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# Course Descriptions Summer 2005 - Spring 2006

## Contact

## Summer 2005

- Math 7350: Complex Variables
- 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.

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, and harmonic functions.

## Fall 2005

- 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. Hoffman .
- 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: Topics in Galois Theory
- Instructor: Prof. Verrill
- Prerequisite: Math 7210: Algebra I
- Text: Milne's online book from the link Fields and Galois Theory.

This course will start with a review of basic concepts in Galois theory, which students will have seen in Math 7210. The first two weeks will be a review, probably covering the first four chapters of Milne's book. We will probably spend up to two weeks on each of the remaining chapters of the book, which cover algebraic closures, infinite Galois groups and transcendental extensions. Assuming we finish this on schedule, we will continue the course with some Galois cohomology, probably using a book such as Cassels and Frohlich's "Algebraic number theory" book. Alternatively, or additionally, depending on interest, we will discuss the inverse Galois problem, i.e., given a group, how can one determine whether is is the Galois group of a field extension. Students taking the course for credit will be expected to present solutions to exercises (given in the book) to the class.

- MATH 7280: Commutative Algebra & Algebraic Geometry - I
- Instructor: Prof. Hoffman
- Prerequisites: Elementary algebra and elementary topology. That is, familiarity with groups, rings, fields, modules, topological spaces, continuous maps and homeomorphisms.
- Text: Algebraic Geometry, by R. Hartshorne

This is the first semester of a course on algebraic geometry to be continued in the spring by Marco Schlichting. In the fall we will study affine and projective varieties, the correspondence between ideals and subsets of these spaces, the nullstellensatz, morphisms and rational maps. Then there will be an introduction to sheaf theory, followed by a discussion of Spec(A), and the general definition of a scheme. Some commutative algebra will be introduced along the way - primes, localization, etc. Roughly this will correspond to the first two chapters of Hartshorne's book.

- Math 7290: Reflection Groups and Hecke algebras
- Instructor: Prof. Achar .
- Prerequisite: Math 7210
- J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, no. 29, Cambridge University Press, Cambridge, 1990.

A reflection of a real vector space is a linear transformation that keeps some hyperplane fixed, and has eigenvalue -1 on the orthogonal complement of that hyperplane. Reflection groups--that is, groups generated by reflections--have long played a vital role in the structure and representation theory of Lie groups and algebraic groups, as well as in combinatorics, geometry, and other disciplines.

The first part of the course will be devoted to developing the basic theory of these groups, including the classification of finite reflection groups. In the second part, we will study Hecke algebras, which are certain deformed versions of the group algebras of these groups. The main topic here is the ground-breaking 1979 paper of Kazhdan and Lusztig on representations of Hecke algebras. Finally, possible topics for the end of the semester include symmetric algebras, braid groups, complex reflection groups, and cyclotomic Hecke algebras.

- MATH 7311: Real Analysis I
- Instructor: Prof. Davidson
- Prerequisite: Math 4032 or 4035 or the equivalent.
- Text:

- MATH 7360: Probability Theory
- Instructor: Prof. Sundar
- Time and Location: Fall 2005: Tues., Thurs. 1:40 -3:00 in Lockett 113.
- Prerequisite: Math 7311.
- Text: Probability Theory - S.R.S. Varadhan Published by The American Mathematical Society.

The course starts right from the concept of a random variable and proceeds to discuss extension of measures, independence of random variables, strong limit theorems, weak convergence, tightness, Prohorov theorem, characteristic functions, central limit theorems, conditional expectation, Markov chains, martingales, martingale convergence theorem, the Doob decomposition, and 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: Applied Stochastic Analysis
- Instructor: Prof. H-H Kuo
- Prerequisites: Math 4031 and 4055. Math 7311 would be helpful but is not required.
- Text: 1. Kuo, H.-H.: Introduction to Stochastic Integration. (to appear in the Universitext series, Springer-Verlag) 2. Oksendal, B.: Stochastic Differential Equations. 5th edition, Springer-Verlag, 2000

