Course I: Symbolic Dynamical Systems, Information Theory and Coding ( 3 weeks: July 2- 20, 2001)
Professor Ayse Sahin, DePaul
University, IL
TA: Nirit Sandman, University
of California- Berkeley, CA
Course Description: This course will be an introduction to symbolic dynamical systems. This is currently a very active and interesting area of study with many areas of application: information theory, coding problems and data storage.
We will begin by learning the basics of dynamical systems theory. The objects of study will be sequences made up of symbols from a finite alphabet, such as sequences of 0's and 1's. We will consider questions, constructions, and problems from both the theory of dynamical systems and several of the applications mentioned above.
Professor Leila Schneps, Ecole
Normale Superieure, Paris
TA: Kariane Calta, University
of Chicago, IL
Course Description: I
will start with modular arithmetic (i.e. arithmetic in Z/nZ) and then cover
some of Fermat's contributions to number theory: infinite descent, irrationality
of the square root of 2, Fermat's little theorem, and his "last theorem"
in the cases n=3,4. Then we'll cover Kummer's proof of Fermat's last theorem
in the case of regular primes.
Professor Joanna Kania-Bartoszynska,
Boise State University, ID
TA: Kariane Calta, University
of Chicago, IL
Course Description: We will develop the fundamental concepts of Knot Theory, and present some mathematical methods used in its study.
One can think of knots as strings tied in three-dimensional space. Two knots are identified if one can be made to look like another by moving it around. A fundamental question is: how do we tell them apart? Even though the classification of knots is far from complete, we will learn about various tools for distinguishing knots.
A convenient way of working with knots is via diagrams, that is their shadows in the plane. There is a set of moves, called the Reidemeister moves, between the diagrams of equivalent knots. We will focus on diagrammatic methods for studying knots.
A knot invariant is an object assigned to a knot such that if the objects are different, then the knots also have to be different. We will introduce the following invariants:
1. 3-coloring
2. linking number
3. Alexander polynomial,
4. Kauffman bracket and Jones polynomial; and examine their properties.
Knot Theory is closely connected to the study of 3-manifolds - one of
the most active areas of mathematical research today. Knot Theory also
has numerous applications outside mathematics: to biology, chemistry and
physics.
Course IV: Mathematical Logic ( 2 weeks: July 23- August 3, 2001)
Professor Tami Hummel, Allegheney College, PA
TA: Nirit Sandman, University
of California- Berkeley, CA
Course Description: Mathematical logic developed as an attempt to explore the foundations of mathematics. Central to understanding the basic concepts of mathematics is understanding the basic notion of proof. In this course, we will consider the notions of mathematical truth and provability, and begin to explore the relationship between them.
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