Graduate classes, Spring 2008, Mathematics
MATH 512: Analysis II  Credits: 4  − Description  − Sections 
Content: Topics will include: Measure and integration theory on the real line as well as on a general measure space, Bounded linear functionals on L^p spaces. If time permits, Sobolev spaces and Fourier transforms will be introduced. Prerequisites: Students are expected to have the background of Math 411412 sequence or the equivalent.  000  MSC: W306  MWF 11:45am  12:35pm  David Borthwick   MATH 516: Numerical Analysis II  Credits: 4  − Description  − Sections 
Content: This course covers fundamental concepts of numerical analysis and scientific computing. Material includes numerical methods for curve fitting (interpolation, splines, least squares), differentiation, integration, and differential equations. It is assumed that students have a strong background in numerical linear algebra. Prerequisites: Math 515, undergraduate course work in multivariable calculus and ordinary differential equations. An undergraduate course in numerical analysis would help, but is not absolutely essential.  000  MSC: W304  TuTh 10:00am  11:15am  Alessandro Veneziani   MATH 522: Algebra II  Credits: 4  − Description  − Sections 
Content: Continuation of 521. Topics: Modules, especially modules over a principal ideal domain, fields, Galois theory, representation of finite groups, Commutative algebra. Prerequisites: Math 521.  000  MSC: E408  TuTh 11:30am  12:45pm  Eric Brussel  max 15  MATH 532: Graph Theory II  Credits: 4  − Description  − Sections 
Content: Topics include: independence of vertices and edges (matchings), factorizations and decompositions, coloring (both vertices and edges), and classic external theory. Prerequisites: Mathematics 531.  000  MSC: E406  MWF 3:00pm  3:50pm  Ron Gould  max 15  MATH 544: Algebraic Topology II  Credits: 4  − Description  − Sections 
Content: Singular, simplicial and cellular homology, long exact sequences in homology, MayerVietoris sequences, excision, Euler characteristic, degrees of maps, BorsukUlam theorem, Lefschetz fixed point theorem, cohomology, universal coefficient theorem, the cup product, Poincare duality  000  MSC: W306  MWF 9:35am  10:25am  Emily Hamilton  max 15  MATH 550: Functional Analysis  Credits: 4  − Description  − Sections 
Content: An introduction to concepts and applications including: metric and normed spaces. Sobolev spaces, linear operators, and functionals, compactness in metric and normed spaces. Fredholm's solvability theory, spectral theory, calculus in metric and normed spaces, selected application.  000  MSC: E406  TuTh 1:00pm  2:15pm  Michele Benzi  max 15  MATH 558: Partial Differential Equations  Credits: 4  − Description  − Sections 
Content: This course will introduce some of the basic techniques for studying and solving partial differential equations (PDE's) with special emphasis on applications. Included in the course are the following topics:
1. Basic concepts, sample problems, motivation
2. Maximum principles for elliptic and parabolic equations
3. Basic concepts of the theory of distributions
4. Method of fundamental solutions; Green's functions
5. Fourier transform
6. Variational methods, eigenvalues and eigenfunctions
7. Applications; Maxwell's equations, diffusion, geometric flows, image processing  000  MSC: W301  TuTh 2:30pm  3:45pm  Vladimir Oliker  max 15  MATH 578R: Seminar in Algebra  Credits: 1  12  − Description  − Sections 
Content: Research topics in algebra of current interest to faculty and students.  000  MSC: W303  Tu 4:00pm  5:00pm  Skip Garibaldi   MATH 590: Teaching Seminar  Credits: 4  − Description  − Sections 
Content: This seminar will concentrate on effective teaching techniques in mathematics. Topics included will include:
General advice for new TA's. General advice for International TA's. Students will present several practice lectures over different levels of material. They will receive practice on quiz and test preparation. Syllabus information on courses most likely to be taught by new TA's will be supplied. General professional development information will also be included.  000  MSC: W306  W 2:00pm  2:50pm  James Nagy   MATH 597R: Directed Study  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)   MATH 599R: Master's Thesis Research  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)   MATH 741: Geometric Topology  Credits: 4  − Description  − Sections 
Content: In this course we will study discrete groups from a geometric perspective. (Discrete groups
are groups which are defned by generators and relations, such as free groups, free abelian
groups, braid groups, etc.) By treating groups as geometric objects, one can solve many
algebraic problems which are much more difficult without the geometry.
Introductory topics include:
Free groups, group presentations, Cayley graphs and Cayley complexes, Dehn's problems, fundamental groups, hyperbolic geometry, EilenbergMacLane spaces and some algebraic topology.
Advanced topics may include:
Coxeter and Artin groups, BassSerre theory (of groups acting on trees and other spaces), CAT(0) geometry, CAT(0) groups, hyperbolic groups, coarse geometry (quasiisometries etc), ends of groups, boundaries of groups, Gromov's polynomial growth theorem.  000  MSC: E408  TuTh 1:00pm  2:15pm  Aaron Abrams  max 15  MATH 772: Numerical Partial Differential Equations  Credits: 4  − Description  − Sections 
Content: Examples and classification of PDE's, initial and boundary value problems, wellposed problems, the maximum principle, finite difference methods, variational formulations for elliptic PDE's, finite element methods, and iterative solution methods. Prerequisites: Mathematics: 511512, 515516.  000  MSC: E308A  TuTh 11:30am  12:45pm  Eldad Haber  max 15  MATH 787R: Topics in Combinatorics: Extremal Combinatorics & Optimization  Credits: 4  − Description  − Sections 
 000  MSC: E406  TuTh 10:00am  11:15am  Vojtech Rodl  max 15  MATH 787R: Topics in Combinatorics: Random Structures  Credits: 4  − Description  − Sections 
Content: The course will cover several advanced topics from the theory of random graphs, hypergraphs and other random structures, like random subsets of integers. Prerequisites: No prerequisite is required, but some knowledge of probability, graph theory and combinatorics is anticipated  000  MSC: W304  TuTh 8:30am  9:45am  Andrzej Rucinski  max 15  MATH 788R: Topics in Algebra: Algebraic Geometry  Credits: 4  − Description  − Sections 
Content: After an introduction to affine and projective varieties, we shall cover the following topics on algebraic curves: Bezout's theorem, resolution of singularities for curves, Riemann Roch theorem. Necessary materials from commutative algebra will also be covered.  000  MSC: E406  TuTh 11:30am  12:45pm  Raman Parimala  max 15  MATH 797R: Directed Study  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)   MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 
 00P    Faculty (TBA)  
