Math Seminar with Charles Katerba (MSU) "On the AJ-conjecture, Part 1"
- Monday, September 10, 2018 at 4:00pm
- Wilson Hall, 1-144 - view map
Over the past 35 years, knot theory has witnessed an explosion of powerful new invariants such as the colored Jones polynomial and the HOMFLY-PT polynomial. The origin of many of these invariants can be unified using quantum groups. One of the fundamental problems in modern knot theory is to relate these quantum knot invariants to the topology of a knot complement and the AJ-conjecture gives one possible connection. In this two part series of lectures, we will endeavor to understand the statement of the AJ-conjecture, its current states of affairs, and attempt to pinpoint the main difficulties in its resolution.
In part 1, we will discuss the 'A' in the AJ-conjecture, which refers to the A-polynomial of a knot. The A-polynomial is defined using representations from the fundamental group of a knot complement to SL(2, C) and has seen broad applications in 3-manifold topology. To understand its connection to topology, we will describe representation/character varieties and schemes and Culler-Shalen theory, which uses the algebraic geometry of a character varieties to build special surfaces in 3-manifolds.
These lectures will be expository in nature and should be well suited to introduce graduate students to the topics described above.
- Department of Mathematical Sciences