Past seminars

December 16, 2022 (Friday) 10:30–11:30 @KIAS

 Donghoon Jang (Pusan National University) 

Title: Almost complex torus manifolds - graphs, Hirzebruch genera, and a problem of Petrie type

Abstract: Let a k-dimensional torus act on a 2n​-dimensional compact connected almost complex manifold M​​ with isolated fixed points. There is a multigraph that contains information on weights at the fixed points and isotropy submanifolds. If k=n​, that is, M​​ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. For an almost complex torus manifold, the coefficients of its Hirzebruch genus are non-zero. Using these ideas, we show that if k=n​ and there are n+1​ fixed points, many invariants of M​​ agree with those of a linear action on the complex projective space; if in addition, the action is equivariantly formal, their equivariant cohomologies also agree.

December 2, 2022 (Friday) 10:30–11:30, 13:30–14:30 @Chungbuk National University S1-1-450

 Donggun Lee (Seoul National University) 

Title:  Birational geometry of generalized Hessenberg varieties and the generalized Shareshian–Wachs conjecture

Abstract:  Hessenberg varieties are interesting objects in both algebro-geometric and combinatorial perspectives. There is a naturally defined action of the symmetric group on their cohomology. The Shareshian–Wachs conjecture connects their characters with the chromatic quasi-symmetric functions of the associated graphs, which are symmetric function refinements of the chromatic polynomials. In this talk, we study the birational geometry of Hessenberg varieties via blowups by introducing generalized Hessenberg varieties. Based on explicit geometry, we obtain two new recursive algorithms for the characters. Moreover, we provide an elementary proof of the Shareshian–Wachs conjecture and its natural generalization.

In the first half of this talk, we will explain the Shareshian–Wachs conjecture, and briefly describe the geometry behind our algorithms and our proof of the conjecture. In the second half, we will discuss the birational geometry of Hessenberg varieties in detail.

This is a joint work with Prof. Young-Hoon Kiem.

December 11 (Friday) 4:00–5:30

Seonjeong Park (KAIST)  slides

Title:  Torus orbit closures in flag varieties and Coxeter matroids  

Abstract:   A subset M​​ of a finite Coxeter group W​​ is a Coxeter matroid if it satisfies the minimality property, that is, for every u​​ in W​​, there is a unique element in v​​ in M​​ such that u-1v≤ u-1w​​ in Bruhat order. We call v​​ the u​​-minimal element. It is known that the torus fixed point set of a torus orbit closure in the flag variety G/B​​ is a Coxeter matroid. However, not every Coxeter matroid can be realized as a torus orbit closure. In this talk, we discuss some geometric and algebraic properties of the u​​-minimal element and describe the fan of a torus orbit closure in flag varieties using the minimality property. This talk is based on joint work with Eunjeong Lee and Mikiya Masuda. 

December 4 (Friday) 4:00–5:30 

Seonjeong Park (KAIST)  slides

Title:  Poincare polynomials of torus orbit closures in the flag variety 

Abstract:  The closure of a generic torus orbit in the flag variety G/B of type A is known to be a permutohedral variety and its Poincare polynomial agrees with the Eulerian polynomial. In this talk, I first introduce that the Poincare polynomial of a generic torus orbit closure in a Schubert variety in G/B can be expressed as a certain generalization of the Eulerian polynomial. After that, we discuss some open problems related to the generic torus orbit closure in a Richardson variety. This talk is based on joint work with Eunjeong Lee, Mikiya Masuda, and Jongbaek Song. 

