View Proposal #262
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| ID | 262 |
|---|---|
| First Name | Louis |
| Last Name | Kauffman |
| Institution | University of Illinois at Chicago |
| Speaker Category | invited |
| Title of Talk | Introduction to Knot Theory |
| Abstract | The theory of knots is a recent part of mathematics. It originated in the tabulation of tables of knots by the mathematicians Tait, Kirkman and Little in the 19th century. These tables were prepared at the behest of Lord Kelvin (Sir William Thompson) who had developed a theory that atoms were three dimensional knotted vortices in the luminiferous aether. Along with these speculations came the development of geometry and topology in the hands of Gauss, Riemann, Poincare and others. As the knotted vortex theory declined (it has never entirely disappeared!), the mathematics of topology ascended, and the theory of knots came into being as part of the study of low dimensional manifolds, using the fundamental group of Poincare and early versions of homology theory. Max Dehn used the fundamental group to show that a trefoil knot and its mirror image are topologically distinct. J. W. Alexander in the 1920's found a polynomial invariant of knots that bears his name to this day. Kurt Reidemeister, in the 1920's, discovered a set of moves on diagrams for knots that made their classification a (difficult) combinatorial problem. In the 1980's there came a rebirth of these combinatorial schemes in the discovery of the Jones polynomial invariant of knots and links (and its relatives and descendants). Along with the new combinatorial invariants came new relationships with physics and with many fields of mathematics (combinatorics, graph theory, Hopf algebras, Lie algebras, von Neumann algebras, functional integration, category theory) and new kinds of mathematics such as higher categories and categorification. This talk will discuss the history of knot theory and then it will concentrate on describing the Jones polynomial, its relationships with physics, and recent developments related to categorification. |
| Subject area(s) | MAA George Polya Lecturer |
| Suitable for undergraduates? | ??? |
| Day Preference | |
| Computer Needed? | Y |
| Bringing a laptop? | |
| Overhead Needed? | Y |
| Software requests | |
| Special Needs | |
| Date Submitted | 9/20/2009 |
| Year | 2009 |