Spring 2005 COURSES

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[All courses meet at the Graduate Center; Rms. TBA]

MATH 70200: Functions of a Real Variable
M & W, 12:30-1:45 p.m.   4.5 credits Prof. Richard Churchill

Basic theory of Banach and Hilbert spaces; applications to Fourier analysis; implicit and inverse function theorems; existence theory for ordinary differential equations; the recurrence theorem for ordinary differential equations; the recurrence theorem for divergence-free vector fields; additional topics as time allows.

MATH 70400: Functions of Complex Variable
T & Th, 12:30-1:45 p.m.  4.5 credits Prof. David Fisher

This is the second semester of a year long course in complex analysis. I will begin by covering some topics in complex function theory (primarily from Ahlfors' ``Complex Analysis", Chapter 5) and various versions of the Riemann mapping theorem, (primarily Ahlfhors, Chapter 6). Afterwards I will cover a variety of topics concerning Riemann surfaces, using Narasimhan's ``Compact Riemann Surfaces" as a primary reference.

MATH 70600: Algebra I
T & Th, 2:00-3:15 p.m. 4.5 credits Prof. Alphonse Vasquez

This is the second semester of a year long course in Algebra. The entire year's course should be adequate preparation for the Qualifying Exam in Algebra.

The book, Basic Abstract Algebra by Bhattacharya, Jain, and Nagpaul has a very large intersection with the year's work.

The spring semester will assume the fall semester's topics and proceed from there. It will certainly contain and probably begin with the structure theorem for finitely generated modules over a principal ideal domain (PID). Since any abelian group is, in a unique way, a module over the ring of integers, this theorem includes as a special case the structure theorem for finitely generated abelian groups. It has another very important application. Let T be a linear map from a finite dimensional vector space V (over the field k) into itself. One can then view V as a module over the ring R=k[X] of polynomials over k by letting Xv mean T(v). V, so considered, is a finitely generated module over R. The structure theorem applies in this case and elucidates notions such as T's characteristic and minimal polynomials and the Cayley-Hamilton theorem. It leads to the rational canonical form for n x n matrices over k. When the PID is a Euclidean ring (such as the two just mentioned) this material can be described algorithmically in terms of the Smith normal form. Galois theory will certainly be an important segment of the course.

MATH 70800: Topology I
T & F, 10:15-11:30 a.m. 4.5 credits Prof. Joseph Roitberg

Brief introduction to higher homotopy groups; homology and cohomology groups, including the Eilenberg-Steenrod axioms, calculations for CW-complexes, and applications (such as the Jordan-Brouwer theorem). References include "Algebraic Topology, A First Course" by Greenberg & Harper, "Algebraic Topology" by Hatcher, "Elements of Algebraic Topology" by Munkres, "An Introduction to Algebraic Topology" by Rotman, and "Algebraic Topology" by Spanier.

MATH 71000: Differential Geometry I
M & W, 2:00-3:15 p.m.    4.5 credits Prof. Jуzef Dodziuk

The theme of the course will be to reinforce and illustrate concepts already introduced in the first semester. The theme will be the interaction between analysis, geometry and topology. Some topics to be covered:

• Integration of differential forms
• Stokes' theorem
• Poincare lemma
• de Rham cohomology (including the proof of de Rham theorem)
• Hodge theory
• Relations between curvature of Riemannian manifolds and topology (theorems of Hadamard, Bonnet-Myers, Bochner, ...)

MATH 71200: Logic I
M & W, 4:15-5:30 p.m.   4.5 credits Prof. Russell Miller

This course is a continuation of Math 71100, which was offered in the fall semester and should be considered a prerequisite. We will examine and prove Godel's Incompleteness Theorems, and then continue with an introduction to the theory of computability (also known as recursion theory, or recursive function theory) and to the axioms and basic results of set theory. The text for the course is Shoenfield's *Mathematical Logic*, specifically chapters 6, 7, and 9. At times Shoenfield's presentation can be terse, so for further background, students are encouraged to consult Enderton (*A Mathematical Introduction to Logic*) or Mendelson (*Introduction to Mathematical Logic*) on incompleteness, Soare (*Recursively Enumerable Sets and Degrees*) on computability, or Kunen (*Set Theory: An Introduction to Independence Proofs*) on set theory.

MATH 82530: Algebraic Topology for Geometry, Algebra and Analysis
T 10:30 am-12:30 pm, 3 credits, Professor Sullivan

The idea of this two semester course is to select from all of algebraic topology those parts that are quite useful in geometry, algebra and analysis.

Homology theory is presented in terms of geometric cycles and homologies mapping into a space. Two related topics are then spherical cycles relating to the Hurewicz theorems and the interplay of the intersection product in manifolds with the dualities of Alexander, Lefschetz, and Poincaré.

