조윤형 교수

Motivated by the study of Nishinou-Nohara-Ueda on the Floer thoery of Gelfand-Cetlin systems over complex partial flag manifolds, we provide a complete description of the topology of Gelfand-Cetlin fibers. We prove that all fibers are \emph{smooth} isotropic submanifolds and give a complete description of the fiber to be Lagrangian in terms of combinatorics of Gelfand-Cetlin polytope. Then we study (non-)displaceability of Lagrangian fibers. After a few combinatorial and numercal tests for the d

Yunhyung Cho and

Let [Formula: see text] be a six-dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian [Formula: see text]-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then [Formula: see text] is [Formula: see text]-equivariantly symplectomorphic to some Kähler Fano manifold [Formula: see text] with a certain holomorphic [Formula: see text]-action. We also give a complete list of all such Fano manifolds and describe al

This is the second of the series of papers on the classification of six-dimensional closed monotone symplectic manifold admitting a semifree Hamiltonian [Formula: see text]-action. In [Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Int. J. Math. 6 1950032], we dealt with the case where at least one of the extremal fixed point is isolated and proved that every such manifold is Kähler Fano. In this paper, we show that if the maximal and

We prove that if there exists a $c_1$-preserving graded ring isomorphism between integral cohomology rings of two Fano Bott manifolds, then they are isomorphic as toric varieties. As a consequence, we give an affirmative answer to McDuff's question on the uniqueness of a toric structure on a Fano Bott manifold.

The unimodality conjecture posed by Tolman in [L. Jeffrey, T. Holm, Y. Karshon, E. Lerman and E. Meinrenken, Moment maps in various geometries, http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf ] states that if [Formula: see text] is a [Formula: see text]-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers [Formula: see text] is unimodal, i.e. [Formula: see text] for every [Formula:

Let $(M, ω)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\dim M^{S^1} \leq 2$. Assume that $ω$ is integral with a generalized moment map $μ$. We first prove that the action is Hamiltonian if and only if $b_2^+(M_{\red})=1$, where $M_{\red}$ is any reduced space with respect to $μ$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\red}) \geq 2$. Secondly, we focus on the case when the action is semifree and Hamilto

Let G be a semisimple algebraic group over [Formula: see text]. For a reduced word [Formula: see text] of the longest element in the Weyl group of G and a dominant integral weight [Formula: see text], one can construct the string polytope [Formula: see text], whose lattice points encode the character of the irreducible representation [Formula: see text]. The string polytope [Formula: see text] is singular in general and combinatorics of string polytopes heavily depends on the choice of [Formula:

In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact Kähler manifold $(M,\omega ,J)$ and a symplectic form $\sigma$ on $M$ which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form $\omega$. As a consequence, we can give an answer to the question posed by Khesin and McDuff as follows. According to sympl

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya–Oh–Ohta–Ono’s bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n≥3) with a monotone Kirillov–Kostant–Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S3-fiber in Fl(3) is nondisplaceable, answering a question raised by Nohara and U

The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the longest element $w_0$ in the symmetric group which encodes the character of a certain irreducible representation of a Lie group of type $A$. In this paper, we provide a recursive formula for the number of reduced words of $w_0$ such that the corresponding string polytopes are combinatorially equivalen

Using the bulk-deformation of Floer cohomology by Schubert cycles and non-Archimedean analysis of Fukaya--Oh--Ohta--Ono's bulk-deformed potential function, we prove that every complete flag manifold $\mathrm{Fl}(n)$ ($n \geq 3$) with a monotone Kirillov--Kostant--Souriau symplectic form carries a continuum of non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the Hausdorff limit. In particular, the Lagrangian $S^3$-fiber in $\mathrm{Fl}(3)$ is non-displaceable, answering