# 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 411-412 sequence or the equivalent.
000MSC: W306MWF 11:45am - 12:35pmDavid 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.
000MSC: W304TuTh 10:00am - 11:15amAlessandro 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.
000MSC: E408TuTh 11:30am - 12:45pmEric Brusselmax 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.
000MSC: E406MWF 3:00pm - 3:50pmRon Gouldmax 15
 MATH 544: Algebraic Topology II Credits: 4 − Description − Sections
Content: Singular, simplicial and cellular homology, long exact sequences in homology, Mayer-Vietoris sequences, excision, Euler characteristic, degrees of maps, Borsuk-Ulam theorem, Lefschetz fixed point theorem, cohomology, universal coefficient theorem, the cup product, Poincare duality
000MSC: W306MWF 9:35am - 10:25amEmily Hamiltonmax 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.
000MSC: E406TuTh 1:00pm - 2:15pmMichele Benzimax 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
000MSC: W301TuTh 2:30pm - 3:45pmVladimir Olikermax 15
 MATH 578R: Seminar in Algebra Credits: 1 - 12 − Description − Sections
Content: Research topics in algebra of current interest to faculty and students.
000MSC: W303Tu 4:00pm - 5:00pmSkip 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.
000MSC: W306W 2:00pm - 2:50pmJames Nagy
 MATH 597R: Directed Study Credits: 1 - 12 − Description − Sections
00PFaculty (TBA)
 MATH 599R: Master's Thesis Research Credits: 1 - 12 − Description − Sections
00PFaculty (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, Eilenberg-MacLane spaces and some algebraic topology. Advanced topics may include: Coxeter and Artin groups, Bass-Serre theory (of groups acting on trees and other spaces), CAT(0) geometry, CAT(0) groups, hyperbolic groups, coarse geometry (quasi-isometries etc), ends of groups, boundaries of groups, Gromov's polynomial growth theorem.
000MSC: E408TuTh 1:00pm - 2:15pmAaron Abramsmax 15
 MATH 772: Numerical Partial Differential Equations Credits: 4 − Description − Sections
Content: Examples and classification of PDE's, initial and boundary value problems, well-posed problems, the maximum principle, finite difference methods, variational formulations for elliptic PDE's, finite element methods, and iterative solution methods.
Prerequisites: Mathematics: 511-512, 515-516.
000MSC: E308ATuTh 11:30am - 12:45pmEldad Habermax 15
 MATH 787R: Topics in Combinatorics: Extremal Combinatorics & Optimization Credits: 4 − Description − Sections
000MSC: E406TuTh 10:00am - 11:15amVojtech Rodlmax 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
000MSC: W304TuTh 8:30am - 9:45amAndrzej Rucinskimax 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.
000MSC: E406TuTh 11:30am - 12:45pmRaman Parimalamax 15
 MATH 797R: Directed Study Credits: 1 - 12 − Description − Sections
00PFaculty (TBA)
 MATH 799R: Dissertation Research Credits: 1 - 12 − Description − Sections
00PFaculty (TBA)