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 not absolutely necessary. Below are some items to be covered in this course:

- Brownian motion
- Construction of Brownian motion
- Wiener integrals
- Ito's integrals
- Stochastic integrals for martingales
- The Ito formula
- Girsanov theorem
- Wiener-Ito theorem
- Stochastic differential equations
- Hedging portfolio
- Arbitrage and option pricing
- Black-Scholes analysis

- MATH 7380-2: Applied Analysis in the Materials Sciences
- Instructor: Prof. Lipton
- Prerequisite: Math 7311 - Real Analysis I.
- Text: The first part of the course will be taken from the text Partial Differential Equations by Lawrence C. Evans; the second part of the course will be based upon lecture notes. A good reference for the second half of the course is the text Topics in the Mathematical Modeling of Composite Materials Edited by A. Cherkaev and R.V. Kohn.

In this course we introduce a novel blend of the theory of the calculus of variations and the basic theory of elliptic PDE to recover the modern mathematical theory of materials design. The first 4 weeks of the course provides the introduction of the basic existence and uniqueness theory for second order elliptic PDE. The next three weeks develop the fundamental mathematical notions of G and H convergence of solution operators for elliptic PDE. During the next three weeks we trace the approach of Murat and Tartar to exhibit the connection between G and H convergence and the notion of effective material properties that were used and developed by a significant community of scientists including J.C. Maxwell and A. Einstein. The next few weeks are used to identify special classes of microstructures with extremal effective properties. The remainder of the course shows how these extreme microstructures can be employed in a numerical scheme to identify novel designs of functionally graded microstructures for optimal thermal management. This course is self contained and provides the theoretical underpinnings enabling access the the rapidly developing literature in the theory of optimal design and multiscale modeling.

- MATH 7390-1: Calculus of Variations & Control Theory
- Instructor: Prof. Wolenski
- Prerequisite:
- Text:

This course will cover the classical theory of the calculus of variations from the viewpoint of modern optimal control theory. Thus the first part of the course will cover the following topics.

- Examples and History: The Brachistochrone and Minimum distance problems.
- Necessary Conditions for an extremum: The Euler--Lagrange equation and Erdmann corner conditions; the Weierstrass condition; Jacobi field conditions and conjugate points.
- Sufficient conditions: Field theory and embedding theorems; Verification functions and the Hamilton--Jacobi equation.
- Hamilton--Jacobi Theory: Principle of least action; Noether's theorem.
- Existence theory: Examples; Tonelli's existence theorem; The Lavrentiev phenomenon.

The second part of the course will cover basic results in optimal control theory, including the Pontryagin Maximum Principle, existence and relaxation theorems, and stability.

- MATH 7390-2: Harmonic Analysis I: Classical Fourier Analysis and Distributions
- Instructor: Prof. Fabec
- Prerequisite: Math 7311 - Real Analysis
- Text: The text is being prepared and can be accessed by LSU graduate students.

This is the first of a two semester sequence of courses which begins with Fourier analysis on Euclidean space. Topics include the Fourier transforms of both functions and distributions, the Fourier inversion formula, and the Plancherel theorem. It includes the Stone-von Neumann theorem and its meaning in terms of the Schrödinger model in quantum mechanics. The Stone-von Neumann theorem focused attention on the Heisenberg group (the group of upper triangular matrices with ones on the diagonal) and its relation to classical Fourier analysis. The Heisenberg group provides a natural introduction to noncommutative groups, homogeneous spaces and representation theory - subjects which will be developed further in the second semester.

- MATH 7490: Matroid Theory II
- Instructor: Prof. Oxley
- Prerequisite: A course in matroid theory or permission of the department.
- Text: J.G. Oxley, Matroid Theory, Oxford, 1992. Topics to be covered will be taken from Chapters 6 through 13.

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. Baldridge
- Prerequisite: Math 2057 or the equivalent.
- Text: James R. Munkres, Topology, 2nd edition.

