Graduate classes, Fall 2006, Mathematics

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MATH 511: Analysis ICredits: 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: E406TuTh 10:00am - 11:15amShanshuang Yangmax 20
MATH 515: Numerical Analysis ICredits: 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: W301MWF 12:50pm - 1:40pmEldad Habermax 20
MATH 521: Algebra ICredits: 4− Description− Sections
Content: Groups, homomorphisms, the class equation, Sylow's thoerems, rings and modules.
000MSC: E408MWF 2:00pm - 2:50pmRaman Parimalamax 20
MATH 535: Combinatorics ICredits: 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.
000MSC: E406MWF 10:40am - 11:30amRon Gould
MATH 541: Topology ICredits: 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.
000MSC: E406MWF 9:35am - 10:25amWilliam Mahavier
MATH 545: Introduction to Differential Geometry ICredits: 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 3-space to illustrate key concepts.
Particulars: Open to undergraduates with permission of the instructor.
000MSC: E408TuTh 11:30am - 12:45pmGideon Maschler
MATH 558: Partial Differential EquationsCredits: 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: E408TuTh 2:30pm - 3:45pmVladimir Oliker
MATH 597R: Directed StudyCredits: 1 - 12− Description− Sections
MATH 599R: Master's Thesis ResearchCredits: 1 - 12− Description− Sections
MATH 733: Probabilistic MethodsCredits: 4− Description− Sections
Content: This course will develope discrete probabilistic aspects of topics begun in Math 535-536. Included will be: The probabilistic method of Erdos. Modifications: the deletion method. Refinements: the Lovasz-Local Lemma and its applications. The second-moment method. Large deviation inequalities and Derandomization.
Particulars: Specific prerequisites: Math 535-536 Combinatorics I & II or permission of the instructor.
000MSC: E408TuTh 8:30am - 9:45amVojtech Rodl
MATH 737: Random GraphsCredits: 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 531-532 Graph Theory I and II, Math 500, or permission of the instructor.
000MSC: E408TuTh 10:00am - 11:15amFaculty (TBA)
MATH 788R: Topics in Algebra: Number TheoryCredits: 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 polynomial-time 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 521-522.
000MSC: E406TuTh 1:00pm - 2:15pmEric Brussel
MATH 789R: Comp. Methods for Image RestorationCredits: 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.
000MSC: W304MWF 2:00pm - 2:50pmEldad Haber
MATH 797R: Directed StudyCredits: 1 - 12− Description− Sections
MATH 799R: Dissertation ResearchCredits: 1 - 12− Description− Sections