  • Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.

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.

  • MATH 7210: Algebra I
  • Instructor: Prof. Yakimov.
  • Prerequisite: Math 4200 or the equivalent.
  • Text:  Grove: Algebra and Serge Lang: Algebra. (Revised third edition)

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.

  • Math 7280-1: Commutative Algebra and Algebraic Geometry I.
  • Instructor: Prof. Madden.
  • Prerequisite: Math 7210 or the equivalent.
  • Text: D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry.

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.

  • Math 7280-2: Elliptic Curves and Modular Forms.
  • Instructor: Prof. Hoffman.
  • Prerequisite: Math 7210 or the equivalent.
  • Text: There are two main texts, recommended but not required: A first course in modular forms by Diamond and Shurman and Elliptic curves by James Milne. Also, the use of the MAGMA software system will be explained and utilized. There will be homework assignments collected and graded.

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).

  • MATH 7290-1: Algebraic Number Theory.
  • Instructor: Prof. Perlis.
  • Prerequisites: Prerequisites for MATH 7290-1 are a working knowledge of groups, rings, abstract vector spaces, and fields, including Galois theory, at the level of MATH 7210 and MATH 7211. A student who has not had the prerequisite material should talk with the instructor before signing up for this course.
  • Text: We will follow the text Algebraic Number Theory by A. Frohlich and M. Taylor, Cambridge Studies in Advanced Math 27, published by the Cambridge University Press. The book is available in paperback. We will cover the first seven chapters.

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.

  • MATH 7290-2: Lie Theory.
  • Instructor: Prof. Achar.
  • Prerequisites: Math 7211 (Algebra II).
  • Text: P. Tauvel and R.W.T. Yu, Lie Algebras and Algebraic Groups.

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.

  • MATH 7350: Complex Analysis
  • Instructor: Prof. Estrada.
  • Prerequisite: Math 7311 - Real Analysis I, or the equivalent.
  • Text: Narasimhan, R. and Nievergelt, Y., Complex Analysis in One Variable, second edition, Birkhauser, Boston, 2001.

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. .

  • MATH 7360: Probability Theory
  • Instructor: Prof. Kuo.
  • Prerequisite: Math 7311 (Real Analysis I) or the equivalent.
  • Text: John W. Lamperti: Probability, second edition, John Wiley and Sons, Inc., 1996

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

  • MATH 7380-1: Radon Transforms and Integral Geometry.
  • Instructor: Prof. Rubin.
  • Prerequisite: Math 7311 or the equivalent.
  • Text: Professor's notes.

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.

  • Math 7380-2: Numerical solution of Stochastic Partial Differential Equations.
  • Instructor: Prof. Wan
  • Prerequisites: Math 4032 or the equivalent.
  • Text: Professor's notes.

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.

  • Math 7380-3: Applied Stochastic Analysis.
  • Instructor: Prof. Sengupta.
  • Prerequisites: Math 7311 or equivalent.
  • Text: Lecture notes, and Introduction to Stochastic Integration, by H.-H. Kuo, Springer-Verlag (2006).

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.

  • Math 7386: Theory of Partial Differential Equations.
  • Instructor: Prof. Almog.
  • Prerequisites: .
  • Text: .

.

  • Math 7390-1: Harmonic Analysis-I.
  • Instructor: Prof. Olafsson.
  • Prerequisites: Math 7311 or equivalent.
  • Text: We will mainly use Lecture notes by R. Fabec and G. Olafsson available at http://www.math.lsu.edu/harmonic.

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, L2-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.

  • MATH 7490: Combinatorial Optimization.
  • Instructor: Prof. Ding.
  • Prerequisite: Math 4171 or equivalent.
  • Text: None.

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.

  • MATH 7510: Topology I
  • Instructor: Prof. Litherland.
  • Prerequisite: MATH 4031 and 4200 or equivalent.
  • Text: Topology (2nd ed.) by James R. Munkres.

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 .

  • MATH 7520: Algebraic Topology
  • Instructor: Prof. Stoltzfus.
  • Prerequisite: MATH 7510 and 7512, or equivalent.
  • Text: Algebraic Topology by A. Hatcher

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.

  • MATH 7590: 4-Manifolds.
  • Instructor: Prof. Baldridge.
  • Prerequisite: Math 7512 (Topology II) or permission of instructor.
  • Text: .

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.

Spring 2011

  • MATH 4997-1: Vertically Integrated Research: Vertically integrated research: Equivariant cohomology---Algebra and the shape of space.
  • Instructor: Profs. Sage and Achar.
  • Prerequisites: For graduate students: 7210 and 7510. For undergraduates: 4200 and 2085, or permission of the instructor.
  • References: .

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.)

  • MATH 4997-2: Vertically Integrated Research: Mathematical Problems in Quantum Information Theory.
  • Instructor: Prof. Lawson.
  • Prerequisite: A reasonably good background in linear algebra (e.g.Math 2085) and some basic knowledge of probability should be sufficient background for the course.
  • Text: Quantum Information by Stephen Barnett, Oxford Press, 2009.

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.

  • MATH 4997-3: Vertically Integrated Research: Cluster Algebras.
  • Instructor: Prof. Yakimov and Dr. Muller.
  • Prerequisite: Permission of Instructor.

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.

  • MATH 7002: Communicating Mathematics II
  • Instructor: Prof. Oxley .
  • Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.
  • MATH 7211: Algebra II
  • Instructor: Prof. Adkins
  • Prerequisite: Math 7210: Algebra I.
  • Text: Larry C. Grove, "Algebra," Academic Press, 1983; reprinted by Dover, 2004; ISBN 0-486-43947-X (paperback).

