Math Seminar with Charles Katerba (MSU) "On the AJconjecture, Part 1"
When:
 Monday, September 10, 2018 at 4:00pm
Where:
 Wilson Hall, 1144  view map
Description:

Over the past 35 years, knot theory has witnessed an explosion of powerful new invariants such as the colored Jones polynomial and the HOMFLYPT 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 AJconjecture gives one possible connection. In this two part series of lectures, we will endeavor to understand the statement of the AJconjecture, 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 AJconjecture, which refers to the Apolynomial of a knot. The Apolynomial is defined using representations from the fundamental group of a knot complement to SL(2, C) and has seen broad applications in 3manifold topology. To understand its connection to topology, we will describe representation/character varieties and schemes and CullerShalen theory, which uses the algebraic geometry of a character varieties to build special surfaces in 3manifolds.
These lectures will be expository in nature and should be well suited to introduce graduate students to the topics described above.
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