SPRING 2003 COURSES

SPRING 2003 COURSES

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Mon.	Tues.	Wed.	Thurs.	Fri.
10:00-11:15, Mon, Wed 81800 Topology II Prof. Thompson Rm. 6417		10:00-11:15, Mon, Wed 81800 Topology II Prof. Thompson Rm. 6417	10:00-12:00, Thu 87600 Topics in Algebraic Number Theory Prof. Kolyvagin Rm. 4419	10:00-12:00 SEMINAR Group Theory II Seminar
11:45-1:00, Mon, Wed 71000 Differential Geometry I Prof. Moskowitz Rm. 5417	11:45-1:00, Tue, Thu 70800 Topology I Prof. Vasquez Rm. 5417	11:45-1:00, Mon, Wed 71000 Differential Geometry I Prof. Moskowitz Rm. 5417	11:45-1:00, Tue, Thu 70800 Topology I Prof. Vasquez Rm. 5417	2:00-3:45, Tue; 10:00-11:45, Fri 81400 Topics in Complex Variables: Complex Dynamical Systems Prof. Yunping Jiang Rm. 5417
2:00-3:15, Mon, Wed 70600 Algebra I Prof. Rocha Rm. 6417	2:00-3:45, Tue; 10:00-11:45, Fri 81400 Topics in Complex Variables:Complex Dynamical Systems Prof. Yunping Jiang Rm. 5417	2:00-3:15, Mon, Wed 70600 Algebra I Prof. Rocha Rm. 6417		11:00-12:30 SEMINAR Complex Analysis Student Seminar Rm. TBA
2:00-3:15, Mon, Wed 85800 Partial Differential Equations Prof. Velling Rm. 6421	2:00-3:15, Tue, Thu 70200 Functions of a Real Variable Prof. Randol Rm. 8405	2:00-3:15, Mon, Wed 85800 Partial Differential Equations Prof. Velling Rm. 6421	2:00-3:15, Tue, Thu 70200 Functions of a Real Variable Prof. Randol Rm. 8405	11:45-1:45, Fri 87200 Group Theory Prof. Miasnikov Rm. 5417
4:15-5:30, Mon, Wed 70400 Functions of a Complex Variable Prof. Kaplan Rm. 8405	4:15-5:30, Tue, Thu 71200 Logic I Prof. Fitting Rm. 4214.03	4:15-5:30, Mon, Wed 70400 Functions of a Complex Variable Prof. Kaplan Rm. 8405	4:15-5:30, Tue, Thu 71200 Logic I Prof. Fitting Rm. 4214.03	1:45-3:45 SEMINAR Complex Analysis Seminar Rm. TBA
4:15-6:15, Mon 83540 Algebra II (Intro. to Algebra, Geometry, & Number Theory) Prof. Szpiro Rm. 6417	2:00-6:00 SEMINAR Einstein Chair Seminar:Topology & Quantum Objects Room 6417	4:15-6:15, Wed 80020 Combinatorial Geometry Prof. Pach Rm. 6417	2:00-4:00 SEMINAR Arithmetic Geometry Seminar Room 6417	2:00-3:30 SEMINAR CUNY Logic Workshop Rm. TBA
	4:00-5:00 SEMINAR Lie Theory Seminar Rm. TBA	Mon 6:00-7:15 Wed 4:15-5:30 80200 Model Theory Prof. Kossak Rm. TBA	3:30-5:00 SEMINAR Number Theory Seminar Rm. TBA	4:00-5:00 SEMINAR Group Theory Seminar Rm. TBA
	4:00-5:30 SEMINAR Probability Seminar Rm. TBA	5:00-6:00 SEMINAR Topology Seminar Rm. TBA		5:30-7:00 SEMINAR History of Mathematical Sciences Seminar Rm. TBA
Mon 6:00-7:15 Wed 4:15-5:30, 80200 Model Theory Prof. Kossak Rm. TBA		6:30-7:30 SEMINAR Combinatorics Seminar Rm. TBA

Mathematics Course Descriptions:

85800 Partial Differential Equations
Using Gilbarg & Trudinger's Elliptic Partial Differential Equations of Second Order as text, and several recent papers as models, we will examine how basic existence and uniqueness questions in geometry can be posed and addressed in the language of PDEs. It is hoped that guest lectures and student participation will be regular features of the class
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80020 Combinatorial Geometry
Can one plant n trees in an orchard, not all along the same line, so that every line determined by two trees will pass through a third? This question raised by Sylvester in the last century, has generated a lot of interest among professional and amateur mathematicians. It led to the birth of a new mathematical discipline, which even Euclid would appreciate and which has close ties to convexity, number theory, and applications to computational geometry, computer graphics, robotics, etc. This course offers an introduction to this rapidly developing field, where combinatorial and probabilistic (counting) methods play a crucial role. We will learn how to apply combinatorial methods to geometric problems and algorithms. Some familiarity with elementary combinatorics and probability theory is required. However, we will not build heavily on the material covered in my course given in the Fall semester.
TOPICS: Extremal graph theory, Repeated distances in space, Arrangements of lines and curves, Geometric graphs, Epsilon nets, Discrepancy theory, Applications in computational geometry.
TEXTBOOK: J. Pach and P. Agarwal: Combinatorial Geometry, Wiley, New York, 1995
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80200 Model Theory This will be a one semester course covering basic results on
nonstandard models of arithmetic and related theories. Topics will
include:

