THE MALLAT FAMILY FUND FOR RESEARCH IN MATHEMATICS
invites you to a
SPECIAL LECTURE SERIES
to be presented by
University of Texas at Austin
The lectures will be held in
Amado Mathematics Building
Technion - Israel Institute of Technology
Lecture I: Monday, 28 April 2003 at 15:30
THE CLASSIFICATION OF KNOTS
The systematic study of knots began in the latter half of the 19th century, although rigorous proofs only became possible later with the advent of topological methods. An algorithmic solution of the unknotting problem was obtained by Haken around 1960, using 3-dimensional geometric topology, and his methods ultimately led to the solution of the general knot problem in the late 1970's. More intuitive results for alternating knots were obtained using the Jones polynomial, discovered in 1984, culminating in the proof of Tait's flyping conjecture by Menasco and Thistlethwaite. Milnor's conjecture on the unknotting number of torus knots was proved by Kronheimer and Mrowka in 1993, using results of Donaldson about smooth 4-manifolds. Nevertheless, the unknotting number remains an elusive invariant. We will discuss these developments in the history of knot theory, as well as some elementary questions about knots that remain unanswered.
Lecture II: Wednesday, 30 April 2003 at 15:30
DEHN SURGERY OF KNOTS I
Lecture III: Thursday, 1 May 2003 at 15:30
DEHN SURGERY OF KNOTS II
Dehn surgery is a way of constructing 3-manifolds from knots. Generically, the properties of the knot persist in the resulting manifold; for example, most Dehn surgeries on most hyperbolic knots (and "most" knots are hyperbolic) give hyperbolic 3-manifolds. In the light of recent developments, it seems that it may eventually be possible to completely describe the exceptions. We will discuss progress towards this goal.
The first talk is intended for a general mathematical audience. The second and third talks are connected.