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Graduate Course Outlines, Summer 2008-Spring 2009

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Summer 2008

  • Introduction to Gröbner Bases
  • Instructor: Prof. Lax
  • Prerequisite: Math 7200, or 7210, or the equivalent.
  • Text: Introduction to Gröbner Bases by Adams and Loustaunau (AMS Graduate Studies in Math. vol. 3, 1994).

This course will focus primarily on ideals in polynomial rings. A Gröbner basis of such an ideal is a generating set that has certain properties with respect to a partial order on the monomials. An algorithm for finding a Gröbner basis was developed by Buchberger in his dissertation in the early 70's, but was largely ignored until advancements in computer speed and memory made the algorithm practical in many cases. Gröbner bases are useful in solving systems of polynomial equations, and, in some cases, they provide a generalization for the method of Gaussian elimination that is used for systems of linear equations. We will study the basic ideas in the theory, and consider applications to problems in commutative algebra, affine algebraic geometry, and algebraic coding theory. We will also see how to use software such as Mathematica, Macaulay2, and SAGE to compute Gröbner bases, and students will be asked to use such software for some of the exercises.

Fall 2008

  • 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 7280: Combinatorial Commutative Algebra
  • Instructor: Prof. Madden
  • Prerequisite: Math 7210 or the equivalent.
  • Text: Miller and Sturmfels, Combinatorial Commutative Algebra.

Combinatorial commutative algebra. This course concerns the interplay between combinatorial geometry of convex bodies with integral vertices and the geometry of algebraic varieties defined by monomials or differences of monomials. This is an active and accessible 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 the basic theory of affine semigroups, and toric varieties and a selection of topics including free resolutions of binomial ideals, combinatorial descriptions of these objects, and applications.

  • MATH 7290-1: Algebraic Groups I
  • Instructor: Prof. Achar
  • Prerequisites: Math 7211 Algebra II.
  • Text: J. Humphreys, Linear Algebraic Groups, Springer-Verlag, 1975.

An algebraic group is an affine variety that is also a group, such that the group multiplication map and the inverse map are morphisms of varieties. Thus, algebraic groups are the algebraic-geometry version of Lie groups. In this course, we will cover the basic structure theory of reductive algebraic groups (Borel subgroups, maximal tori, Weyl groups, Bruhat decomposition), and we will classify the simple algebraic groups. If time permits, we may begin discussing the representation theory of algebraic groups. Note: Algebraic geometry is not a prerequisite for this course; we will cover as much algebraic geometry as we need during the course.

  • MATH 7290-3: Mathematical Logic
  • Instructor: Prof. Delzell
  • Prerequisites: Mathematical maturity (no specific courses).
  • Text: Herbert B. Enderton, A Mathematical Introduction to Logic, 2nd edition, Harcourt/Academic Press, 2001.

Sentential logic deals with formulae such as "(p&q) --> (not r)," where p,q,r are "sentence symbols," each capable of being assigned the value "true" or "false." We shall prove the compactness theorem for sentential logic. First-order logic deals with formulae such as: "for all x, there exists a y such that y is a prime number and x is less than y." First-order logic (unlike sentential logic) is admirably suited to expressing and analyzing the deductions encountered in mathematics. We shall prove the soundness of first-order logic, and its converse, viz., Goedel's completeness theorem for first-order logic. Finally, we shall (if the teacher and students move quickly enough) prove Goedel's (first) incompleteness theorem, viz., number theory is not recursively axiomatizable.

  • MATH 7311: Real Analysis I
  • Instructor: Prof. Richardson
  • Prerequisite: Math 4032 or 4035 or the equivalent.
  • Text: Measure and Integration: A Concise Introduction to Real Analysis, Richardson, John Wiley & Sons. The revised prepublication edition will be available at Barnes and Noble in the Union as a Course Pack by mid August.

