Course Outlines: Summer 2007 - Spring 2008

Summer 2007

  • Math 7380: Matrix Groups & Control Theory
  • Instructor: Prof. Lawson
  • Prerequisite: Linear Algebra (Math 4153 or 7200 or equivalent), Differential Equations (Math 4027 or 7320 or equivalent)
  • Text: Lecture Notes .

This course is something of a hybrid course, with a goal of providing students with at least an elementary knowledge of two important mathematics disciplines: control theory and Lie theory. Hence the course will provide an introduction to basic ideas of control theory and basic ideas of Lie theory (in the context of linear matrix groups), and develop connections between the two areas. Topics to be covered include matrix Lie groups and their Lie algebras and exponential maps, control systems, particularly linear control systems and control systems on matrix groups, notions of controllability, observability, optimality, and stability, Riccati equations and optimal control on linear systems with quadratic cost, control sets and semigroups, and the interpretation of certain mechanical systems as control systems.

Fall 2007

  • 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 7210: Algebra I
  • Instructor: Prof. Sage .
  • Prerequisite: Math 4200 or the equivalent.
  • Text: Algebra, Grove.

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: Topics in Commutative Algebra
  • Instructor: Prof. Schlichting
  • Prerequisite: Math 7200.
  • Text:Introduction to commutative algebra" by Atiyah and Macdonald, with complements taken from Commutative ring theory, by Matsumura.

This course is an introduction to "commutative algebra" which - besides being an area in its own right - is the basis for algebraic number theory and algebraic geometry. Topics include: Rings and modules, Spec of a commutative ring, tensor products, Nakayama's lemma, Integral extensions (going-up and going-down), Nullstellensatz, Noetherian and Artinian rings, Discrete valuation rings, Dedekind domains, Dimension theory, regular local rings, Kaehler differentials, smooth and étale extensions, UFD's, etc. I will regularly assign exercises and expect some of them to be turned in.

  • MATH 7290: Lie Algebras
  • Instructor: Prof. Achar
  • Prerequisites: Math 7210 (Algebra I).
  • Text: Lie Algebras by Anthony Henderson (not yet published; it will be distributed in PDF format)

Lie algebras are essential in many areas of mathematics and theoretical physics. In this course, after covering the definition and basic properties of Lie algebras, we will study the structure theory of semisimple Lie algebras and the classification of simple Lie algebras, and then take a brief look at their representation theory. Throughout the course, we'll keep in mind the example of sl(2), the smallest semisimple Lie algebra. It's small enough (only 3-dimensional) to easily do explicit calculations in, and at the same time interesting enough to give us a good idea of what goes on in higher-dimensional Lie algebras.

  • MATH 7311: Real Analysis I
  • Instructor: Prof. Richardson
  • Prerequisite: Math 4032 or 4035 or the equivalent.
  • Text: A Little Book of Measure and Integration, written by the teacher, will be available at the LSU Bookstore in the Union on or about August 10th. It will be an inexpensive course-pack.

We will treat measure theory and integration on measure spaces. The cases of the real line and Euclidean space will be emphasized. 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, the Radon-Nikodym theorem, Vitali's covering theorem, bounded variation, and the Riesz-Markov-Saks-Kakutani theorem if there is sufficient time. Visit the class website for further information: https://www.math.lsu.edu/personal/rich/7311info .

  • MATH 7330: Functional Analysis
  • Instructor: Prof. Rubin
  • Prerequisite: Math 7311 - Real Analysis I.
  • Text: Functional Analysis by George Bachman and Lawrence Narici, Dover Publications, Inc. 2000, ISBN 0-486-40251-7

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.

This is a second graduate course in complex analysis, in one and several complex variables. The first part of the course will cover results in one variable, including the modular function, the Picard theorems, the boundary behavior of conformal maps, the proof of the Bieberbach conjecture and the theory of Hp spaces. The second part will introduce the basic ideas from the analysis of functions of several complex variables.