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 & Arrangements
- Instructor: Prof. Cohen
- Prerequisite: MATH 7512: Topology II
- Text: None

A hyperplane arrangement is a finite collection of (n-1)-dimensional subspaces in an n-dimensional vector space, such as lines in a plane, planes in 3-space, etc. Arrangements arise in a variety of mathematical contexts, and in applications ranging from mathematical physics to robotics.

The complement of an arrangement, what is left of space after the hyperplanes have been removed, is an object of fundamental interest in the topological study of arrangements. For instance, a collection of lines cuts the real plane into pieces, and understanding the topology of the complement amounts to counting the pieces. This depends on combinatorial aspects of the collection of lines such as parallelism, how many of the lines intersect at a given point, and so on. This example illustrates a central theme in the subject, the relationship between combinatorial and topological aspects of arrangements.

A complex hyperplane does not disconnect space, and the complement of an arrangement in a complex vector space has rich topological structure. The focus of this course will be on the (low-dimensional) topology of these spaces. Specific topics include braid groups, both as primary objects of study and as tools, configuration spaces, the Fox calculus, and Alexander polynomials and generalizations. In particular, we will develop general algorithms for computing these latter invariants, and investigate the extent to which these invariants of arrangements are combinatorially determined.

The topological aspects of the course will assume familiarity with the fundamental group, while other (combinatorial) aspects will be self-contained.

- MATH 7590-2: Introduction to Quantum Topology
- Instructor: Prof. Gilmer
- Prerequisite: Math 7510 and 7512
- Text: Some class notes, also some accessible research papers and portions of some texts available on the net.

We will study the Jones polynomial of knots & links and its descendants:

- the Kauffman bracket polynomial
- skein modules of 3-manifolds
- skein algebras of surfaces
- Witten-Reshetihkn-Turaev invariants of 3-manifolds
- Topological quantum field theory (this is easier than it sounds)

Some applications of these tools to questions in geometric topology. Background material will be presented as needed.

## Spring 2006

- 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. Delzell
- Prerequisite: Math 7200: Geometric and Abstract Algebra.
- Text: Abstract Algebra, third edition, by David Dummit and Richard Foote, John Wiley and Sons, 2004.

This is the second-semester core course in Algebra, treating field theory and Galois theory. (It has a dual purpose of preparing students for the optional Core-2 Algebra comprehensive Exam. Students can see past Core-2 Algebra Comprehensive Exams at the link Graduate Exams: by selecting Past Comprehensive Exams.)

The subject has its roots in antiquity. Parts of the subject led to proofs of the impossibility of various geometric constructions with straightedge and compass (constructions that had been attempted by the Greeks and by others for 2000 years), such as the problems of trisecting an angle, duplicating the cube, and "squaring the circle." Other parts of the subject led to a proof of the insolvability of the general quintic equation by means of radicals. But the subject goes beyond these classical results.

Topics on extensions of fields:

algebraic extensions, transcendental extensions, separable and inseparable extensions, normal extensions, Galois extensions, composite extensions, simple extensions, cyclotomic extensions, solvable extensions, radical extensions, and Abelian extensions.

Other topics on fields and Galois theory:

classical straightedge-and-compass constructions, splitting field of a polynomial,

algebraic closure of a field, cyclotomic polynomials, Galois groups of polynomials, fundamental theorem of Galois theory, finite fields, solvable groups, insolvability of the quintic, and computation of Galois groups over Q.

- MATH 7280: Lattices and Codes
- Instructor: Prof. Morales .
- Prerequisite: Algebra I (such as MATH 7210), Complex Analysis (such as MATH 7350). A graduate-level course in algebraic number theory is recommended.
- Text: For most (~ 75 %) of the course we will use W. Ebeling's book "Lattices and Codes", Vieweg (1994).

This is an introductory course on the theory of lattices and its interplay with the theory of error-correcting linear codes.

A lattice in this course is a full-rank discrete subgroup of Rn. The main problem for lattices is their classification up to rotation (or rotation and scaling) in Rn. The theory of lattices is also closely related to the problem of sphere packing (i.e. finding the most efficient way of packing spheres of equal size in Rn).