We will treat measure theory and integration on measure spaces. The examples of the real line and of Euclidean space will be emphasized throughout. Topics will include the Hopf extension theorem, completion of the Borel measure space, Egoroff's theorem, Lusin's theorem, Lebesgue dominated convergence, Fatou's lemma, product measures, Fubini's theorem, absolute continuity, bounded variation, Vitali's covering theorem, Lebesgue differentiation theorems, and the Radon-Nykodim theorem. Applications to Lp and its dual, and the Riesz-Markov-Saks-Kakutani theorem will be presented if there is sufficient time. Visit the class website for further information.

Functional Analysis is the language of modern mathematics. The course provides an introduction to the theory of normed spaces and metric spaces, linear operators and linear functionals, Banach and Hilbert spaces, spectral theory, and other important topics. The students will become familiar with such notions as isometry, completeness, orthogonal projections, compactness, duality, and many others.

  • MATH 7360: Probability Theory
  • Instructor: Prof. Sundar
  • Prerequisite: Math 7311.
  • Text: A Course in Probability Theory - K.L. Chung (Academic Press)

The course starts right from the concept of a random variable and proceeds to discuss independence of random variables, strong limit theorems, weak convergence, characteristic functions, the central limit theorem, conditional expectation, and basics of the Brownian motion. This is a basic and standard course in modern probability theory and is based on the development and contributions of A. N. Kolmogorov and J. L. Doob.

  • Math 7380-1: Free discontinuity problems in image processing and fracture mechanics
  • Instructor: Prof. Bourdin
  • Prerequisites:
  • Text:

 

  • Math 7380-2: Finite Difference Methods
  • Instructor: Prof. Brenner.
  • Prerequisites:
  • Text:

This is an introduction to finite difference methods for the numerical solution of differential equations. We will derive many fundamental finite difference schemes and develop mathematical tools for their analysis. Topics will include:

  • Ordinary Differential Equations: Euler's methods; Consistency, Stability and Convergence; Runge-Kutta Methods; Multistep Methods; Adaptive Methods; Absolute Stability and Stiff Equations; Centered and Upwind Schemes for Two-point Boundary Value Problems.
  • Partial Differential Equations: Advection Equations; Courant-Friedrichs-Lewy Condition; Von Neumann Stability Analysis; Crank-Nicolson Scheme for Parabolic Equations, Centered Difference Scheme and Discrete Maximum Principle for Elliptic Boundary Value Problems.

 

  • MATH 7380-3: Applied Stochastics
  • Instructor: Prof. Kuo
  • Prerequisites: Math 3355 and Math 4031.
  • Text: Kuo, H.-H.: Introduction to Stochastic Integration, Universitext, Springer, 2005.

We will study the basic theory of stochastic integration with applications to finance. Many concrete examples will be used to motivate the concepts and theorems. We will assume the advanced calculus and elementary probability theory. Basic knowledge of measure theory and Hilbert space will be helpful, but is not absolutely necessary. Below are some items to be covered in this course:

  1. Brownian motion
  2. Construction of Brownian motion
  3. Wiener integrals
  4. Ito's integrals
  5. Stochastic integrals for martingales
  6. The Ito formula
  7. Girsanov theorem
  8. Wiener-Ito theorem
  9. Stochastic differential equations
  10. Hedging portfolio
  11. Arbitrage and option pricing
  12. Black-Scholes analysis

 

  • MATH 7390-1: Applied Harmonic Analysis
  • Instructor: Prof. Olafsson
  • Prerequisite: Math 7311 - Real Analysis, I.
  • Text: Lecture Notes by Prof. Olafsson.

The heat equation is one of the more important partial differential equations. We will discuss its relation to complex analysis and infinite dimensional analysis. The main topic are:

  1. Basic Introduction to the heat equation on Rn.
  2. Short introduction to Fourier analysis.
  3. The solution of the heat equation using Fourier analysis. The heat kernel.
  4. Holomorphic functions and the Fock space of holomorphic functions on Cn.
  5. The Segal-Bargmann transform as an unitary isomorphism of L2-spaces into to the Fock space.
  6. The infinite dimensional case.
  7. If there is still time, then we will discuss some generalizations, in particular related to root systems, multiplicity functions, finite reflection groups.

