Tuesday, October 23 |
07:00 - 09:00 |
Breakfast (Vistas Dining Room) |
09:00 - 10:00 |
Martha Precup: The Betti numbers of Hessenberg varieties ↓ This talk will begin with a survey of results describing the Betti numbers of Hessenberg varieties. There are many geometric and combinatorial applications of these formulas. In the second half of the talk, I will report on recent joint work with M. Harada in which we prove an inductive formula for the Betti numbers of certain regular Hessenberg varieties called abelian Hessenberg varieties. Using a theorem of Brosnan and Chow, this formula yields a proof of the Stanley-Stembridge conjecture for this special case. (TCPL 202) |
10:00 - 10:30 |
Coffee Break (TCPL Foyer) |
10:30 - 11:30 |
Erik Insko: Singularities of Hessenberg varieties ↓ Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. The local geometric structure of Hessenberg varieties can often be studied using cell decompositions, group actions, patch ideals, and the combinatorics of the symmetric group. In this talk, we will give a survey of results regarding the singularities of Hessenberg varieties. This is based on joint works with Alex Yong and Martha Precup. (TCPL 202) |
11:30 - 13:30 |
Lunch (Vistas Dining Room) |
13:15 - 14:15 |
Mikiya Masuda: The cohomology rings of regular semisimple Hessenberg varieties for h=(h(1),n,…,n) ↓ I will talk about an explicit ring presentation of the cohomology ring of the regular semisimple Hessenberg variety in the flag variety Fl(Cn) with Hessenberg function h=(h(1),n,…,n) where h(1) is an arbitrary integer between 2 and n. This is joint work with Hiraku Abe and Tatsuya Horiguchi. (TCPL 201) |
14:30 - 15:30 |
Tatsuya Horiguchi: The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials ↓ A regular nilpotent Hessenberg variety Hess(N,h) is a subvariety of a flag variety determined by a Hessenberg function h. I will explain a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials fi,j were introduced by Abe-Harada-Horiguchi-Masuda. In this talk, I will show that every polynomial fi,j is an alternating sum of certain Schubert polynomials Sw(i,j)k (k=1,2,...,i-j). Moreover, we can interpret the permutations w(i,j)k from a geometric viewpoint under the circumstances of having a codimension one regular nilpotent Hessenberg variety Hess(N,h′) in the original regular nilpotent Hessenberg variety Hess(N,h) where the Hessenberg function h' is obtained from the original Hessenberg function h and (i,j). (TCPL 201) |
15:15 - 15:45 |
Coffee Break (TCPL Foyer) |
16:00 - 17:00 |
James Carrell: Cohomology algebras of varieties with a 'regular' (Ga,Gm)-action ↓ This is an expository talk on "regular (Ga,Gm)-actions" on projective varieties (over the complexes). A (Ga,Gm)-action is the same thing as an action of a Borel subgroup of SL(2).
We call a (Ga,Gm)-action regular when the maximal unipotent subgroup Ga has a unique fixed point. There are lots of examples of varieties with a regular (Ga,Gm)-action such as
Schubert varieties. Of particular interest are regular nilpotent Hessenberg varieties. The main point is that the cohomology algebra of a smooth projective variety X with (Ga,Gm)-action
is isomorphic with the fixed point scheme of the Ga action, and this is also true of many singular (Ga,Gm)-stable subvarieties such as the two classes mentioned above. The Gm-equivariant cohomology admits a nice description which I will also explain. (TCPL 201) |
17:30 - 19:30 |
Dinner (Vistas Dining Room) |