MacDowell-Specker theorem and partial classification of
countable models of PA

Arithmetized Completeness Theorem

Gaifman's Splitting Theorem and consequences of
Matiyasevic-Robinson-Davis-Putnam theorem

Minimal types and their applications

Models generated by indiscernibles

Satisfaction classes, resplendency, and recursive saturation

Automprphism groups of models of PA

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71200 Logic I
This course will cover the cluster of fundamental logic results grouped under the general heading of incompleteness and undecidability: Gцdel's and Rosser's theorems on incompleteness of formal systems of mathematics, Church's theorem on the undecidability of first-order logic, Tarski's theorem on the undefinability of truth, Turing's theorem on the undecidability of the halting problem. Along the way, the basics of computability (recursion) theory will be developed. Some familiarity with first-order logic is needed, along with the ability to follow mathematical arguments. The basic text for the course will be a set of notes, available from my web site, comet.lehman.cuny.edu/fitting; click on "Books & Papers," then on "Unpublished Books," and finally on "Notes on Incompleteness and Undecidability."

Web Page: comet.lehman.cuny.edu/fitting

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81800 Topology II
We will develop cobordism theory. This is a very rich topic which leads to a variety of applications in topology and geometry. We will start with applications of cobordism theory to prove some index theorems. We will look at methods for computing cobordism rings, e.g. Steenrod operations and the Adams Spectral Sequence. We will develop more applications to manifolds such the Conner -Floyd theorem and talk about the immersion conjecture. Along the the way it will be natural to introduce generalized cohomology theories like K-theory, complex cobordism, and elliptic cohomology.

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83540 Algebra II (Intro. to Algebra, Geometry, & Number Theory)
The course will be an introduction to Algebraic Geometry and Number Theory. We will study Hilbert Nullstellenzatz, Zariski topology, The Picard Group, Affine and Projective varieties and schemes, Metrized Picard group and the basic theorems of Algebraic number theory. The presentation will unify the geometric aspects and the number theoretic aspects with the point of view of Dedekind and Grothendieck modernized by Arakelov.

Recommended books:
R. Hartshorne. Algebraic Geometry (Springer)
D. Eisenbud, J. Harris. Schemes: The Language of Modern Algebraic Geometry (Wadsworth and Brooks/Cole)
L. Szpiro. Basic Arithmetic Geometry (Notes to be distributed)

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89901 Math - Independent Research in Analysis 1-12 cr

89902 Math - Research in Algebra & Numerical Theory 1-12 cr

89903 Math - Independent Research in Geometry and Topology 1-12 cr

89904 Math - Independent Research in Logic 1-12 cr

89905 Math - Independent Research in Applied Mathematics 1-12 cr

99000 Math - Dissertation Supervision 1 cr.

Computer Science Courses:

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SEE ALSO

CSC 80040 Algebraic and Numerical Computations
Prof. Victor Pan
Tuesdays, 6:30 - 8:30

Algebraic and numerical computing is the cornerstone of modern computations in sciences, engineering, and signal processing. It is a huge cache of topics for study and research in both computer science and mathematics. Systematic comparison of algebraic and numerical techniques for algorithm design and analysis simplifies study in both areas. The students from the CS and Math Programs learn fundamentals as well as selected research topics, eventually leading to PhD defenses (15 in the last 7 years in both programs).

Because of the variety of the available topics the subjects in the seminar can be partly adjusted to the students interests and background. Sample topics considered for the Spring include:

a) Structured (e.g., Toeplitz, Hankel, Cauchy, and Pick) matrices, omnipresent in modern computing, signal and image processing, and many areas of math, and closely related to fundamental algebraic computations
with polynomials and rational functions. This subject is treated both algebraically and numerically based on the instructor's recent book and several papers, many of them joint with his PhD students.

b) Polynomial and rational interpolation.

c) Solving a polynomial equation (the central and most influential problem in math for 4 millennia and still highly important in computer algebra), with possible extension to fundamentals of the solution of systems of multivariate polynomial equations.

d) Algebraic techniques for coding and cryptography.

The seminar resumes with new topics and new students every semester. The students are divided into the entry level group and the advanced group, the instructor meets separately for 2 hours per week with each group. The students in the entry level group study the fundamentals and eventually join the group of advanced students. The instructor supplies survey and research papers as handouts, in addition to his books currently on display by both Math and CS Programs. Good progress in learning is sufficient for high grades, but Computer Science students are also encouraged to implement new algorithms devised in the seminar, math. students to solve the relevant open problems in math.

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Last Updated on 1/30/03