Title & Abstract
Byunghee AN (Kyungpook National University)
Quasi-isometric classification of graph 2-braid groups
For a compact (weakly) special square complex, the intersection complex which is a certain complex-of-group decomposition structure of its fundamental group is a well-defined invariant under quasi-isometry. In this talk, we use this invariant to classify quasi-isometry types of 2-braid groups of circumference 1 graphs. As an application, we also classify quasi-isometry types of 4-braid groups of trees. This is a joint work with Sangrok O.
Youngjin BAE (Incheon National University)
Lagrangian cobordism of positroid links
All positroid strata of the complex Grassmannian are realized as augmentation varieties of Legendrians called positroid links. We discuss that the partial order on strata induced by Zariski closure also has a symplectic interpretation, given by exact Lagrangian cobordism. This is joint work with Johan Asplund, Orsola Capovilla-Searle, Marco Castonovo, Caitlin Leverson, and Anjela Wu.
Yunhyung CHO (Sungkyunkwan University)
Unique toric structure on a Fano manifold
A symplectic manifold is called toric if it admits an effective Hamiltonian action of a compact torus whose dimension is half the dimension of M. Delzant proved that every compact symplectic toric manifold M is a smooth projective toric variety. In this talk, I will talk about the following conjecture posed by Dusa McDuff: If M is Fano, then M admits at most one toric structure. If time permits, I will explain some positive results related to the conjecture. This is based on joint work with Eunjeong Lee, Mikiya Masuda, and Seonjeong Park.
Jaehyun HONG (IBS-CCG)
Cohomology of line bundles on Lusztig varieties
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.
Finite sets of points in projective spaces and some applications
Assume that S is a finite set of points in n-dimensional space. In algebraic geometry, it is interesting to ask when the points of the set S impose independent linear conditions on polynomials of degree at most d. The most basic and useful is to take points in n-dimensional complex projective space and to ask about homogeneous forms of degree d instead of polynomials of degree at most d. In commutative algebra, a unique factorization domain is a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. An affine algebraic variety is called factorial if its coordinate ring is UFD. For a projective algebraic variety, one can define the factoriality in a similar way. We plan to investigate how the factoriality of singular 3-folds depends on the number of singular points.
Hyeontae JANG (Ajou University)
The classification of PL-spheres for Toric Manifolds of Picard number 4
A non-singular complete fan can be characterized by the underlying simplicial complex K of the fan and the set of primitive ray vectors of 1-dimensional cones. In particular, K is a PL-sphere, and the set of the ray vectors gives a map λ from the vertices to the set of integer vectors such that the image of each face under the map is a unimodular set. Such λ is called a characteristic map over K, and a simplicial complex admitting a characteristic map is called toric colorable. Although not every characteristic map gives a fan, classifying toric colorable simplicial complexes can be a first step for classifying toric manifolds. In this talk, we provide a combinatorial method which allows us to construct a powerful strategy for obtaining the complete list of PL-spheres which can be the orbit of a toric manifold of Picard number 4, and some applications of the result. This is joint work with Suyoung Choi and Mathieu Vall\'ee.
K-stability of ample divisors of del Pezzo surfaces
In Kahler geometry, searching for canonical metrics on a given Kahler manifold is an important problem. Proving the existence of the twisted Kahler-Einstein metric on a compact Kahler manifold is challenging, but recent progress has provided strong tools to address this problem. In this talk, we study how to prove the existence of the twisted Kahler-Einstein metrics on the del Pezzo surface of degree 5.
Shinyoung KIM (IBS-CGP)
On the problem of the characterizations of smooth projective horospherical varieties of Picard number one
We would like to explain the variety of minimal rational tangents (VMRT) of irreducible smooth Fano spherical varieties as an orbit closure for some known examples. And we talk about the characterization problem of these varieties under the condition of Picard number one using VMRTs. In particular, if X is a smooth projective horospherical variety of Picard number one, then we show that any uniruled projective manifold with Picard number one is biholomorphic to X if the variety of minimal rational tangents at a general point is projectively equivalent to that of X. This talk is mainly based on joint work with J. Hong.
