《Lecture Schedules for categorified quantum groups and their applications》

(ECNU, May 4—25, 2012)
(1) on May 10, Lecture Room 129
1:30---3:30pm, Lauda
4:00---5:00pm, Russell

(2) on May 11, Lecture Room 126
9:30---11:30am, Lauda
1:30---2:30pm, Russell

May 12, morning, departure time 8:30am at the head of Math Building, bus travel to the east Zhejiang Province, bubble hot spring (with young colleagues).

(3) on May 14, Lecture Room 126
1:30---3:30pm, Lauda
4:00---5:00pm, Russell

(4) on May 15, Lecture Room 129
1:30---3:30pm, Lauda
4:00---5:00pm, Russell

May 17---May 18, free
(5) on May 16, Lecture Room 129
9:30---11:30am, Lauda

May 19 or May 20, travel to Suzhou (with Prof. M. Rosso), ancient city, for one day.

(6) on May 21, Colloquium talk, Lecture Room 102
1:30---2:30pm, Marc Rosso A recollection on quantum groups: from Drinfeld-Jimbo to Woronowicz and back.
2:40---3:40pm, A. Lauda Categorical Lie algebra actions and categorified quantum groups.
4:00---5:00pm, H. Russell Odd cohomology of type A Springer varieties.


Lecture Contents:
1) An introduction to the ideas of categorification taking examples from knot theory
2) An introduction to 2-categories as a tool for turning algebra into planar diagrammatics.
3) An introduction to quantum groups and their categorification. How to invent the definitions yourself.
4) A closer look at the categorification of the positive half of the quantum group for sl2. This theory is based on an algebra called the odd nilHecke algebra. Here we see interesting structures like symmetric functions, schur polynomials, and cohomology rings of Grassmannians.
5) Categorifying irreducible representations of quantum groups with an emphasis on examples. This includes what are sometimes called cyclotomic quotients. These ideas lead directly to Webster's work categorifying knot invariants.
6) Categorifying the entire quantum group using 2-categories.
7) Braid group actions arising from actions of categorified quantum groups. Here we would review the work of Chuang-Rouquier constructing equivalences using categorical sl2-actions. We then look at work of Cautis-Kamnitzer-Licata extending these ideas to construct geometric
knot homology theories using a beautiful algebraic idea called skew Howe duality.
8) Odd categorifications of quantum groups. Here we explain that the categorification of the Jones polynomial is not unique. This suggests looking for additional categorifications of quantum sl2. This theory leads to surprising new structures including a noncommutative version of symmetric functions and cohomology rings for Grassmannians.

(II) An introduction to Springer varieties, Khovanov homology and knot theory.
By Prof. H. Russell, U. Sounthern California
1) An introduction to Springer varieties and Springer theory in Type A
2) Connections between Springer varieties, Khovanov homology, and related knot theory constructions
3) Springer varieties and the combinatorics of Kuperberg webs.
4) Oddification of the cohomology of Springer varieties.