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.  000  MSC: W306  TuTh 8:30am  9:45am  Andrzej Rucinski  max 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.  000  MSC: E408  MWF 11:45am  12:35pm  David Borthwick  max 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.  000  MSC: W304  TuTh 10:00am  11:15am  Michele Benzi  max 20  MATH 521: Algebra I  Credits: 4  − Description  − Sections 
Content: Groups, homomorphisms, the class equation, Sylow's thoerems, rings and modules.  000  MSC: E408  TuTh 11:30am  12:45pm  Eric Brussel  max 20  MATH 531: Graph Theory I  Credits: 4  − Description  − Sections 
Content: I will introduce basic graphtheoretical 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.  000  MSC: E406  MWF 2:00pm  2:50pm  Ron Gould  max 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  000  MSC: E406  MWF 10:40am  11:30am  Emily 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  000  MSC: W303  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  Eric Brussel  max 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 stateoftheart numerical methods for solving unconstrained, largescale optimization problems. Algorithm development will be emphasized, including efficient and robust implementations. In addition, students will be exposed to stateoftheart software that can be used to solve optimization problems. Prerequisites: Mathematics 511512, 515516.  000  MSC: E406  MW 9:00am  10:15am  Eldad Haber  max 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.  000  MSC: E406  MWF 11:45am  12:35pm  Dwight Duffus  max 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 NagelLutz Theorem, elliptic curves over finite fields, MordellWeil theorem.  000  MSC: E406  TuTh 10:00am  11:15am  Raman Parimala  max 15  MATH 789R: Topics in Analysis: Geometric Partial Differential Equations  Credits: 4  − Description  − Sections 
Content: No description available.  000  MSC: W303  TuTh 10:00am  11:15am  Vladimir Oliker  max 15  MATH 797R: Directed Study  Credits: 1  12  − Description  − Sections 
 MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 