October 30 (Friday) 10:00-11:30 AM

Eunjeong Lee (IBS-CGP) (Notes)

Title: On the combinatorial description of Białynicki-Birula decomposition of regular semisimple Hessenberg varieties

Abstract: Hessenberg varieties are subvarieties of the flag variety which provide a fruitful connection between geometry, representation theory of finite groups, and combinatorics. Indeed, the symmetric group acts on the cohomology of a regular semisimple Hessenberg variety, and studying this representation is related to the Stanley–Stembridge conjecture on chromatic symmetric functions. In this talk, we study a basis of the equivariant cohomology of a regular semisimple Hessenberg variety obtained from the Białynicki-Birula decomposition and present a combinatorial description of the support of a basis element. This leads us to compute the symmetric group actions on the cohomology. If time permits, we will study how this provides a geometric construction of permutation module decomposition for the equivariant cohomology of permutohedral varieties. This talk is based on joint work with Soojin Cho and Jaehyun Hong.

October 30 (Friday) 2:00–3:30 PM

Kyeong-Dong Park  (KIAS) 

Title:  Algebraic moment polytopes and K-stability of Fano spherical varieties 

Abstract:  For a Fano spherical variety, an algebraic moment polytope is a polytope encoding the structure of representation of G on the spaces of sections of tensor powers of the anticanonical line bundle. Many geometric properties of Fano spherical varieties can be described from the (algebraic) moment polytopes. In particular, Delcroix gave a combinatorial criterion for K-stability of a smooth Fano spherical variety in terms of the barycenter of its moment polytope with respect to the Duistermaat-Heckman measure and some data associated to the corresponding spherical homogeneous space. This criterion enables us to prove that all smooth Fano symmetric varieties with Picard number one admit Kaehler-Einstein metrics.   

October 16 (Friday) 4:00–5:30 PM

Kyeong-Dong Park  (KIAS) (Slide) 

Title:  Colored fans and classification of spherical embeddings   

Abstract:  Spherical varieties admit a simple combinatorial description in a given equivariant birational class. The normal equivariant embeddings of a given spherical homogeneous space are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. In this talk, after reviewing spherical lattices, valuation cones, color map, I explain colored cones for simple spherical embeddings and colored fans for the classification of spherical varieties in the same birational class.  

September 18 (Friday) 4:00–5:30 PM

Kyeong-Dong Park  (KIAS)  (slide) 

Title:  Introduction to spherical varieties: Characterizations of sphericality and examples   

Abstract:  For a reductive algebraic group G, a normal G-variety is called spherical if it contains an open B-orbit, where B is a fixed Borel subgroup of G. The class of spherical varieties contains several important families which were studied independently, for example, toric varieties, horospherical varieties, symmetric varieties, and group embeddings. In this talk, I give several equivalent definitions for spherical varieties and present some examples of spherical varieties.  

July 31 (Friday) 4:00–5:30 PM

Jinhyung Park (Sogang University) (Slide) 

Geometry of secondary fans and toric Mori theory

The secondary fan (or GKZ decomposition) gives a combinatorial way to describe the variation of geometric invariant theory of affine space by algebraic torus, which is the same to the birational geometry of toric variety. In this final lecture, I give the combinatorial construction of the secondary fan, which is equivalent to the Mori chamber decomposition.

July 24 (Friday) 4:00–5:30 PM

Jinhyung Park (Sogang University) (slide) 

The total coordinate ring of a toric variety and toric GIT

This is the second lecture. Every simplicial projective toric variety is a Mori dream space, and the Cox ring is the total coordinate ring. Thus every toric variety is a GIT quotient of affine space by algebraic torus. I talk about the GIT constructions of toric varieties, and introduce the secondary fans.

July 17 (Friday) 4:00–5:30 PM

Jinhyung Park (Sogang University)

Introduction to Mori dream spaces (slide) 

The goal of the three lectures is to explain the geometry of secondary fans and toric Mori theory. This is a special case of a more general theory about Mori dream spaces. We begin with the basic review of geometry of Mori dream spaces. In this talk, I explain the main results of Hu-Keel: every Mori dream space is a GIT quotient of affine variety of a Cox ring by algebraic torus, and the variation of geometric invariant theory of the affine variety by algebraic torus is the same to the birational geometry of the given Mori dream space.