  • MATH 7380-2: Elliptic solvers: advanced parallel implementation & related topics.
  • Instructor: Prof. Bourdin
  • Prerequisite: Basic numerical analysis, and PDE. Familiarity with the Finite Element Method is a plus but not required.
  • Text: Lecture notes provided.

This course focusses on the numerical implementation of elliptic solvers on parallel supercomputers, using finite differences and finite element methods. I will briefly review the numerical analysis of these methods, together with the required mathematical tools, if necessary. I will present some peripheral issues (mesh generation, iterative solvers, visualization). I will give some elements of parallel programming, using MPI and PETSc, which will be the basis of the numerical project. Time permitting, I will also discuss domain decomposition methods and their implementation, and related problems. The numerical part of this class will be based on MPI (Message Passing Interface) and PETSc (Portable, Extensible Toolkit for Scientific Computation developed at Argonne National Laboratories) and will require some basic programming skills in C or Fortran. Numerical project and homework problems will include the parallel implementation of finite difference and finite element method.

  • MATH 7380-3: Applied Stochastics
  • Instructor: Prof. Ambar Sengupta
  • Prerequisite: Math 7311 Real Analysis I and elementary probability theory
  • Text: Notes will be provided; books will be recommended.

This course will cover topics in stochastic analysis with a view towards applications in finance. In the first few weeks we will go over the basic framework of probability theory (probability spaces, random variables) and essential tools. We will then study Brownian motion and other stochastic processes, examining the beautiful and surprising relationship between such processed and partial differential equations. A typical question of interest is: how long does it take for a random process to escape from a given region? In this context we will study the Feynman-Kac formula, which arose in physics but has application in finance. On the financial side, we will see a very simple approach to the celebrated Black - Scholes - Merton formula for pricing stock options. We will examine the structure of certain financial products such as bond options, swaps, and credit derivatives. Pricing and risk management of these derivatives will be examined. Credit derivatives form a currently highly active area both in the "real world" of finance and in development of the theory.

  • MATH 7380-4: Fast Solvers.
  • Instructor: Prof. Brenner
  • Prerequisites: .
  • text: .

Numerical methods for partial differential equations generate large ill-conditioned systems. Since direct methods are too expensive, such systems are usually solved by iterative methods. In this course we will discuss two classes of modern iterative methods for numerical PDEs: multigrid and domain decomposition. The goal is to gain insight through the analysis and implementation of various algorithms. The audience for this course should have some background in finite element methods. Experience in MATLAB is desirable but not required.

  • MATH 7390-1: Harmonic Analysis I - Fourier Transforms and Distributions
  • Instructor: Prof. Olafsson
  • Prerequisite: Math 7311 - Real Analysis, I.
  • Text: Lecture Notes by R. Fabec and G. Olafsson, available at https://www.math.lsu.edu/harmonic. You can log in using your Departmental ID.

This is the first part of a one year course on Harmonic Analysis and Representation Theory. It is also an introductory course on Fourier Series and Integrals. Topics include: Periodic functions, Fourier series, classical functions spaces, in particular the space of rapidly decreasing functions, the Fourier transform and its application to differential equations (heat equation and the wave equation), Hermite polynomials and functions, distributions, and the continuous wavelet transform. More advance topics might be introduced if there is time. Further information is available at https://www.math.lsu.edu/~olafsson/sy7390.html

  • MATH 7390-2: Optimal Control and the Calculus of Variations
  • Instructor Prof. Wolenski
  • Prerequisite: Advanced Calculus (Math 4031, 4032, or 4035), Linear Algebra (2085 or equivalent), and ordinary differential equations.
  • Text: No formal text is required

This course will begin with a review of continuous optimization in Euclidean space, and quickly move to the main topic, so-called dynamic optimization. The first half will cover the classical material of the calculus of variations, including topics such as the Euler-Lagrange equation, Weierstrass maximality condition, Erdmann corner conditions, and Jacobi conjugate points. Plenty of examples will be covered. There is a natural transition into optimal control, which will be the focus of the rest of the course from a neo-classical point of view.

