Title & abstract
Hiraku Abe (Okayama University of Science)
Line bundles over the regular semisimple Hessenberg variety for h=(2,3,4,...,n,n)
Abstract: Hessenberg varieties are certain subvarieties of the flag variety. Among them, I will consider the toric one; the regular semisimple Hessenberg variety associated to the Hessenberg function h=(2,3,4,...,n,n). In this talk, I will "discuss" how to describe line bundles over this Hessenberg variety. This is joint work in progress with Haozhi Zeng.
Soojin Cho (Ajou University)
Permutation module decomposition of the cohomology of a regular semisimple Hessenberg variety
Abstract: We consider the module structure of the cohomology of regular semisimple Hessenberg varieties (of type A) under the symmetric group action. We use BB(Bialynicki-Birula) basis of the cohomology space to construct permutation module decomposition of the cohomology for the Hessenberg varieties associated with some lollipop graphs and the second-degree cohomology for general cases. This talk is based on the works done with Jaehyun Hong and Eunjeong Lee, and the ongoing research with JiSun Huh and Seonjeong Park.
Yunhyung Cho (Sungkyunkwan University)
Mutations of polytopes and Diophantine equations
Abstract: It is known by Hacking-Prokhorov and Akhtar-Kasprzyk that each weighted projective plane corresponds to some Diophantine equation (called Markov-type equation) whose integer solutions are in one-to-one correspondence with its Q-Gorenstein toric deformation classes. In this talk, we analyze this using theory of mutations of polytopes and Newton-Okounkov bodies and also discuss its generalization to higher dimensions. This is ongoing joint work with Yoosik Kim and Jongbaek Song.
Suyoung Choi (Ajou University)*
Toric wedge induction and its applications
Abstract: In this talk, we introduce an effective method known as toric wedge induction to prove certain properties of (real) toric manifolds that have a specific Picard number. This method is inspired by the research on classifying toric manifolds by the speaker and Hanchul Park, which uses a process called the wedge operation in combinatorics. We will share some examples where the toric wedge induction method has been used to address various unresolved issues with (real) toric manifolds that have a Picard number of 4 or less.
Jaehyun Hong (IBS CCG)
Cohomology of line bundles on Lusztig varieties
Abstract: Lusztig varieties are subvarieties of flag manifolds G/B induced from the intersections of conjugacy classes and the closures of double cosets in G. In this talk, we investigate their singularities and show the vanishing of the cohomology in positive degree of line bundles associated with regular dominant weights. This is a joint work in progress with Donggun Lee.
Tatsuya Horiguchi (Ube College)
Regular nilpotent partial Hessenberg varieties
Abstract: Hessenberg varieties are subvarieties of full flag varieties, which were introduced by De Mari, Shayman, and Proceci. Among them, the cohomology ring of regular nilpotent Hessenberg varieties are related with hyperplane arrangements. One can generalize regular nilpotent Hessenberg varieties in full flag varieties to subvarieties in partial flag varieties. We call them regular nilpotent partial Hessenberg varieties. In this talk, I will talk about a cohomology relation between regular nilpotent Hessenberg varieties and regular nilpotent partial Hessenberg varieties.
Donghoon Jang (Pusan National University)
Circle actions on four-dimensional almost complex manifolds with discrete fixed point sets
Abstract: During this talk, we consider a circle action on a 4-dimensional compact almost complex manifold with a discrete fixed point set (isolated fixed points.) We give a necessary and sufficient condition for weights at the fixed points. We do this by using a combinatorial object, a (labeled directed) multigraph. We show that there is a graph that contains information on weights at the fixed points, and there are a minimal graph and operations that attain any such graph. As an application, we provide a necessary and sufficient condition for the Chern numbers of such an action. We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a 𝕋2-action. We discuss applications to circle actions on complex/symplectic 4-manifolds.
