FALL 2002 COURSES

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Computer Science Courses:

Seminar: Combinatorial Computation, Prof. Pach, Wed. 6:30-8:30pm

Public Key Cryptography, Prof. Anshel, Fri. 2:00-4:00pm

Algebraic Numerical Computation, Prof. Pan, Tues. 6:30-8:30pm

Topics in Algorithm Analysis: Parallel Alogorithms and Complexes, Prof. Zachos, Tues. 2:00-4:00pm

Mathematics Course Descriptions:

MATH 81200: Topics in Complex Variables
Prof. Nikola Lakic
T , 2:00 - 3:15pm; F, 9:35-11:15

This course will be based on some introductory topics in advanced complex analysis. We will talk about non-euclidean geometry and hyperbolic metric with their applications to complex analysis. These applications provide a link between two groups of mathematical topics. On the one hand, we have hyperbolic geometry, Riemann surfaces and geometric function theory, and on the other hand, we have Teichmuller theory, quadratic differentials and holomorphic motions with their applications to complex dynamics. The course will contain the introduction to all these topics.

MATH 83530: Algebra II: Intro. To Alg. Geom. & Number Theory
Prof. Lucien Szpiro
M, 4:15 - 6:15pm

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. Books recommended:

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).

MATH 84700: Geometric Graph Theory
Prof. Janos Pach
W, 4:15 - 6:15pm

During the past decade Geometric Graph Theory yielded some striking results that have proved to be instrumental for the solution of a variety of problems in combinatorial and computational geometry. These include the k-set problem, proximity problems, and bounding the number of incidences between points and lines. This course is designed to introduce a broad range of fundamental and recent results in Geometric Graph Theory to senior researchers, junior researchers and graduate students, in combinatorial and computational geometry, graph theory, topology, theoretical computer science and graph drawing. Some familiarity with combinatorics and probability theory is required. TOPICS:

TEXTBOOK: Notes and manuscripts distributed in class.

SUGGESTED READING: G. di Battista, P. Eades, R. Tamassia, I. G. Tollis: Graph Drawing, Prentice Hall, 1999.

CSC 80030: Quantum Cryptographic Algorithms
Prof. Michael Anshel
Fri, 2:00 - 4:00pm

Quantum Cryptography is concerned with the conceptualization, definition and construction of quantum algorithms that address security concerns. Researchers have focused on defining cryptographic tasks and solving cryptologic problems using quantum algorithms. After providing a foundation for the study of quantum cryptographic algorithms we examine in depth:the Bennett-Brassard 84 quantum key distribution protocol; Shor's quantum algorithm for integer factorization; Grover's quantum search algorithm; and fault-tolerant quantum algorithms via quantum error correction. In the process we discuss the emergence of scalable quantum algorithms and their implications for code making and code breaking.

CSC 80040: Algebraic and Numerical Computation
Prof. Victor Pan
T, 6.30 - 8.30pm

Algebraic and numerical computing is the cornerstone of modern computations in sciences, engineering, and signal processing. This means a huge cache of topics of study and research in both computer science and computational mathematics. Some of these topics can be easily accessed by students with little background in either field, the study of other topics would benefit from some knowledge of most basic techniques of algorithm design and/or computations with matrices and polynomials. Respectively, the students are usually divided into two groups with which the instructor meets separately for 2 hours per week each. The students in the entry group study the fundamentals of polynomial and matrix computations. They may eventually join the group of students who study more advanced topics, potentially the theses topics (15 defenses in the last 7 years.) Students in Computer Science Program are also encouraged to work on computer implementation of the algorithms they develop with the instructor. For the students in the Math Program, the seminar gives a chance to use nontrivial math tools for solving problem of importance in modern computing. The instructor supplies handouts of relevant journal papers and also uses his books, the most recent being the volume: V. Y. Pan, Structured Matrices and Polynomials: Unified Superfast Algorithms, Birkh/Springer, Boston/New York, 2001. Because of the variety of the available topics, the subjects in the seminar can be partly adjusted to the students interests and background. The topics of the recent semesters included univariate and multivariate polynomial rootfinding, polynomial and rational interpolation, polynomial computations in finite fields, algebraic encoding/decoding, fundamental computations with Toeplitz, Hankel, Cauchy and other structured matrices, and the techniques for exploring and exploiting the correlation between algebraic and numerical computations in algorithm design. The latter two topics are in the hot research areas anare likely to be under demand in the upcoming semester as well.

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