  • MATH 7390-3: Homogenization Theory in Micro Structured Media
  • Instructor: Prof. Lipton
  • Prerequisite: Math 4035, 4038, or their equivalent.
  • Text: Lecture Notes; Reference texts: Mechanics of Composite Materials, R.M. Christensen; Theory of Composites, G.W. Milton; Random Heterogeneous Materials, S. Torquato,

The course provides an introduction to the field of homogenization theory as well as a guide to the current research literature useful for understanding the mathematics and physics of complex heterogeneous media. The course is suitable for both mathematicians and engineers. The first part of the course develops the variational tools and asymptotic analysis necessary for characterizing the macroscopic behavior of heterogeneous media. The techniques introduced in this course include the Hashin-Shtrikman variational principles, analytic continuation methods, and the method of two-scale asymptotic expansions. Explicit solutions of field equations inside special microstructures will be developed including the space filling coated spheres construction and confocal coated ellipsoid construction. The second part of the course focuses on very recent developments related to multi-scale inverse problems known as de-homogenization theory. The techniques introduced here include the maximum principle for harmonic functions, applications of the Hölder inequality, and convex duality. These techniques provide a characterization of microscopic field behavior inside complex materials in terms of applied macroscopic fields and the statistics of the microgeometry.

  • MATH 7490: Matroid Theory II
  • Instructor: Prof. Oxley
  • Prerequisite: A course in matroid theory or permission of the department.
  • Text: Lecture notes and the book "Matroid Theory", by James Oxley, Oxford Univ. Press, reprinted with corrections, 2006.

The study of matroids is a branch of discrete mathematics with basic links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics. This course presents more advanced topics in the theory of matroids. In particular, the course will consider problems relating to the representability of matroids over fields especially over small finite fields; constructions for matroids; higher connectivity for matroids; and the structure of certain classes of matroids.

  • MATH 7510: Topology I
  • Instructor: Prof. Stoltzfus
  • Prerequisite: .
  • Text: Notes by McCleary & A. Hatcher, Algebraic Topology, Cambridge University Press, available at Alg. Top.

Topology I has been revised to cover the fundamentals of basic point-set topology as well as homotopy and the fundamental group.

  • MATH 7520: Algebraic Topology
  • Instructor: Prof. Cohen
  • Prerequisite: MATH 7200 and 7510, or equivalent.
  • Text: A. Hatcher, Algebraic Topology, Cambridge University Press, available at http://www.math.cornell.edu/~hatcher/AT/ATpage.html .

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, universal coefficient theorems, cup products, and Poincare duality. Geometric examples, including surfaces, projective spaces, lens spaces, etc., will be used to illustrate the techniques.

  • MATH 7590: Gromov-Witten Theory
  • Instructor: Prof. Baldridge
  • Prerequisite: A differential geometry course and/or Riemannian geometry course.
  • Text: J-Holomorphic Curves and Symplectic Topology by Dusa McDuff and Dietmar Salamon.

The purpose of this course is to introduce students to the basic concepts and results in symplectic geometry/topology and develop Gromov's theory of pseudo-holomorphic curves. The content of the course will be on pseudo-holomorphic curves, including: basic definitions and local properties of pseudo-holomorphic curves, Fredholm theory and transversality of moduli space, Gromov's compactness theorem, pseudo-holomorphic curves in dimension 4, some applications in symplectic topology.

Spring 2008

  • 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. Achar
  • 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-Hölder theorem; tensor products and Hom for modules; noetherian rings; the Hilbert Basis Theorem; and algebras over a field, including Wedderburn's and Maschke's Theorems.