More information is posted at http//www.math.lsu.edu/~olafsson.

  • MATH 7390-2: Elliptic PDEs on Nonsmooth Domains
  • Instructor Prof. Sung
  • Prerequisite:
  • Text:

 

  • 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.

Topology I has been revised to cover the fundamentals of basic point-set topology as well as homotopy and the fundamental group. The revised syllabus for Topology I is available at Math 7510 Topics.

  • MATH 7520: Algebraic Topology
  • Instructor: Prof. Gilmer
  • Prerequisite: MATH 7510 and 7512, or equivalent.
  • Text: Hatcher, [i]Algebraic Topology[/i]

A fundamental problem in topology is that of determining whether or not two spaces 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. 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, cohomology. and homotopy. We will start off approximately where last springs 7512 left off. We will study topics from chapters 3 and 4 of Hatcher's book, including Poincare duality. Geometric examples, including surfaces, projective spaces, lens spaces, etc., will be used to illustrate the techniques. We may also discuss characteristic classes from Milnor's book.

  • MATH 7590: Hyperbolic Geometry
  • Instructor: Prof. Litherland
  • Prerequisite: Math 7550.
  • Text: Notes available online. (This is currently a work in progress.)

Hyperbolic geometry has been an important tool for the study of three-manifolds since the groundbreaking work of Thurston in the 1970s. In particular, Thurston's geometrization conjecture, recently proved by Perelman, is a far-reaching generalization of the Poincare conjecture. This course will discuss the geometry of hyperbolic space and the definition of a hyperbolic structure on a manifold. Examples of hyperbolic structures on 2- and 3-manifolds will be given, and we will discuss the idea of hyperbolic Dehn surgery.
A good source (though not a required text) for this material and more is John G. Ratcliffe, Foundations of Hyperbolic Manifolds (2nd ed.), Springer GTM 149. Also of interest: William P. Thurston, Three-Dimensional Geometry and Topology (ed. Silvio Levy), Princeton Mathematical Series 35.

Spring 2009

  • MATH 4999-1: Vertically Integrated Research: Mathematical Problems in Relativity Theory
  • Instructor: Prof. Lawson.
  • Prerequisite for graduate students who will enroll: Math 7311. (For undergraduates: Math 4031 would be nice, but Math 2057 and some linear algebra are sufficient.)
  • References: 1. Special Relativity (M.I.T. Introductory Physics Series) by A. P. French. Paperback: 304 pages Publisher: W. W. Norton; 1 edition (August 19, 1968), ISBN-10: 0393097935 or ISBN-13: 978-0393097931.
    2. Special Relativity (Springer Undergraduate Mathematics Series) by N.M.J. Woodhouse. Paperback: 192 pages, Publisher: Springer, ISBN-10: 1852334266 or ISBN-13: 978-1852334260

This is an LSU VIR course, or vertically integrated research course, for graduate and undergraduate students. It will be taught as Math. 4999-1, to allow both undergraduate and graduate students to participate for credit. The topic is the foundations of special relativity and the related mathematics, including algebra, analysis, and geometry. Topics to be covered include 4--dimensional space--time, reference frames and coordinates, Lorentz transformations, simultaneity, time dilation and length contraction, causality and prohibition of motion faster than light, and composition of velocities. Geometric ideas will be stressed, and there will be an attempt to make connections with hyperbolic geometry. Students will have the opportunity to discuss selected topics in smaller groups and then present them to the others. The course serves as an introduction to research. The expectation is therefore, that the smaller groups will discuss open problems and work on them as we get more familiar with the topic.

  • MATH 4999-2: Vertically Integrated Research: Computing the Bandwidth of Graphs.
  • Instructor: Prof. Ding and and Dr. Bilinski .
  • Prerequisite: Math 2020.

This is a 3-hour research course with graduate credit. Students will read research papers and solve open problems. More details on the topic, Computing the bandwidth of graphs, can be found in Prof. Ding's website.

  • MATH 4999-3: Vertically Integrated Research: .
  • Instructor: Prof. Dasbach and Prof. Stoltzfus.
  • Prerequisite: Differential geometry of curves and surfaces in space.

