Special Lecture Series: Webster -- Representation theory through the lens of categorical actions

UPDATE (June 4): Slides and hopefully also recordings of Ben Webster’s lectures
are being posted here:

http://www.maths.usyd.edu.au/u/AlgebraSeminar/15abstracts/webster15.html
 
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UPDATE (May 25): Ben Webster’s visit will now start on June 1, so his scheduled talk
on May 29 will be cancelled. His series will now consist just of the four talks on
Wednesday afternoons at 4pm in June. Some talks may now extend beyond one hour,
so I have updated the calendar entries to show a nominal finishing time of 5.30pm.

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UPDATE (May 18): Ben Webster has had to delay his arrival in Sydney for family reasons,
so some postponement/recheduling of his talks will be necessary. His first scheduled
talk on May 22 will definitely not take place then, and May 29 is doubtful at this stage.
A revised schedule will be posted when we know his new dates.

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Ben Webster (University of Virginia) will be visiting the School from May 18 to June 26,
funded by the University of Sydney International Research Collaboration Award scheme.
As part of his visit he will be giving a series of 6 lectures on representation theory.
The first two will take place in the usual Friday Algebra Seminar time-slot, and will be
accessible to a broad algebraically-oriented audience.  The subsequent lectures will be
on Wednesday afternoons in the Access Grid Room.  If you can attend any or all of this
series, please take advantage of this extended visit by a major young American
mathematician.  

Anthony Henderson 

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Speaker: Ben Webster (University of Virginia)
 
Web-page: http://people.virginia.edu/~btw4e/ 

Lecture times: 

1) Friday May 22, 12-1pm, Carslaw 375 (Algebra Seminar) ***CANCELLED***
2) Friday May 29, 12-1pm, Carslaw 375 (Algebra Seminar) ***CANCELLED***
3) Wednesday June 3, 4-5.30pm, Carslaw 829 (Access Grid Room) 
4) Wednesday June 10, 4-5.30pm, Carslaw 829 (Access Grid Room) 
5) Wednesday June 17, 4-5.30pm, Carslaw 829 (Access Grid Room) 
6) Wednesday June 24, 4-5.30pm, Carslaw 829 (Access Grid Room) 

Title: Representation theory through the lens of categorical actions 

Abstract: 

Representation theorists (and other mathematicians) have come to appreciate
that a vector space or set having self-maps that satisfy certain relations can tell one
a great deal about the object being studied.  These talks will be about the categorified
version of this principle: that a category is much easier to study when it carries an
action of functors which satisfy certain relations (where, of course, one must be more
careful about what relations means).  

In a remarkable self-similarity, many categories that are of great interest to
representation theorists are themselves representations of a more complicated type.
Specifically, they are special cases of a more general notion of an "action of a Lie
algebra on a category." Thus, the entire representation category of all symmetric 
groups over a finite field is actually a representation of an affine Lie algebra.
This is a very powerful notion, which both leads us to unexpected structures 
(such as gradings on algebras without an obvious homogeneous presentation)
and to a new perspective on older results (for example, the identification of
decomposition numbers with Kazhdan-Lusztig polynomials).  

I’ll start by discussing the case of the symmetric group and general linear groups 
over a finite field, and expand outward to less familiar objects like categories O 
for Lie algebras and Cherednik algebras. In further lectures, I’ll discuss other 
applications of these ideas to areas like canonical bases, algebraic geometry and 
knot homology.