  • MATH 7280-1: Algebraic Geometry
  • Instructor: Prof. Hoffman
  • Prerequisite: Ideally, the student entering this course has had some background in Commutative Algebra. The Fall 2007 course by Marco Schlichting covers everything we need. It is possible to follow this course without having taken Marco's, but this presumes a willingness to pick up the necessary background when needed. A minimal background is familiarity with groups, rings, modules as well as the basics of topological spaces and continuous maps.
  • Text: Algebraic Geometry, by Robin Hartshorne, recommended but not required.

Roughly speaking we will cover the first two chapters of Hartshorne's book, but we will supplement with other material. For instance we will do some introduction to software systems (Macaulay2 and Singular) for computing things. Thus we will do an introduction to affine and projective varieties, sheaves and schemes.

  • MATH 7280-2: Numerical Linear Algebra This course can serve as a Core-2 option in Spring 2008.
  • Instructor: Prof. Sung
  • 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: Galois Cohomology
  • Instructor: Prof. Morales.
  • Prerequisite: Algebra I, Algebra II. Some familiarity with the formalism of homological algebra is helpful but is not required, as we will start from scratch.
  • Text: There is no official textbook for this course.

This is an introductory course to the cohomology of groups and Galois cohomology. We will begin with basic homological algebra and define the cohomology of a group G with coefficients in a G-module M. We will discuss methods for explicit computation and the classical interpretations of the low-dimensional cohomology groups. The second part will be devoted to the cohomology of profinite groups and Galois cohomology. We will give some standard applications such as the cohomological interpretation of the Brauer group. We will also discuss Galois cohomology with nonabelian coefficients and the theory of torsors, with applications to algebra and algebraic geometry (invariants for quadratic forms, elliptic curves, Severi-Brauer varieties).

  • MATH 7290-2: Positive Polynomials.
  • Instructor: Prof. Delzell.
  • Prerequisite: first-year graduate-level algebra (e.g., the basic theory of commutative rings and fields).
  • Text: Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, by A. Prestel and C. Delzell, Springer Monographs in Mathematics, 2001.

Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of Rn which itself is often given by polynomial inequalities. The main objective of the course is to give useful characterizations of such polynomials. It takes as a starting point Hilbert's 17th Problem (which was to prove that if a polynomial f in n variables with real coefficients takes only nonnegative values on Rn, then f can be written as a sum of squares of real rational functions (the converse being obvious)), and explains how E. Artin's solution of that problem (1926) eventually led to the development of "real algebra" towards the end of the 20th century. Topics include ordered fields, real closed fields, semialgebraic sets, elimination of quantifiers, valuation theory, quadratic forms over real fields, the Positivstellensatz and the real Nullstellensatz, the real spectrum of a commutative ring, and Schmüdgen's celebrated theorem (1991) on polynomials that are positive on compact semialgebraic sets in Rn.

  • MATH 7320: Ordinary Differential Equations
  • Instructor: Prof. Almog
  • Prerequisite: Advanced Calculus and Linear Algebra (preferably Math 7311 and 7200).
  • Text: Differential equations, Dynamical Systems & An introduction to Chaos, by Morris W. Hirsch. Stephen Smale, Robert L. Devaney. Elsevier - Second edition. Also, Eugene Speer's notes.

Topics:

  • Introduction - first order equations
  • Existence and uniqueness theory, Peano's theorem, extension theorem
  • Grönwall's inequality
  • Dependence on initial conditions
  • Linear systems
  • Stability of equilibrium points - Lyapunov functions
  • Phase plane analysis
  • MATH 7350: Complex Analysis
  • Instructor: Prof. Olafsson
  • Prerequisite: Math 7311.
  • Text: .
  • MATH 7380: Introduction to Mathematical Control Theory.
  • Instructor: Prof. Malisoff
  • Prerequisites: Elementary differential equations and linear algebra; some background in real analysis (e.g. Math 7311) 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. More advanced topics may also be included depending on the backgrounds of the students. It is intended for PhD students in applied mathematics and engineering graduate students with a basic background in real analysis and nonlinear ordinary differential equations. The course is designed to help students prepare for 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: Harmonic Analysis II (Basic Representation Theory)
  • Instructor: Prof. Davidson
  • Prerequisites: Math 7311 (Real Analysis I) or the equivalent.
  • Text: .