The study of lattices involves several areas of mathematics: Analysis (modular forms and theta functions), Geometry (root systems, reflection groups), Number Theory (number fields, ideal class groups).

Of particular interest are the so-called even unimodular lattices, which exist only in dimensions that are multiples of 8. We will classify these lattices in dimensions up to 24. In particular, we will give Niemeier's enumeration of such lattices in dimension 24, and (at least) one construction of the exceptional Leech lattice L24, whose automorphism group is a two-fold central extension of the largest of the three Conway sporadic simple groups.

We will also see the relation between the theory of lattices with that of linear error-correcting codes. There are very interesting formal analogies between the theory of theta functions of self-dual even lattices and the theory of weight enumerators of self-dual doubly even linear codes. These analogies are in fact more than formal as there is (almost) a two-way dictionary between lattices and codes. For instance, the so-called Hamming code H8 corresponds to the 8-dimensional lattice E8, and the Golay code G24 correspond to the Leech lattice L24 mentioned above.

Time permitting, we will discuss recent applications by Bayer, Oggier and Viterbo of the theory of ideal lattices (lattices constructed as ideals in number fields) to the construction of signal constellations with specified modulation diversity for transmissions over a fading channel (such as wireless communications).

- MATH 7290-1: Algebraic Geometry II
- Instructor: Prof. Schlichting .
- Prerequisite: MATH 7280: Commutative Algebra & Algebraic Geometry - or equivalent (at least sections II.1, II.2) of Hartshorne's book, and some commutative algebra.
- Text: Algebraic Geometry, by R. Hartshorne.

This is the continuation of MATH 7280 "Commutative Algebra & Algebraic Geometry" taught by Prof. Hoffman in Fall 05. The course will cover selected topics from Chapters II, III, IV of Hartshornes book, with emphasis on chapter III (Cohomology). We will start off with first properties of schemes (separated, projective, and proper morphisms, divisors), cover some of the main results on Zariski Cohomology (definition, calculations for projective spaces, affine spaces, Serre duality, higher direct images), and finish off with applications to curves (Riemann-Roch, classification of curves).

- MATH 7290-2: Hopf Algebras and Quantum Groups
- Instructor: Prof. Sage .
- Prerequisite: The first year graduate algebra sequence or similar familiarity with graduate algebra.
- Text: Kassel, Quantum groups.

Quantum groups first arose in theoretical physics in the 1980's. They have excited great interest because of their unexpected connections to numerous areas of mathematics, including knot theory, the representation theory of algebraic groups and Hecke algebras, noncommutative geometry, and mathematical physics.

This course is designed to give an introduction to some of the ideas behind quantum groups. Quantum groups are in fact not groups at all, but rather Hopf algebras. These are a type of algebra which includes group algebras and universal enveloping algebras of Lie algebras as basic examples. The course will begin by discussing Hopf algebras. This will be followed by a detailed presentation of the quantum groups associated with the matrix group SL(2) consisting of 2 by 2 matrices with determinant 1; these may be viewed as deformations of SL(2). We will then show how to generalize these constructions. The course will conclude with applications, either to representation theory or to knot theory, depending on the interest of the participants.

- MATH 7312: Measure and Integration
- Instructor: Prof. Sengupta
- Prerequisite: Math 7311
- Text: Lecture notes.

This course presents the basic framework of ideas and results in measure theory and integration in the abstract. After setting up the basic concepts and definitions, we will cover results on interchange of limits and integrals, and then proceed to Fubini's theorem on integration on product spaces, and the Radon-Nikodym theorem on derivatives for measures. We will also cover fundamental results for integration on locally compact spaces. Towards the later part of the course we will study fundamental facts about Hilbert spaces and Banach spaces.

- MATH 7320: Ordinary Differential Equations
- Instructor: Prof. Almog
- Prerequisite: Advanced Calculus and Linear Algebra (preferably Math 7311 and 7200)
- Text: Philip Hartman, Ordinary Differential Equations, Second Edition, Classics in Applied Mathematics, No. 38, SIAM.