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-Hölder theorem; tensor products and Hom for modules; noetherian rings; the Hilbert Basis Theorem; and algebras over a field.

  • MATH 7280-1: Numerical Linear Algebra
  • Instructor: Prof. Walker.
  • Prerequisite: Linear Algebra and Advanced Calculus.
  • Text: Fundamentals of matrix computation by David S. Watkins (2nd edition) 2002.

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.

  • MATH 7280-3: Algebraic Geometry-II.
  • Instructor: Prof. Sage.
  • Prerequisite: The first year algebra sequence---Math 7210 and Math 7211. The commutative algebra and algebraic geometry class taught by Prof. Madden last semester is not a prerequisite.
  • Text: K. Ueno, Algebraic Geometry 1 and 2.

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. .

  • MATH 7290-1: Class Field Theory.
  • Instructor: Prof. Morales.
  • Prerequisite: Math 7211 or equivalent, Complex Analysis (undergraduate-level should be enough), some exposure to Algebraic Number Theory (the basic number theory needed will be reviewed the first couple of sessions).
  • Text: Class Field Theory, by Nancy Childress, Springer Universitext, 2009. (Available free of charge to LSU students on the SpringerLink site.)

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.

  • MATH 7320: Ordinary Differential Equations
  • Instructor: Prof. Malisoff.
  • Prerequisite: Undergraduate courses on ordinary differential equations, linear algebra, and advanced calculus.
  • Text: Carmen Chicone, Ordinary Differential Equations with Applications, Second Edition, Springer Texts in Applied Mathematics, 2006.
This introductory course on the theory of ordinary differential equations will cover basic topics such as existence and uniqueness theorems for the initial value problem, linearization and linear theory, the Lyapunov approach to stability, omega limit sets, Poincaré-Bendixson theory, and invariant manifolds. Time permitting, it may also include the Hopf bifurcation theorem and some theory of order-preserving dynamical systems. It aims to cover much of Chapters 1-3 from the text.
  • MATH 7330: Functional Analysis.
  • Instructor: Prof. Lipton.
  • Prerequisite: Math 7311 or its equivalent.
  • Text: Functional Analysis by George Bachman and Lawrence Narici, (1966, 2000) now published by Dover.

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.

  • MATH 7380-1: Singular Integrals.
  • Instructor: Prof. Nguyen.
  • Prerequisite: MATH 7311 and MATH 7350.
  • Text: Loukas Grafakos, Classical Fourier Analysis (second edition), Springer, GTM 249.
  • Recommended Reference: Elias Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.

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.

  • MATH 7380-2: Convex Optimization.
  • Instructor: Prof. Zhang.
  • Prerequisite: Basic Optimization Theory, Advanced Calculus and Linear Algebra (Math 4025, 4032, 4035, 4153).
  • Text: Convex Analysis and Optimization, by Dimitri P. Bertsekas, 2003, and Convex Optimization, by Stephen Boyd, 2004, and Lecture Notes .

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.

  • MATH 7384: The variational Approach to Fracture
  • Instructor: Prof. Bourdin.
  • Prerequisites: Permission of the Instructor.

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.

  • MATH 7390-1: Harmonic Analysis-II (Classical Groups).
  • Instructor: Prof. He.
  • Prerequisites: Math 7311 Real Analysis.
  • Text: SL2(R) by Serge Lang, Graduate Texts in Mathematics, Springer. Reference (optional): Harmonic Analysis in Phase Space by G. Folland, Annals of Mathematics Studies, Princeton University Press..

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 SL2(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.

  • MATH 7390-2: Stochastic Analysis.
  • Instructor: Prof. Kuo.
  • Prerequisites: Math 7311 or equivalent
  • Texts: H.-H. Kuo: Introduction to Stochastic Integration, Universitext, Springer, 2006; H.-H. Kuo: Gaussian Measures in Banach Spaces, Lecture Notes in Math. vol. 463, Springer 1975 (Reprinted by Amazon, 2006); and H.-H. Kuo: White Noise Distribution Theory, CRC Press, 1996

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.

  • MATH 7400: Graph Theory
  • Instructor: Prof. Oporowski
  • Prerequisite: The prerequisites for the course are very modest---all graduate students in Mathematics should be able to follow the lectures.
  • Recommended References: Graph Theory by Reinhard Diestel, Third Edition, Springer, 2006, which is available both as a paperback (for about $43 + shipping from various online stores), or as a free download from the author's homepage. Another good book on the subject is Introduction to Graph Theory by Douglas B. West, Prentice Hall, 1996.
  • MATH 7490: Matroid Theory.
  • Instructor: Prof. Oxley.
  • Prerequisite: Permission of the department.
  • Text: J.G. Oxley, Matroid Theory, Oxford, 1992, reprinted in paperback with corrections, July, 2006.

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.

  • MATH 7512: Topology - II
  • Instructor: Prof. Dasbach.
  • Prerequisite: MATH 7510.
  • Text: Algebraic Topology by Allen Hatcher.

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.

  • MATH 7550: Differential Geometry
  • Instructor: Prof. Dani.
  • Prerequisites: Math 7210 and 7510.
  • Text: John M. Lee, Introduction to Smooth Manifolds, Springer, GTM 218.

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.

  • MATH 7590-1: Geometric Topology---Skein Theory.
  • Instructor: Prof. Gilmer.
  • Prerequisite: 7510 Topology I.
  • Text: Class notes plus parts of various research papers.

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-Turaev invariants of 3-manifolds from the skein theory point of view.

  • MATH 7590-2: Contact Topology.
  • Instructor: Prof. Russell.
  • Prerequisite: Math 7512.
  • Text: .