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UID:20200120T100255CET-49401D0KMr@ical.php
DTSTAMP:20200120T090155Z
CLASS:PUBLIC
DESCRIPTION:Abstract: 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 moder
n knot theory is to relate these quantum knot invariants to the topology o
f a knot complement and the AJ-conjecture gives one possible connection. I
n this two part series of lectures\, we will endeavor to understand the st
atement of the AJ-conjecture\, its current states of affairs\, and attempt
to pinpoint the main difficulties in its resolution. \nIn part 2 of this
series\, we will define the A-polynomial of a knot using the SL(2\, C) ch
aracter variety machinery developed last time. We will briefly discuss th
e famous 'boundary slopes are boundary slopes' theorem\, then move on to t
he 'J' in the AJ-conjecture\, which refers to the colored Jones polynomial
of a knot. Next we'll turn to the recurrence ideal of a knot and frame o
ur discussion in terms of skein modules. This perspective illuminates the
plausibility of the AJ-conjecture. The lecture will finish with the stat
ement of a theorem of Le and Zhang which highlights the primary difficulty
in resolving the conjecture. \nThese lectures will be expository in natur
e and should be well suited to introduce graduate students to the topics d
escribed above.
DTSTART:20180917T160000
DTEND:20180917T160000
LOCATION:Wilson Hall\, 1-144
SUMMARY:Mathematics Seminar: 'On the AJ-conjecture\, Part 2'
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