The course will cover the basic theory of ODE together with some more advanced material. No prior knowledge of ODE is assumed. The following is a list of topics, with the first section of the list being topics definitely to be covered, and the second less possible additional topics.

A list of compulsory topics:

0. Introduction:

0a)preliminaries and motivation,

0b)solution techniques for first order scalar ODE.

0c)solution techniques for second order linear scalar ODE.

1. Existence and uniqueness theory:

1a) Picard-Lindelof theorem

1b) Peano's existence theorem

1c) Extension theorem and global existence

1d) Gronwall's inequality

1e) Uniqueness theorems.

2. Linear Systems of ODE:

2a) The solution space for a homogeneous system

2b) Constant coefficients

2c) Inhomogeneous systems, the exponential matrix

3. Dependence on initial conditions and parameters:

3a) Continuity

3b) Differentiability

4. Analytic theory of ODE: solution by power series.

Additional topics (the audience can have some influence here):

1. Floquet theory and the Mathieu equation as an example

2. Regular Singular points for linear second order ODE

3. Phase plane analysis: Poincare-Bendixon Theorem

4. Stability analysis: Lyapunov functions

- MATH 7380-3: Singular Integral Equations
- Instructor: Prof. Estrada
- Prerequisite: Math 7350 Complex Analysis or equivalent.
- Text: Estrada,R. and Kanwal,R.P., Singular Integral Equations, Birkhauser, Boston, 2000.

This is a first graduate course in singular integral equations. The first part of the course will cover basic results from the theory of regular integral equations, and then we will proceed to singular equations. Topics to be studied include Abel's equations, Cauchy type integral equations, Carleman type equations, distributional solutions of integral equations, equations with logarithmic kernels and Wiener-Hopf equations.

- MATH 7380-2: Analytical theory of differential equations
- Instructor: Prof. Antipov
- Prerequisite: Some 4000-level analysis, such as Math 4027, 4031, 4032, 4035, 4038, or 4340.
- Text: Lecture notes

This course aims to give a unified presentation of systems of differential equations that have a Hamiltonian and Lagrangian structure. Analytical insights will be used to describe qualitative properties of solutions.

Content: Lagrangian and Hamiltonian systems, phase space, phase flow, variational principles and Euler-Lagrange equations, Hamilton's principle of least action, Legendre transform, Liouville's theorem, Poincare recurrence theorem, Noether's theorem. Examples are from analytical and quantum mechanics.

- MATH 7390: Harmonic Analysis II
- Instructor: Prof. Olafsson
- Prerequisite: You may take this class even if you have not taken Harmonic Analysis I. However, you may need to refer to notes available at the above site. More importantly, you need a background in topology and functional analysis (Hilbert spaces, bounded operators, Lp-spaces).
- Text: Online Lecture Notes.

The four main topics in this class are: Topological groups, homogeneous spaces, basic representation theory and representation theory of compact groups. Representations of topological groups play an essential role in several areas of mathematics. Amongst these are number theory where representations are used to study automorphic functions and forms, geometry where representation theory is used to construct important vector bundles and differential operators, and in the study of Riemannian symmetric spaces where representation theory provides the framework to generalize many of the significant results of classical harmonic analysis. In this class, we look to some examples important in analysis. These include the sphere, Grassmanians, the upper half plane and their generalizations. Each of these is a special instance of a homogeneous manifold on which representations occur naturally. Representations are used in branches of applied mathematics where notions such as generalizations of the windowed Fourier transform and wavelet transforms fit into a representation theoretic setup. Several examples of applications in analysis and geometry will be discussed in the class.

More information can be found at the harmonic analysis web-page: https://www.math.lsu.edu/harmonic. In particular the site contains lecture notes for this class and lecture notes on several other topics important in harmonic analysis and representation theory.

- MATH 7390-2: Deep Reading Seminar - Introduction to Geometric Analysis
- Instructor: Prof. Olafsson
- Students may sign up for this seminar for their choice of 1 to 3 credit hours.
- Text: S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces.