July 10 (Friday) 4:00–5:30 PM

Yonghwa Cho (KIAS)

Toric surfaces and exceptional collections

We study the toric surfaces that admit Q-Gorenstein smoothing. Given such a smoothing, we may associate a full exceptional collection on the general fiber. Furthermore, it turns out that the toric surfaces can be recovered from these collections. However, there are examples of surfaces having multiple number of full exceptional collections that are "combinatorially equivalent" (meaning that they define the same toric surface), but corresponding Q-Gorenstein degenerations are not equivalent. In this talk, we will discuss our ongoing attempts understand this phenomenon via relative minimal model program.

July  3 (Friday) 9:00–10:30 AM *  slides

July 10 (Friday) 9:00–10:30 AM *  slides

July 17 (Friday) 9:00–10:30 AM *  slides

Peter Crooks (Northeastern University) 

The cohomology rings of regular Hessenberg varieties I, II, III

Hessenberg varieties form a distinguished class of closed subvarieties in the flag variety, and this class includes all Grothendieck--Springer fibres, the Peterson variety, and the projective toric variety associated to the Weyl chambers. Several important results and conjectures in algebraic combinatorics amount to statements about the cohomology rings of regular Hessenberg varieties. My first lecture will develop this idea, providing the relevant background on Hessenberg varieties in the process. The second lecture will outline some recent joint work with Ana Balibanu on Hessenberg varieties and their cohomology rings.    

June 19 (Friday) 4:00–5:30 PM, Zoom meeting    slides

Dongkwan Kim (University of Minnesota)

Robinson-Schensted correspondence for unit interval orders 

Stanley-Stembridge conjecture, currently one of the most famous conjectures in algebraic combinatorics, asks whether a certain generating function with respect to a natural unit interval order is a nonnegative linear combination of complete homogeneous symmetric functions. There are many partial progress on this conjecture, including its connection with the geometry of Hessenberg varieties. Here, instead we study its Schur positivity, which is originally proved by Haiman and Gasharov. We define an analogue of Knuth moves with respect to a natural unit interval order and study its equivalence classes in terms of D graphs introduced by Assaf. Then, we show that if the given order avoids certain two suborders then an analogue of Robinson-Schensted correspondence is well-defined, which proves that the generating function attached to each equivalence class is Schur positive. It is hoped that it proposes a new combinatorial aspect to investigate the Stanley-Stembridge conjectures and cohomology of Hessenberg varieties. This work is joint with Pavlo Pylyavskyy.

June 5 (Friday) 4:00–5:30 PM, Zoom meeting

Jongbaek Song (KIAS)

Pearson Conjecture: Toric orbifolds and Hessenberg varieties. 

A nilpotent Hessenberg variety and a semisimple Hessenberg variety with fixed Hessenberg function have a cohomological relationship, namely the cohomology ring of the former is isomorphic to the ring of invariant of the latter with respect to the symmetric group action. One extreme case of this relationship is given by Peterson varieties and permutohedral varieties, where the latter are smooth toric varieties. In this talk, we introduce a certain class of toric varieties having orbifold singularities and discuss how these objects interact with Hessenberg varieties from the cohomology ring point of view. This is a joint work with M. Masuda and T. Horiguchi. 

May 22 (Friday) 4:00–5:30 PM, Zoom meeting

Jaehyun Hong (KIAS)

Schur rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one   

Given a rational homogeneous manifold S=G/P​​ of Picard number one and a Schubert variety S0​​ of S​​, the pair (S,S0)​​ is said to be homologically rigid if any subvariety of S​​ having the same homology class as S0​​ must be a translate of S0​​ by the automorphism group of S​​. The pair (S,S0)​ is said to be Schur rigid if any subvariety of S​​ with homology class equal to a multiple of the homology class of S0​​ must be a sum of translates of S0​​. Earlier we completely determined homologically rigid pairs (S,S0)​​ in case S0​​ is homogeneous and answered the same question in smooth non-homogeneous cases. In this seminar, we consider Schur rigidity, proving that (S,S0) exhibits Schur rigidity whenever S0​​ is a non-linear smooth Schubert variety.