# Graduate classes, Fall 2007, Mathematics

 MATH 500: Probability Credits: 4 − Description − Sections
Content: This course will begin the development of fundamental topics in probability theory and its applications in combinatorics and algorithms. Included will be: events and their probabilities, random variables and their distributions, limit theorems, martingales, concentration of probability, random walks and Markov chains.
000MSC: W306TuTh 8:30am - 9:45amAndrzej Rucinskimax 20
 MATH 511: Analysis I Credits: 4 − Description − Sections
Content: An introduction to fundamental analytic concepts including: The complex number system, geometry and topology of the complex plane, analytic functions, conformal mappings, complex integration, and singularities.
000MSC: E408MWF 11:45am - 12:35pmDavid Borthwickmax 20
 MATH 515: Numerical Analysis I Credits: 4 − Description − Sections
Content: The course will cover fundamental concepts of numerical analysis and scientific computing. Material includes numerical methods for 1. Interpolation 2. Differentiation 3. Integration 4. Linear algebra 5. Ordinary differential equations 6. Partial differential equations This is a "hands on" course and students will be required to demonstrate their understanding of the concepts through programming assignments (help will be given for the novice programmer).
Particulars: Background in calculus, linear algebra and ODE's is assumed. Some knowledge of computer architectures, applied mathematics and elementary numerical analysis would help but is not absolutely essential.
Prerequisites: Background in calculus, linear algebra and ODE's is assumed. Some knowledge of computer architectures, applied mathematics and elementary numerical analysis would help but is not absolutely essential.
000MSC: W304TuTh 10:00am - 11:15amMichele Benzimax 20
 MATH 521: Algebra I Credits: 4 − Description − Sections
Content: Groups, homomorphisms, the class equation, Sylow's thoerems, rings and modules.
000MSC: E408TuTh 11:30am - 12:45pmEric Brusselmax 20
 MATH 531: Graph Theory I Credits: 4 − Description − Sections
Content: I will introduce basic graph-theoretical concepts, graphs, trees, networks, cycles, independence number, chromatic number, planarity and genus, paths and cycles, etc. I will emphasize "extremal" problems and counting techniques.
Particulars: Grades will be based on written assignments.
000MSC: E406MWF 2:00pm - 2:50pmRon Gouldmax 15
 MATH 543: Algebraic Topology I Credits: 4 − Description − Sections
Content: Homotopy theory, the fundamental group, free products of groups with amalgamation, Van Kampen's Theorem, covering spaces, classification of surfaces, classifying spaces, higher homotopy groups
000MSC: E406MWF 10:40am - 11:30amEmily Hamilton
 MATH 557: Partial Differential Equations I 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: W303TuTh 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 - 5:00pmEric Brusselmax 15
 MATH 597R: Directed Study Credits: 1 - 12 − Description − Sections
 MATH 599R: Master's Thesis Research Credits: 1 - 12 − Description − Sections
 MATH 771: Numerical Optimization Credits: 4 − Description − Sections
Content: This course will provide students with an overview of state-of-the-art numerical methods for solving unconstrained, large-scale optimization problems. Algorithm development will be emphasized, including efficient and robust implementations. In addition, students will be exposed to state-of-the-art software that can be used to solve optimization problems.
Prerequisites: Mathematics 511-512, 515-516.
000MSC: E406MW 9:00am - 10:15amEldad Habermax 15
 MATH 787R: Topics in Combinatorics: Ordered Combinatorial & A Credits: 4 − Description − Sections
Content: The course will have two components: (1) A series of lectures on the basics of order theory, including an introduction to finite and infinite partially ordered sets and lattices, topics from combinatorics/set systems such as Sperner theory, uses of ordering to study classes of graphs, digraphs and other relational systems. The length of time spent on this will depend upon the backgrounds and interests of participants. (2) Focus on topics from "Graphs and Homomorphisms" by Hell and Nesetril, particularly the lattice of graph types ordered by homomorphism. There will be time spent on recent papers motivated by the Hedetniemi Conjecture, diverse notions of chromatic number [fractional, circular], etc. A set of coherently organized papers will be made available to participants before the beginning of the seminar. The emphasis will be on several open problems [of varying degrees of accessibility]. Participants will be expected to take an active role in presenting material and in [skeptically] attending all presentations.
Particulars: There will be no text but, in addition to the Hell/Nesetril source, basic material on orderings, set systems, combinatorics and ordered sets, and lattice theory can be found in: (1) Combinatorics of Finite Sets, I. Anderson (2) Combinatorial Theory, M. Aigner (3) Introduction to Lattices and Order, B. Davey and H. Priestley
Prerequisites: The introductory sequence in combinatorics and basic knowledge of group theory and graph theory will be assumed.
000MSC: E406MWF 11:45am - 12:35pmDwight Duffusmax 15
 MATH 788R: Topics in Algebra: Elliptic Curves Credits: 4 − Description − Sections
Content: This will be an introductory course on elliptic curves. The course content will include the following topics: geometry of cubic curves, Weierstrass normal form, groups of rational points, torsion points and Nagel-Lutz Theorem, elliptic curves over finite fields, Mordell-Weil theorem.
000MSC: E406TuTh 10:00am - 11:15amRaman Parimalamax 15
 MATH 789R: Topics in Analysis: Geometric Partial Differential Equations Credits: 4 − Description − Sections
Content: No description available.
000MSC: W303TuTh 10:00am - 11:15amVladimir Olikermax 15
 MATH 797R: Directed Study Credits: 1 - 12 − Description − Sections
 MATH 799R: Dissertation Research Credits: 1 - 12 − Description − Sections