Course Descriptions

Summer 2006 - Spring 2007

Summer 2006

Math 7290: Number Theory

Instructor: Prof. Morales
Prerequisite: Math 7200 or the equivalent.
Text: Algebraic Number Theory by A. Froehlich and M. Taylor, Cambridge University Press, 1991.

This course is intended to be a bridge between undergraduate and graduate-level number theory and should appeal to a broad audience. Due to time constraints during the summer term, this is not intended to be a full-fledged research-level number theory course , but rather a complete introduction to the standard algebraic and analytic techniques that are used in more advanced courses.

We will develop in some detail algebraic tools such as the theory of Dedekind domains, discrete valuation rings, p-adic numbers, completions, etc., and also some analytic tools such as zeta functions. We will also discuss applications of analytic methods to classical results such as the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.

The "official" textbook is "Algebraic Number Theory" by A. Froehlich and M. Taylor (Cambridge University Press 1991), but we will also use other sources for some parts of the course.

Fall 2006

MATH 7001: Communicating Mathematics I
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.

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 7200: Geometric and Abstract Algebra
Instructor: Prof. Adkins .
Prerequisite: Math 4200 or the equivalent.
Text: Algebra: an approach via module theory, by Adkins and Weintraub

This course will cover the basic material on groups, rings, fields and modules with special emphasis on linear algebra.

MATH 7211: Representations of Finite & Compact Groups
Instructor: Prof. Sage
Prerequisite: Math 7200 or comparable familiarity with groups and linear algebra
Text: Simon, Representations of finite and compact groups

Representation theory is the study of the ways in which a given group may act on vector spaces. Intuitively, it investigates ways in which an abstract group may be interpreted concretely as a group of matrices with matrix multiplication as the group operation. Group representations are ubiquitous in modern mathematics. Indeed, representation theory has significant applications throughout algebra, topology, analysis, and applied mathematics. It also is of fundamental importance in physics, chemistry, and material science. For example, it appears in quantum mechanics, crystallography, or any physical problem in which one studies how symmetries of a system affect the solutions.

This course is designed to give an introduction to the representation theory of finite and compact groups. It will start with the representations of finite groups over the complex numbers. In particular, we will discuss how finite-dimensional representations break up into sums of irreducible representations, Schur's lemma, the group algebra, character theory, induced representation and Frobenius reciprocity. We will give many examples, culminating in the representation theory of the symmetric groups and the remarkable combinatorics associated with it. We will then turn to the representation theory of compact groups, which is formally very similar to the finite group situation. The course will end with a discussion of the representation theory of the unitary groups and its close relationship with the representation theory of the symmetric groups.

MATH 7280: Homological Algebra
Instructor: Prof. Hoffman
Prerequisites: Math 7200 or the equivalent.
Text: An Introduction to Homological Algebra by Charles A. Weibel, Cambridge University Press, ISBN: 0521559871

Homological Algebra is a tool that appears in almost all branches of mathematics, ranging from mathematical physics, PDEs, Lie groups and representation theory, through topology (where it originiated) and on to number theory and combinatorics. We will cover the basics of homology of chain complexes, elementary category language, derived functors, Ext and Tor, spectral sequences, simplicial methods. There are various further topics and directions which will depend on the available time and interests of the class. Each student will have a project to do - these can be chosen from an application of homological methods in the student's area of interest.

Math 7290: Toric Geometry and Combinatorial Commutative Algebra
Instructor: Prof. Madden.
Prerequisite: Math 7210
Text: Ezra Miller, Bernd Sturmfels: Combinatorial Commutative Algebra (Springer Graduate Texts in Mathematics, Paperback)

The theme of this course is the interplay between the combinatorial geometry of convex bodies in Rⁿ with integral vertices and the algebraic geometry of toric varieties, algebraic varieties that are locally defined by differences of monomials. This is a very active area of current research, with a surprisingly diverse range of connections, including applications in computational algebra, statistical modeling, singularity theory and elsewhere. This course will cover the following topics:

basic theory of commutative monoids, monoid algebras and affine toric varieties
selected topics in toric geometry (selected from Ewald: Combinatorial Convexity and Algebraic Geometry)
advanced topics, including resolutions and syzygies of binomial ideals, combinatorial descriptions of these objects, and applications (as in Part II of Miller and Sturmfels)

MATH 7311: Real Analysis I
Instructor: Prof. Fabec
Prerequisite: Math 4032 or 4035 or the equivalent.
Text: Real Analysis, Measure Theory, Integration, and Hilbert Spaces by E. M. Stein and Rami Shakarchi, Princeton University Press, 2005.