We will discuss the differential geometry of curves and surfaces in space. The course will also incorporate the use of Mathematica in the computation and visualization. As applications we will offer projects in Computer Graphics and 3-dimensional positioning systems (location detection in space with the help of two cameras).

Project: Each student would complete a project (possibly in groups) and make a presentation on a geometric topic.

  • MATH 4999-4: Vertically Integrated Research: Equivariant cohomology: Algebra and the Shape of Space.
  • Instructor: Prof. Sage.
  • Prerequisite: For graduate students: 7210 and 7510. For undergraduates: 4200 and 2085, or permission of the instructor.

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.

  • MATH 4999-5: Vertically Integrated Research: Complex Materials and Fluids.
  • Instructor: Prof. Shipman.
  • Prerequisite: 2057 for undergraduate students; 7311 for graduate students.

This seminar will investigate the dynamical behavior of complex materials and fluids and the behavior of fields that they host. One of the topics will be the interaction of electromagnetic waves with composite materials consisting of natural and artificial components. This includes the creation and characterization of meta-materials and the phenomena of resonance. The mathematics will involve a wide range of topics in analysis and partial differential equations, including homogenization, scattering theory, special functions, and integral equations.

  • 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 7211: Algebra II
  • Instructor: Prof. Adkins
  • Prerequisite: Math 7210: Algebra I.
  • Text: Algebra, by Larry C. Grove.

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. 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.

  • MATH 7280-1: Numerical Linear Algebra
  • Instructor: Prof. Zhang
  • Prerequisite: Linear Algebra and Advanced Calculus.
  • Text: Fundamentals of Matrix Computations, 2nd edition, by David S. Watkins, Wiley-Interscience .

Numerical linear algebra is central to scientific computing. A good understanding of the basic algorithms (derivation, applicability, efficient implementation) in numerical linear algebra is indispensable to anyone who wants to do research involving large scale computation. The following topics will be covered in this course: LU factorization, Cholesky factorization, QR factorization, Singular Values, Condition Numbers, Backward Stability, Rayleigh Quotient Iteration, QR Algorithm, Jacobi Iteration, Gauss-Seidel Iteration, SOR Iteration, Steepest Descent, Conjugate Gradient.

  • MATH 7290-1: Algebraic Number Theory
  • Instructor: Prof. Morales.
  • Prerequisite: Math 7210 and 7211, or the equivalent.
  • Text: The textbook for this course is "Algebraic Number Theory" by A. Fröhlich and M. Taylor (Cambridge University Press 1991).

This is an introductory course. About 2/3 of the course will be devoted to algebraic tools in number theory. These include Dedekind domains, module theory and ideal class groups, discrete valuation rings, p-adic numbers, completions. The remaining 1/3 will include analytic tools such as zeta functions and applications to classical results such as the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.

  • MATH 7290-2: Algebraic Groups II.
  • Instructor: Prof. Sage.
  • Prerequisite: Algebraic groups I, or permission of the instructor.
  • Text: Chriss and Ginzburg, Complex Geometry and Representation Theory.

Geometric methods play a central role in modern representation theory. Indeed, geometry often reveals unexpected connections between disparate phenomena. For example, the irreducible representations of the Weyl group of SL(n,C) are parametrized by partitions of n. Nilpotent conjugacy classes in the Lie algebra of SL(n,C) are also parametrized by partitions of n. Is this just a combinatorial coincidence? No--it is possible to define a natural bijection between these two sets via a geometric construction, and this bijection (the Springer correspondence) has been of fundamental importance in representation theory.

In this course, we will discuss several examples of geometric methods in representation theory. In particular, we will study flag varieties and the nilpotent cone and how they can be used to study representations (of Weyl groups, simple algebraic groups, Hecke algebras, etc) from a geometric perspective. Time permitting, we will also examine the building of a simple algebraic group.