Cohomology appears naturally as obstructions in inductive constructions of mappings and cross sections of bundles. Characteristic classes are an immediate corollary as are Postnikov systems. Cohomology also appears naturally as obstructions to the term by term construction of deformations of algebraic structures such as (differential, graded) associative algebras, commutative algebras or lie algebras. These obstruction theories only obtain full expression when applied to the derived version of the deformation problem where exact equations like associativity or Jacobi or commutativity are replaced by Stasheff type hierarchies of chain homotopies correcting the inequations.

The latter ideas have appeared at the deeper aspects of the interface of geometry and algebra in differential topology, symplectic topology, and holomorphic topology. (mirror conjectures)

Differential forms relate analysis to cohomology and rational homotopy. The above algebraic ideas are useful for extending the partial product on currents to a meaningful structure.

The course will emphasize the fundamental aspects of these issues. Grades will be based on (graded by me) homework assignments, class participation and the exams.

 

MATH. 83200: Probability Theory
T & Th, 2:00-3:15 p.m. 4.5 credits Prof. Elena Kosygina

This is a continuation of Probability Theory I from the Fall, which covered roughly the first 3 chapters of the book by R. Durrett, Probability: Theory and Examples.
Prerequisites: a course in Real Analysis and familiarity with the material mentioned above.
Recommended books:

• R. Durrett, Probability: Theory and Examples, 2nd or 3d edition, ISBN 0-534-24318-5,
• I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, ISBN: 0-387-97655-8.

MATH. 86800: Functional Analysis
4.5 credits M & W, 10:15-11:30 a.m.  Prof. Stanley Kaplan

This is the second half of a one-year course. Topics to be covered: Introduction to the theory of C*-algebras and von Neumann algebras with applications to the spectral theorem for bounded and unbounded operators in Hilbert space, as well as a survey of Fredholm theory.

References: Pedersen "Analysis Now" (Springer),Conway "A First Course in Functional Analysis" (Springer) and Kadison and Ringrose "Fundamentals of the Theory of Operator Algebras" Vol. 1 (Academic Press)

MATH. 87000: Algebra II
M, 4:15-6:15 p.m. 3 credits Prof. Lucien Szpiro

The course will be a continuation of the fall 2004 class. In this class we studied : tensor products, localisation, Zariski topology, affine and projective schemes, projective modules, line bundles, Picard group, rings of dimension zero and of dimension one, principal ideal domains, torsion free abelian groups, integral extensions, number fiels and ring of integers. Basic references for this:

• My notes (web site)
• Hartshorne's Algebraic Geometry,
• Atyah, Mac Donald Introduction to Commutative Algebra

MATH 87200: Group Theory
F, 11:45 a.m.-1:45 p.m. 3 credits Prof. Gilbert Baumslag    

Description: The notion of an automatic group grew out of the recognition that the geodesics in a hyperbolic group can be represented by the words in a regular language and that equality of such words and multiplication by generators can be captured again in terms of regular languages, that is by means of finite state automata. The actual definition is due to W. Thurston, following on work of Cannon and Gilman, and represents a marriage of computer science and computational group theory. The object of the course is to provide an introduction to the theory of these automatic groups. The course will be self-contained. However a smattering of knowledge of elementary group theory, including the notion of a finite presentation, will make the subject matter more meaningful.

Reference material:

• Word processing in Groups, by D.B.A. Epstein et. al., Publisher A.K. Peters Ltd.
• Introduction to automata theory, languages and computation, by John E. Hopcroft and Jeffrey D. Ullman, Published by Addison-Wesley

MATH. 87400: Topics in Alg. Number Theory
Th, 1:00-3:00 p.m. 3 credits Prof. Victor Kolyvagin

MATH. 89901 - Ind Resch in Analysis
Rm. TBA, 1-12 credits

MATH. 89902 - Ind Resch in Alg/Num Theory
Rm. TBA, 1-12 credits

MATH. 89903 - Ind Resch in Geom/Topology
Rm. TBA, 1-12 credits

MATH. 89904 - Ind Resch in Logic
Rm. TBA, 1-12 credits

MATH. 89905 - Ind Resch in Applied Math
Rm. TBA, 1-12 credits

MATH 90000 - Dis. Sup.
Rm. TBA, 1 credit

See Also

CSc 80040: Topics in Algorithm Analysis: Alg. & Num. Comp.
T, 6:30- 8:30 p.m. 3 credits Prof. Victor Pan

CSc 80300: Topics in Algorithm Design: Graph & Network Alg.
Th, 2-4 p.m. Prof. Amoz Bar-Noy 3 credits

CSc 85030: Topics in Cryptography and Computer Security: Foundations of Cryptography
F, 2-4 p.m. 3 credits Prof. Michael Anshel

CSc 85040: Topics in Computational Complexity: Approximation Algorithms & Complexity
W, 2-4 p.m. 3 credits Prof. Stathis Zachos

CSc 85200: Seminar in Theoretical Computer Science: Seminar: Computational Logic
T, 2-4 p.m. 1 credit Prof. Sergei Artemov