Graduate classes, Fall 2006, Mathematics
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: E406  TuTh 10:00am  11:15am  Shanshuang Yang  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: W301  MWF 12:50pm  1:40pm  Eldad Haber  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  MWF 2:00pm  2:50pm  Raman Parimala  max 20  MATH 535: Combinatorics I  Credits: 4  − Description  − Sections 
Content: This course will begin the development of fundamental topics in Combinatorics. Included will be: Basic enumeration theory, uses of generating functions, recurrence relations, the inclusion / exclusion principle and its applications, partition theory, general Ramsey theory, an introduction to posets, and matrices of 0's and 1's, systems of distinct representatives, introduction to block designs, codes, and general set systems.  000  MSC: E406  MWF 10:40am  11:30am  Ron Gould   MATH 541: Topology I  Credits: 4  − Description  − Sections 
Content: An introduction to some of the fundamental concepts of topology required for basic courses in analysis. Some of these comcepts are: Baire category, topological spaces, completeness, compact and locally compact sets, connected and locally connected sets, characterizations of arcs, Jordan curves, and Peano continua, completeness, metric spaces, separability, countable bases, open and closed mappings, and homeomorphisms. Other selected topics if time permits from algebraic topology or continua theory.  000  MSC: E406  MWF 9:35am  10:25am  William Mahavier   MATH 545: Introduction to Differential Geometry I  Credits: 4  − Description  − Sections 
Content: An introduction to Riemannian geometry. The main goal is an understanding of the nature and uses of curvature, which is the local geometric invariant that measures the departure from Euclidean geometry. No previous experience in differential geometry is assumed, and we will rely heavily on pictures of surfaces in 3space to illustrate key concepts. Particulars: Open to undergraduates with permission of the instructor.  000  MSC: E408  TuTh 11:30am  12:45pm  Gideon Maschler   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: E408  TuTh 2:30pm  3:45pm  Vladimir Oliker   MATH 597R: Directed Study  Credits: 1  12  − Description  − Sections 
 MATH 599R: Master's Thesis Research  Credits: 1  12  − Description  − Sections 
 MATH 733: Probabilistic Methods  Credits: 4  − Description  − Sections 
Content: This course will develope discrete probabilistic aspects of topics begun in Math 535536. Included will be: The probabilistic method of Erdos. Modifications: the deletion method. Refinements: the LovaszLocal Lemma and its applications. The secondmoment method. Large deviation inequalities and Derandomization. Particulars: Specific prerequisites: Math 535536 Combinatorics I & II or permission of the instructor.  000  MSC: E408  TuTh 8:30am  9:45am  Vojtech Rodl   MATH 737: Random Graphs  Credits: 4  − Description  − Sections 
Content: This course will introduce the fundamental topics in Random Graph Theory. Included will be: Basic Models: binomial, uniform. The method of moments: subgraph counts Thresholds for large subgraphs: connectedness, perfect matchings, hamiltonicity. Phase transitions. Advanced tools: martingales, Janson's inequality. Chromatic Number and other partition and extremal properties. Particulars: Specific prerequisites: Math 531532 Graph Theory I and II, Math 500, or permission of the instructor.  000  MSC: E408  TuTh 10:00am  11:15am  Faculty (TBA)   MATH 788R: Topics in Algebra: Number Theory  Credits: 4  − Description  − Sections 
Content: An introduction to algebraic number theory. Algebraic number theory is the study of number fields (finite field extensions of the rational number field Q), using Galois theory, finite group theory, and basic commutative ring theory. It is a prerequisite for many ``real world'' applications, including the number field sieve, which is the fastest known algorithm for factoring large integers, and the recent discovery of a polynomialtime primality test. We will study unique factorization of ideals in Dedekind domains, the structure of the group of integer units in a number field, the finiteness of the class group, and the Galois ramification theory. Particulars: Prerequisite: Math 521522.  000  MSC: E406  TuTh 1:00pm  2:15pm  Eric Brussel   MATH 789R: Comp. Methods for Image Restoration  Credits: 4  − Description  − Sections 
Content: In this course we review the field of image restoration and its application to medical imaging. We will discuss the mathematical background, using variational techniques and numerical optimization. The course is a "hands on" course and the students will write Matlab code and deal with actual data. Particulars: Prerequisite: Math 516 or consent of the instructor.  000  MSC: W304  MWF 2:00pm  2:50pm  Eldad Haber   MATH 797R: Directed Study  Credits: 1  12  − Description  − Sections 
 MATH 799R: Dissertation Research  Credits: 1  12  − Description  − Sections 