  • MATH 7320: Ordinary Differential Equations
  • Instructor: Prof. Wolenski
  • Prerequisite: Advanced Calculus and Linear Algebra (preferably Math 7311 and 7200).
  • Text:
  • MATH 7325: Finite Element Methods.
  • Instructor: Prof. Sung
  • Prerequisites: Linear Algebra (Math 2085), Advanced Calculus (Math 4031) and Partial Differential Equations.
  • Text: The Mathematical Theory of Finite Element Methods (Third Edition) by S.C. Brenner and L.R. Scott, Springer, 2008

In this course we will develop the mathematical theory of finite element methods, one of the main comuptational tools in science and engineering. The following topics will be covered: Hilbert spaces, variational formulations of elliptic boundary value problems, Ritz-Galerkin methods, constructions of finite element spaces, interpolation error estimates, discretization error estimates, nonconforming methods, mixed methods and adaptive methods.

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

This is 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 7370: Representation Theory of Linear Groups.
  • Instructor: Prof. He
  • Prerequisite: Abstract Algebra (7210) and Real Analysis I (7311)
  • Text: Lie Groups, Lie Algebras and Their Representations, by Varadarajan. Optional Reference: Representation of Linear Groups: An Introduction Based on Examples from Physics and Number Theory, by Rolf Berndt.

This is an introductory course to the representation theory of the real linear groups. The language of representation theory is widely used in mathematical physics, number theory, Harmonic analysis and geometry. We will go over the basic structure theory, discussing Haar measure, Lie subgroups, Lie algebras, exponential maps, universal enveloping algebras, and integration formulas. We will then introduce the representation theory of linear groups. We will further discuss three basic types of groups: nilpotent groups, compact groups, and semisimple groups, and their representations.

  • MATH 7380-1: Applied and Computational Complex Analysis.
  • Instructor: Prof. Antipov
  • Prerequisites: MATH 4036 or a graduate course in complex analysis.
  • Text: Lecture Notes; Reference texts: P. Henrici, Applied and Computational Complex Analysis, Vol. 3; Automorphic functions, L R Ford; F D Gakhov, Boundary value problems.

The whole field of computational complex analysis has grown rapidly during the past decade. Partly, this is due to discovering new methods in the theory of conformal mappings, integral equations and boundary-value problems of analytic functions which have helped to solve long-standing problems in physics and engineering sciences. This course focuses on the theory and computational aspects of conformal mappings for multiply connected regions and the Riemann-Hilbert problem of the theory of automorphic functions and on Riemann surfaces of algebraic functions. We shall discuss elements of the theory of Schottky groups, abelian differentials, the Schottky-Klein prime function, and the Riemann-Roch theorem. Applications: inverse problems in material sciences and fluid mechanics, diffraction of electromagnetic waves, and fracture mechanics.

  • MATH 7380-3: Topics in Matrix Analysis.
  • Instructor: Prof. Lawson
  • Prerequisites: Linear Algebra (2085 or higher), Analysis (4031 or higher).
  • Text: Positive Definite Matrices by Rajendra Bhatia.

Positive matrices play a similar role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines,including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. The course will introduce several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. Specific topics include positive and completely positive maps, monotonicity, operator norms, operator inequalities, and matrix means, including very recent developments. The course text has numerous exercises integrated into the text itself and at the end of each chapter. This makes it suitable for teaching in the style of a "working seminar'' with heavy student involvement and participation accompanied by suitable professorial explanation, guidance, coaching, and filling in of background.

  • MATH 7390-1: Spherical Harmonics and Integral Geometry.
  • Instructor: Prof. Rubin
  • Prerequisites: Real analysis (MATH 7311); Functional Analysis (MATH 7330)
  • Text: Prof. Rubin will provide his notes.

The focus of the course is introduction to the Fourier analysis on the unit sphere (Spherical Harmonics) in the n-dimensional real Euclidean space and its application to integral (or, more generally, pseudo-differential) operators commuting with rotations, in particular, to Minkowski-Funk transform and some other transformations of integral geometry. Operators of that type arise in PDE, harmonic analysis, group representations, mathematical physics, geometry, and many other areas of mathematics and applications.