We will read parts from the book by S. Helgason Topics in Harmonic Analysis on Homogeneous Spaces. Each student will give at least one presentation. Possible topics are listed below. What we end up doing will depend on the number of students and their interests, but the main idea is to discuss the interplay between geometry, group action and analysis.

1) Harmonic Analysis with respect to the Euclidean motion group.

a) The Euclidean Motion group;

b) Invariant differential operators;

c) Spaces of Eigenfunctions

d) The Paley-Wiener theorem and applications.

2) Spherical Harmonics and harmonic analysis on the two-sphere.

3) Harmonic analysis on the Non-Euclidean plane.

a) The differential Geometry of the Non-Euclidean plane. Geodesics and horocycles.

b) The action of SU(1,1).

c) The Fourier transform on the Non-Euclidean plane.

d) Eigenfunctions, the Poission transform and hyperfunctions;

e) The spherical functions and the spherical Fourier transform.

f) Radon Transform

If you are interested in registering for this course, please contact Prof. Olafsson before you register.

- MATH 7400: Graph Theory
- Instructor: Prof. Oporowski
- Prerequisite:
- Text:

- MATH 7512: Topology - II
- Instructor: Prof. Litherland
- Prerequisite: Math 7510, Topology I
- Text: James R. Munkres, Topology (2nd edition), Prentice Hall, 2000.

A topologist can't tell the difference between a doughnut and a coffee cup (because they are homeomorphic spaces), but can tell the difference between either of them and an orange. This course introduces one tool for doing this, the fundamental group. For any topological space X, we shall define a group pi1(X) (its fundamental group), in such a way that homeomorphic spaces have isomorphic fundamental groups. It turns out that a doughnut and an orange (or, more formally, a solid torus and a 3-ball) don't have isomorphic fundamental groups, so they are not homeomorphic. Other applications of the fundamental group include the Brouwer fixed-point theorem, the Jordan curve theorem, and the classification of surfaces.

We shall also define and study the concept of a covering space. There is a close relation between the fundamental group of a space and the collection of its covering spaces. This can be used to give topological proofs of theorems in group theory (for instance, that a subgroup of a free group is free). Covering spaces are also important in the theory of Riemann surfaces.

We shall cover most of the material in chapters 9, 11 and 13 of Munkres, together with a selection from the applications in chapters 10, 12 and 14.

- MATH 7520: Algebraic Topology
- Instructor: Prof. Stoltzfus
- Prerequisite: MATH 7210: Topology I (and preferably II).
- Text: The text will be Elements of Algebraic Topology by James R. Munkres, but I will be also using Allen Hatcher's book, Algebraic Topology which is available online.

I will initially follow the ``third possible syllabus" outlined on page ix of the preface of Munkres. If time permits, I will cover my favorite topic, Poincare duality. The average pace will be one section each class period. As stated in Munkres: ``We assume that the student has some background in both general topology and algebra." For further details, see the Preface, page viii.

Topics to be Covered: I plan to cover both simplicial homology of simplicial complexes and singular homology of topological spaces, with an emphasis on the common methodology. I will also emphasize the rudiments of homological algebra and chain complexes in others areas of mathematics particularly Galois cohomology in number theory and the wealth of applications in algebraic geometry and Lie Algebras (without, of course, developing any of these applications significantly.) The order of topics is tentatively: Historical Roots, Simplicial Complexes and Maps, Review of Abelian Groups, Simplicial Homology, Surfaces, Cones, Relative Homology, Subdivision and Simplicial Approximation, Axiomatic Homology, Singular Cohomology and Products. If time permits, I will cover my favorite topic, Poincare duality.