This course deals with Lebesgue measure on Euclidean space and its analog in the more general setting of measure spaces. We will obtain all the basic properties of measure and integration, including convergence theorems, Fubini's Theorem, the Radon-Nikodymn Theorem, and the Caratheodory extension process. Banach spaces, particulary the L^p spaces will be discussed.

Time permitting, we will cover other topics important in real analysis, for which useful references may include Real Analysis by Royden and Real Analysis by Rudin.

MATH 7360: Probability Theory
Instructor: Prof. Sengupta
Prerequisite: Math 7311.
Text:

Math 7380-1: Applied Stochastic Analysis
Instructor: Prof. Sundar
Prerequisites: Math 4031, Math 4032, and Math 3355.
Text: Arbitrage theory in continuous time by T. Bjork

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

MATH 7380-2: Mathematical Topics in Systems Theory
Instructor: Prof. Malisoff
Prerequisite: Elementary differential equations and linear algebra; some background in real analysis (e.g. Math 7311-7312) is suggested but not required
Text: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition by E.D. Sontag, Springer, New York, 1998. ISBN 0-387-984895

Systems and control theory is one of the most central and fast growing areas of applied mathematics and engineering. This course provides a basic introduction to the mathematics of finite-dimensional, continuous-time, deterministic control systems at the beginning graduate student level. The course is intended for PhD students in applied mathematics, and for engineering graduate students with a background in real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for interdisciplinary research at the interface of applied mathematics and control engineering. This is a rigorous, proof-oriented systems theory course that goes beyond classical frequency-domain or more applied engineering courses. Emphasis will be placed on controllability and stabilization.

MATH 7390-1: Applied Harmonic Analysis: Introduction to Radon transforms
Instructor: Prof. Rubin
Prerequisite: Math 7311 (Real Analysis-I) or equivalent.
Text: Dr. Rubin will use his own notes for this course.

This is an introductory course in the theory of the Radon transform, one of the main objects in integral geometry and modern analysis. 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 unit sphere, some aspects of the Fourier analysis in the context of its application to integral geometry.

MATH 7390-2: Scientific Computing
Instructor: Prof. Aksoylu
Prerequisite: Basic concepts of numerical analysis, basic knowledge of linear algebra, and basic programming skills in C, Matlab or other computer language. Preferably Math 4065, 4066 or equivalents.
Text: Scientific Computing: An introductory survey by Michael T. Heath, McGraw Hill 2002, Lecture notes

This course serves as a foundation for analysis, design, and implementation of numerical methods. In particular, it provides an in-depth view of practical algorithms for solving large-scale linear systems of equations arising in the numerical implementation of various problems in mathematics, engineering and other applications.

The course will cover the following material:

Linear algebra and numerical linear algebra refresher, in particular, solutions to system of linear equations.
Matrix factorizations.
Linear least squares.
Ortogonalization methods.
Eigenvalue problems.
Basic iterative methods.
Krylov subspace methods.
Preconditioning techniques.
Multilevel methods such as multigrid.

The course will focus on both the theoretical aspects and the numerical implementation of these methods. Evaluation will be based upon both theoretical and numerical projects.

MATH 7390-3: Mathematical Models of Superconductivity
Instructor: Prof. Almog
Prerequisite: Some functional analysis and basic PDE and ODE is necessary. Some of the background will be reviewed in class.
Text: A list of articles for reading will be provided. The background material will be taught from: L. C. Evans - Partial Differential Equations and L. Nirenberg - Topics in nonlinear functional analysis.

At first I'll provide an introduction to the theory of Sobolev spaces, to elliptic regularity to direct methods of the calculus of variations, and to linear bifurcation theory. This should include about two thirds of the course. Then, I'll present the Ginzburg-Landau energy functional, prove existence of minimizers and discuss some general properties of the minimizer. I'll then cover some topics from the theory of superconductivity, including: Surface superconductivity, Abrikosov's lattices, radial vortices, self-duality and the Jaffe-Taubes assumption, thin rings, and if time allow the Bethuel-Brezis-Helein theorem and the weak field limit. (I probably won't be able to do that much so you can have influence on the material to be covered).

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 7510: Topology I
Instructor: Prof. Litherland
Prerequisite: MATH 2057 or equivalent.
Text: James R. Munkres, Topology, 2nd edition, Prentice Hall, 2000.

Topology contains at least three (overlapping) subbranches: general (or point-set) topology, geometric topology and algebraic topology. General topology grew out of the successful attempt to generalize some basic ideas and theorems (e.g., continuity, open and closed sets, the Intermediate Value and Bolzano-Weierstrass Theorems) from Euclidean spaces to more general spaces. The core of this course will be a thorough introduction to the central ideas of general topology (Chapters 2 - 5 of Munkres). This material is fundamental in much of modern mathematics. Time permitting, we will also look briefly at the ideas of homotopy and the fundamental group (Chapter 9), subjects that belong to the algebraic-geometric side of topology, and will be covered in greater depth in 7512.