The four main topic in this class are: Topological groups, homogeneous spaces, basic representation theory and representation theory of compact groups. Representations of topological groups are central in several branches of mathematics: In number theory and the study of automorphic functions and forms, in geometry as a tool to construct important vector bundles and differential operators, and in the study of Riemannian symmetric spaces. Finally, those are important tools in analysis, in particular analysis on some special homogeneous manifolds like the sphere, Grassmannians, the upper half plane and its generalizations. Representations even shows up in branches of applied mathematics as generalizations of the windowed Fourier transform and wavelets. Several examples of those applications in analysis and geometry will be discussed in the class.

  • MATH 7390-2: Applications of spectral theory in the material sciences.
  • Instructor: Prof. Shipman
  • Prerequisites: Math 7311 (Real Analysis I).
  • Text: The sources for the material will be research papers as well as various books expounding the spectral theory, especially in the context of the partial differential equations of mathematical physics.

The purpose of this course is to introduce students to structures and techniques in analysis that have far-reaching applications. The concepts and techniques will form part of a necessary basis for a wide variety of specializations in analysis, including variational calculus, numerical analysis, mathematical physics, and many areas of partial differential equations. Participants will gain a working understanding of Hilbert space, operators in Hilbert space, and spectral decomposition of self-adjoint operators, as well as generalized solutions to differential equations, variational problems, and integral equations. The medium for presenting the material will be applications to problems from the material sciences. We will not prove the theorems of spectral theory, for this would require a course in itself, but our course will follow a rigorous development with these theorems as a basis. Some of the possible topics are these:

  • Quantum mechanics of the helium atom, which involves eigenvalues embedded in the continuous spectrum and resonance.
  • Effective conductivities for two-phase composite materials, with extreme values occurring at resonances of the microscopic structure.
  • Coupled systems, in particular between electromagnetic fields and internal molecular degrees of freedom.
  • Photonic crystals and guided electromagnetic or acoustic modes in periodic slabs.
  • 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 http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/GraphTheoryIII.pdf. 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 above, but they are not absolutely necessary. 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. 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. Brendle
  • Prerequisite: Topology 1 Math 7510
  • Text: .
  • MATH 7550: Differential Geometry
  • Instructor: Prof. Litherland
  • Prerequisites: Math 7200 and 7510.
  • Text: Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, GTM 94.

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. Gilmer
  • Prerequisite: Math 7510 and 7512 (previously or concurrently)
  • Text: An Introduction to Knot Theory, by W. B. Raymond Lickorish.

We will study knots and 3 dimensional manifolds. We will study the Jones and Alexander polynomials, branched covers, surgery descriptions of 3-manifolds, Quantum invariants of 3-manifolds.

  • MATH 7590-2: Heegaard Floer Homology
  • Instructor: Prof. Owens.
  • Prerequisite: Math 7510 and 7512.
  • Text: .

Heegaard Floer homology is a recent development due to Peter Ozsvath and Zoltan Szabo. It assigns a complex to a three-dimensional manifold, whose homology is an invariant of that manifold. The definition involves Heegaard diagrams, symmetric products, and holomorphic disks (all of which I will explain in the class). The definition extends to give invariants of 4-dimensional manifolds and also invariants of knots in 3-manifolds. All are defined using holomorphic disks, though a simple combinatorial definition is possible in some special cases (including knots in the three-sphere). This theory is the focus of a great deal of current research and has led to a lot of progress on longstanding problems in knot theory and three-manifolds. I will describe the invariants for three-manifolds and knots and their main properties. (This will involve some discussion of four-manifolds.) I will also discuss some of the applications.