- MATH 7550: Differential Geometry
- Instructor: Prof. Lawson
- Prerequisite: The needed background for the course is an introductory course in general topology and a reasonable understanding of multivariable calculus.
- Text: Lecture Notes (made available as handouts or on the web)

This course is an introduction to the basic notions of differential topology, the field dealing with differentiable functions on differentiable manifolds, and the adjacent field of differential geometry, the study of geometry using calculus. These studies arise naturally from the study of the theory of differential equations. Topics covered include basic constructions (manifolds, submanifolds, products) and their properties, smooth mappings, the tangent bundle, smooth vector fields and (local) flows, the ring of smooth real-valued functions, tensors and smooth bundles, and differential forms and volumes.

- MATH 7590: Knot theory
- Instructor: Prof. Dasbach
- Prerequisite:
- Text:

- MATH 7999-1: Interdisciplinary course on Computational Fluid Dynamics T-Th 4:30-6:00
- Instructor: Prof. Bourdin. Team-taught with D. Nikitopoulos, J. Lynn, M. Tyagi (Mechanical Engineering.), L. Rouse, C. Li (Oceanography), and J. Tohline (Physics and Astronomy)
- Prerequisite:
- Text:

An introduction to fluid dynamics from a multi-disciplinary perspective. The first half of the course will emphasize how the principal concepts that underpin our understanding of fluid flows apply to a wide range of physical scales; the second half of the course will take the form of four separate, discipline-specific modules. Physical concepts initially will be presented without mathematics, using examples drawn from various disciplines. Then, the physical concepts will be expressed in mathematical terms to make them tangible for the solution of numerous problems. Example problems initially will be introduced on the simplest of levels and will be chosen so that they can be extended upon and brought to a higher level (in terms of both formulation and solution method) during the second half of the course. The focus will be on problems with analytical or semi-analytical solutions, but close coordination with XD-II will allow extensions to numerical problems. (Not counted for credit-hours towards the MS degree in Mathematics.)

- MATH 7999-2: Introduction to Computational Methods for Fluid Dynamics T-Th 1:30-3:00 plus 1h/week of computer lab class to be scheduled.
- Instructor: Prof. Bourdin. Team-taught with S. Acharya (Mechanical Engineering), C. D. White (Petroleum Engineering), and G. Allen (Computer Science)
- Prerequisite:
- Text:

A survey course in the numerical methods for the solution of problems arising in computational fluid dynamics and related areas. The course will focus on interpolation, numerical integration and differentiation; the numerical solution of ODE's, elliptic, hyperbolic and parabolic PDE's with fluid flows as examples; finite differences and finite elements methods; parallel methods; special topics related to the implementation of these numerical methods. (Not counted for credit-hours towards the MS degree in mathematics.)

- MATH 7999-3: Introduction to Linear and Nonlinear Evolution Equations
- Instructor: Prof. Beyer
- Prerequisite: Math 7330 or 7390, basic knowledge of semigroups of linear operators.
- Text: Lecture notes, online resources provided during the class.

This is the second part of a course on abstract evolution equations started in spring 2005. The first part gave an introduction to strongly continuous semigroups of linear operators with view towards application to the solution of initial boundary value problems for parabolic and hyperbolic linear partial differential equations (PDEs) with time-independent coefficients. A knowledge of basic results on strongly continuous semigroups of linear operators is a prerequisite for the second part. Such information can be found, for instance, in Pazy, A. 1983, Semigroups of linear operators and applications to partial differential equations, Springer, New York or in the course notes below. The course starts with a short review of these results.

The second part introduces to the treatment of initial boundary value problems of abstract linear and nonlinear (quasi-linear) evolution equations by semigroup methods. Major applications are given on Hermitian hyperbolic systems of PDEs both linear with time-dependent coefficients and quasi-linear. Also throughout the course examples of applications to current problems in relativistic astrophysics and general relativity are provided. Lecture notes for both parts of the course are available online.

The second part closely follows Tosio Kato, Abstract evolution equations, linear and quasilinear, revisited, in: Komatsu. H. (ed.) 1993, Functional analysis and related topics,1991, Proceedings of the International conference in memory of professor Kosaku Yosida held at RIMS,Kyoto University, Japan, July 29-Aug. 2, 1991, Lecture Notes in Mathematics 1540, Springer, Berlin.