Shintaro Kuroki (Okayama University of Science)
Equivariant cohomology of even-dimensional complex quadrics from a combinatorial point of view
Abstract: The complex 2n-dimensional complex quadrics is the manifold defined by a quadratic equation in the (2n+1)-dimensional complex projective space. The 2n-dimensional complex quadrics has the (n+1)-dimensional torus action which satisfies the GKM conditions. In this talk, I will introduce the equivariant cohomology ring of this space by generators and relations which are described by the subgraphs in the GKM graph. This talk is based on the preprint https://arxiv.org/abs/2305.11332.
Eunjeong Lee (Chungbuk National University)
Toric Schubert varieties in flag varieties
Abstract: Let G be a semisimple Lie group, T a maximal torus of G, and B a Borel subgroup of G containing T. For a parabolic subgroup P containing B, the homogeneous space G/P is a smooth projective algebraic variety, called a flag variety. If P is minimal, that is, P = B, then we call G/B a full flag variety. When P is maximal, we call G/P a Grassmannian variety. The left multiplication of T on G induces that on G/P. Schubert varieties are T-invariant subvarieties of the flag varieties. In this talk, we study toric Schubert varieties in flag varieties, especially in Grassmannians, concerning the action of the torus T. Indeed, we present an explicit description of the fan of a Gorenstein toric Schubert variety in a Grassmannian, and we prove that any Gorenstein toric Schubert variety in a Grassmannian is Fano. This talk is based on joint work with Shin-young Kim.
Seonjeong Park (Jeonju University)
Pattern avoidance and the regularity of h-Bruhat graphs
Abstract: A Hessenberg Schubert variety is the closure of a Schubert cell inside a given Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type A. For a Hessenberg Schubert variety Ωw,h, we get a h-Bruhat graph Γw,h from the natural torus action on Ωw,h induced from that on the Hessenberg variety Hess(S, h). Note that Γw,h is regular if Ωw,h is smooth. In this talk, we prove that the regularity of the h-Bruhat graph Γw,h is completely characterized by the avoidance of the patterns we found. This talk is based on a joint work with Soojin Cho and JiSun Huh.
Takashi Sato (OCAMI)
Automorphism groups of almost complex GKM manifolds
Abstract: When a torus acts on a manifold nicely, the (equivariant) cohomology ring of the manifold can be recovered from the GKM graph, where the vertex set of the GKM graph is the fixed point set and edges with labels correspond to invariant two dimensional spheres. GKM graphs are also powerful to determine the automorphism group of the manifold in the following setting. The automorphism group of an almost complex manifold is a Lie group. Let $G$ be a maximal compact Lie subgroup of the connected component of the automorphism group. When the manifold is a GKM manifold, the Weyl group of $G$ is contained in the automporphism group of the GKM graph. As an application of this result, it is shown that the ordinary torus action on a regular semisimple Hessenberg varietiy is maximal. This is a joint work with Donghoon Jang, Shintaro Kuroki, and Mikiya Masuda.
Jongbaek Song (Pusan National University)
Homotopy rigidity for quasitoric manifolds over the product of simplices
Abstract: The standard formulation of the cohomological rigidity problem in toric topology asks if the homeomorphism/diffeomorphism classes of quasitoric manifolds can be classified by their integral cohomology ring. No counterexamples have been produced so far, while many affirmative results have been published. In fact, since the cohomology ring is a homotopy invariant, it is more natural to ask if the cohomology ring classifies the homotopy equivalent classes. In this talk, we give a partial answer for this homotopy version of the cohomological rigidity problem for quasitoric manifolds over a product of simplices. This is a joint work with X. Fu, T. So and S. Theriault.
Haozhi Zeng (Huazhong University of Science and Technology)
The product of fundamental left weak composition quasisymmetric functions
Abstract: Recently, Guo-Yu-Zhao introduced the concept of left weak composition quasisymmetric functions which can be view as a generalization of quasisymmetic functions. In this talk, we will discuss the product of fundamental left weak composition quasisymmetric functions. This is joint work with Bin Zhang.