Infinitely many monotone Lagrangian tori in flag manifolds
In symplectic topology, constructing monotone Lagrangian tori that are not related by symplectomorphisms is interesting. In this talk, I discuss how to use toric degenerations and cluster algebras for constructing infinitely many monotone Lagrangian tori and distinguishing them in complete flag manifolds of arbitrary type (except in some low-dimensional cases).
Kahler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII
Symmetric varieties are normal equivarient open embeddings of (algebraic) symmetric homogeneous spaces G/H, where G is a simply-connected reductive algebraic group and H is the subgroup consisting of elements fixed by an algebraic group involution of G. We can consider a symmetric homogeneous space G/H as a kind of complexification of a Riemannian symmetric space. For example, the symmetric homogeneous space SL(m, C)/S(GL(r, C) \times GL(m-r, C)) of type AIII(r, m) is an open orbit for the diagonal action of SL(m, C) on the product of complex Grassmannians Gr(r, m) and Gr(m-r, m). As we have a combinatorial criterion for K-stability of smooth Fano spherical varieties obtained by Delcroix in terms of algebraic moment polytopes, one can ask the following questions:
(1) Is the wonderful compactification of a symmetric homogeneous space K-polystable?
(2) Which of the blow-ups of wonderful compactifications of symmetric homogeneous spaces along the (unique) closed orbit are K-polystable?
In this talk, we answer the questions in the case of the wonderful compactifications of symmetric homogeneous spaces of type AIII(2, m) and their blow-ups along the closed orbit. This is based on joint work with Kyusik Hong and DongSeon Hwang.
Seonjeong PARK (Jeonju University)
Classification of toric Schubert varieties
In this talk, we consider toric Schubert varieties in flag varieties. First, we show that for type A, two toric Schubert varieties are isomorphic as varieties if and only if their integral cohomology rings are isomorphic as graded rings. Secondly, we consider general Lie types, and we characterize which toric Schubert varieties are Fano or weak Fano. This is joint work with Eunjeong Lee and Mikiya Masuda.
Integral cohomology ring of toric surfaces
It is well-known that the rational cohomology ring of a toric variety with orbifold singularities behaves similarly to the integral cohomology ring of smooth toric varieties. What has been known for the integral cohomology ring for arbitrary toric varieties is somewhat restrictive and complicated for computational purposes. In this talk, we consider toric surfaces, namely the toric varieties of complex dimension 2. The main result determines the integral cohomology ring structure of toric surfaces in terms of “bases” and “relations”, which can be easily read off from the underlying combinatorial data. This is a joint work with Xin Fu (BIMSA) and Tseleung So (Western University).
Boundedness problem of linear systems on algebraic surfaces.
We introduce Seshadri constant which measures positivity of line bundle of algebraic varieties. It is related with old algebro geometric problem, Nagata conjecture, SHGH conjecture and also bounded negativity. And also the constant is involved with Diophantine approximation problem in Number Theory, Kaehler or symplectic packing problem in Geometry. We suggest new point of view to estimate the constant via delta invariant method for K-stability.
A localization in the moduli of representations of the 3-Kronecker quiver with dimension vector (3,3)
The purpose of this work is to compute the intersection cohomology of the moduli M of representations of the 3-Kronecker quiver with dimension vector (3,3). It is known that M is singular. In this talk, we introduce a Kirwan’s localization which is a method to compute the intersection cohomology of a projective variety with a torus action and then apply the method to the moduli M. This is a joint work with Kiryong Chung.
Younghan YOON (Ajou University)
The cohomology rings of real permutohedral varieties
We establish explicit descriptions of the cohomology ring of real permutohedral varieties. In particular, the multiplicative structure is given in terms of alternating permutations.