MATH 7590-1: Geometric Topology
Instructor: Prof. Dasbach
Prerequisite: MATH 7512 (Topology II) or equivalent.
Text: Notes

The course will mainly focus on two aspects of Geometric Topology:

Braid groups as central objects in topology: Representations of braid groups, Linearity of braid groups, knots as closed braids, subgroups of braid groups, solutions to word problems and conjugacy problems, generalization of braid groups
Three manifolds and their groups: Dehn surgery on knots and links

If time permits we will cover related topics in Geometric Topology.

MATH 7590-2: Riemannian Geometry (connections and Gauge theory)
Instructor: Prof. Baldridge
Prerequisite: Math 7510 (Topology I) or the equivalent.
References: Carmo, Riemannian Geometry (Birkhauser, ISBN: 0-8176-3490-8) and Introduction to Symplectic Geometry by Dusa McDuff and Dietmar Salamon (Oxford University Press, ISBN: 0-1985-0451-9 ) will be useful.

Riemannian Geometry (connections and Gauge theory). This course is an introduction to Riemannian geometry: manifolds, metrics, Levi-Civita connections, and curvature. Riemannian geometry is key to understanding Einstein′s general relativity and plays an important role in gauge theory and invariants of smooth 4-manifolds. We will use this technology to introduce and investigate symplectic geometry, another important topic in mathematics that also comes from physics.

Students do not need to be enrolled in the Spring 2006 differential geometry course (Math 7550) to take Math 7590-2 in the fall.

Spring 2007

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.

This course provides practical training in the teaching of calculus, how to write mathematics for publication, how to give a mathematical talk, 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. Nobile
Prerequisite: Math 7200: Geometric and Abstract Algebra.
Text:

MATH 7280: Applications of Homological Algebra
Instructor: Prof. Achar .
Prerequisite:
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MATH 7290: Algebraic Number Theory
Instructor: Prof. Schlichting .
Prerequisite:
Text:

MATH 7320: Ordinary Differential Equations
Instructor: Prof. Estrada
Prerequisite: Advanced Calculus and Linear Algebra (preferably Math 7311 and 7200)
Text:

MATH 7330: Functional Analysis
Instructor: Prof. Davidson
Prerequisite:
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MATH 7350: Complex Analysis. This course will serve as a Core-2 course for the current year.
Instructor: Prof. Olafsson
Prerequisite: Math 7311
Text:

Note: Math 7350 is being offered this year as a Core-2 Course. Students taking this course will have the option of taking a Complex Analysis Core-2 PhD Qualifying Examination.

MATH 7370: Lie Groups and Representation Theory
Instructor: Prof. He
Prerequisites: Lebesgue measure and integration, and a semester each of graduate algebra and point-set topology, or the equivalent.
Text: Lie groups beyond Introduction By A. Knapp.

Lie groups, Lie algebras, subgroups, homomorphisms, the exponential map. Also topics in finite and infinite dimensional representation theory.

MATH 7380: Partial Differential Equations
Instructor: Prof. Tom
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MATH 7390-1: Stochastic Analysis
Instructor: Prof. Kuo
Text:

MATH 7390-2: Finite Element Method: Analysis and Implementation
Instructor: Prof. Bourdin
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MATH 7400: Graph Theory
Instructor: Prof. Oporowski
Prerequisite:
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MATH 7512: Topology - II
Instructor: Prof. Gilmer
Prerequisite:
Text:

MATH 7520: Algebraic Topology
Instructor: Prof. Owens
Prerequisite:
Text:

MATH 7550: Differential Geometry
Instructor: Prof. Cohen
Prerequisite:
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MATH 7590: Mapping Class Groups
Instructor: Prof. Brendle
Prerequisite: Math 7510 (Topology-I or equivalent)
Text: Farb and Margalit, A Primer on Mapping Class Groups (manuscript in preparation)

This course will serve as an introduction to mapping class groups of surfaces. Mapping class groups are a fundamental object of study in topology, as the automorphism groups of 2-manifolds, but also arise naturally in many other fields, such as complex analysis and algebraic geometry. We will survey some basics such as generators and relations for the mapping class group, subgroups important in 3-manifold theory such as the Torelli group and the handlebody subgroup, and representations of mapping class groups. As time permits, and according to the interests of the class, we will also discuss related groups in geometric group theory, Teichmuller theory, and associated combinatorial structures such as the curve complex, among other topics.

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