  • MATH 7390-2: Stochastic Analysis.
  • Instructor: Prof. Kuo
  • Prerequisites:
  • Text:
    1. Kuo, H.-H.: Introductory Stochastic Integration. Universitext, Springer, 2005
    2. Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975
    3. Kuo, H.-H.: White Noise Distribution Theory. CRC Press, 1996.

This course contains three parts:
(1) a brief review of the Ito theory of stochastic integration with applications to finance;
(2) infinite dimensional analysis; and
(3) white noise theory. I will explain the basic knowledge in these three related areas and give a comprehensive outline leading to the current research in stochastic analysis.

  • 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 at Universität Hamburg. Another good book on the subject is Introduction to Graph Theory by Douglas B. West, Prentice Hall, 1996.

The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, Eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, and graph minors. There are many books on graph theory. Prof. Oporowski recommends the references listed above, but they are not absolutely necessaary. He will present the lectures with his own notes, which will be available for download, and which should make good study material. However, the amount of detail in the lecture notes is less than that of either of the mentioned books. If your interest in the subject is anything more than superficial, you would be well advised to get at least one of those books - especially the first one listed. Grades for the course will be based 60% on homework and 40% on two exams (midterm and final). Decisions in borderline cases will be made on the basis of class participation. There will be the total of over twenty problems given as homework, and the two lowest problem scores will be dropped. If you have any questions about the course, do not hesitate to contact Prof. Oporowski.

  • MATH 7512: Topology - II
  • Instructor: Prof. Cohen
  • Prerequisite: MATH 7510 (and MATH 7210), or equivalent.
  • Text: A. Hatcher, Algebraic Topology.

A fundamental problem in topology is that of determining whether or not two spaces 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. 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 and cohomology. To a topological space, we will associate sequences of abelian groups, the homology and cohomology groups. Topics (from chapters 2 and 3 of Hatcher's book) include simplicial, singular, and cellular homology, Mayer-Vietoris sequences, and universal coefficient theorems. Geometric examples, including surfaces and projective spaces, will be used to illustrate the techniques. Discussion of cohomology theory, including products and duality, will (presumably) continue in MATH 7520 Algebraic Topology.

  • MATH 7550: Differential Geometry
  • Instructor: Prof. Dasbach
  • Prerequisites: Math 7200 and 7510.
  • Text:

This course gives an introduction to smooth manifolds, which are spaces that locally resemble Euclidean space and have enough structure to support the basic concepts of calculus. We shall cover the core material in chapters 1, 2 and 4, including submanifolds, tangent vectors and bundles, smooth mappings and their derivatives, the implicit and inverse function theorems, vector fields, differential forms and Stokes's theorem. If time permits, we shall continue with chapter 5, on sheaves, cohomology and the de Rham Theorem.

  • MATH 7590-1: Geometric Topology
  • Instructor: Prof. Stoltzfus
  • Prerequisite: Basic properties of the fundamental group and rudiments of algebraic topology (Math 7512: Topology II).
  • Text: Braid Groups, GTM, Vol. 247 by Kassel and Turaev.

Geometric Topology was conceived to be a broad survey of selected current research areas in topology within our Department: braid groups, arrangements, 3-manifolds, in particular complements of knots/links, and mapping class groups of 2-manifolds. This course will start with an introduction to braid groups (with eventual sideways views of the relationship of active research interests in the department: arrangements/configuration spaces and mapping class groups). We will explore the relationship of braids to knots and links, their homology via the technical tool of the free differential calculus and their representations: Burau, Gassner and Lawrence-Krammer. Faithfulness of the Lawrence-Krammer representation will be shown by the methods of Krammer and Bigelow. We will conclude with a survey of related open problems.

  • MATH 7590-2: Topological Quantum Field Theories
  • Instructor: Prof. Gilmer.
  • Prerequisite: Math 7512.
  • Text: I will follow various research papers and my own notes.

We will discuss TQFTs from an axiomatic point of view, as well as several important examples. Then the course will concentrate on one example related to the Jones polynomial of knots. We will also discuss applications to low-dimensonal topology. We will follow a skein theory approach.