Small covers arising as pullbacks from the simplex
In 1991, Davis and Januszkiewicz introduced small covers and showed that those arising as pullbacks from the linear model have torsion-free integral cohomology rings. Later, Cai, Choi, and Park proved that the converse also holds, and related this condition to the vanishing of the first Steenrod square. In this talk, I will discuss small covers arising as pullbacks from the simplex. This class contains pullbacks from the linear model as a special case, and the main result extends the corresponding cohomological characterizations.
On toric Schubert varieties in flag varieties
A partial flag variety is a smooth projective homogeneous variety G/P admitting an action of a maximal torus T of G, where G is a semisimple algebraic group over ℂ and P is a parabolic subgroup. The Schubert varieties form an interesting family of T-invariant subvarieties of the partial flag varieties. In this talk, we consider toric Schubert varieties in G/P with respect to the action of a quotient of the torus T. This talk is based on an ongoing project with Eunjeong Lee.
Complex cobordism of permutohedral varieties and Catalan numbers
In topology, cobordism is a fundamental concept used to classify manifolds. However, explicitly computing the complex cobordism class for a specific manifold is usually difficult. In this talk, I will talk about our recent computation for the permutohedral variety. This is joint work with Mikiya Masuda. By using Milnor's s-numbers, we explicitly calculated its coefficients. As a result, we found that the Catalan numbers appear in the coefficients. I will briefly introduce the basic idea of cobordism. Through our concrete computation, I would like to show how a geometric problem connects to combinatorics.
Weyl group representations on the cohomology of regular semisimple Hessenberg varieties
Let G be a connected reductive algebraic group and let B be a Borel subgroup of G. Hessenberg varieties are subvarieties of the flag variety G/B that provide a fruitful connection between geometry, representation theory, and combinatorics. In particular, the Weyl group W of G acts on the cohomology of a regular semisimple Hessenberg variety. In this talk, we study the Weyl group representations arising from the cohomology of regular semisimple Hessenberg varieties, using recent results of Brosnan, Hong, and D. Lee. This talk is based on an ongoing project with Patrick Brosnan, Jaehyun Hong, and Donggun Lee.
Permutation module decompositions for Hessenberg varieties associated with lollipops
In this talk, we construct a permutation module decomposition of the cohomology groups of regular semisimple Hessenberg varieties associated with lollipop graphs. This is achieved by proving the Cho–Hong–Lee conjecture for the class of lollipop graphs.This is a joint work with Soojin Cho.