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2024 - Fall semester
수요일 오후, 누구나 환영！
수요일 오후, 누구나 환영！
- September 25, Special Geometry Workshop (aka Chuyu Farewell Workshop) at 과학관 255 from 2:00
Donggun Lee(IBS-CCG) 2:00-2:50
Representations on the cohomology of the moduli space of pointed rational curves.
The moduli space of pointed rational curves has a natural action of the symmetric group permuting the marked points. In this talk, we will present a combinatorial formula for the induced representation on the cohomology of the moduli space, along with a recursive algorithm derived from this formula. These results arise from wall crossing phenomena among birational models of the moduli space, governed by Hassett’s theory of weighted stable curves and Choi-Kiem’s theory of delta-stability of quasimaps.
These results enable us to study positivity and log-concavity phenomena of the representation. Specifically, we will provide partial affirmative answers to the question of whether the representation is permutation representation in each degree. We will also present explicit inductive and asymptotic formulas for the multiplicity of the trivial representation and confirm asymptotic log-concavity of this multiplicity.
If time permits, we will also discuss the multiplicities of other irreducible representations as well. This talk is based on joint works with Jinwon Choi and Young-Hoon Kiem.
Jeong-Seop Kim (KIAS) 3:10-4:00
Positivity of tangent bundle and special rational curves
In this expository talk, I will survey recent results on the positivity of tangent bundles of smooth projective varieties. I will then outline the role of rational curves in these problems, including the concept of total dual VMRTs and the uniruledness of the base loci of certain linear systems.
Chuyu Zhou (Yonsei Univ) 👋😢 4:20-5:10
K-moduli with real coefficients
In this talk, I will construct a proper good moduli space to parametrize K-semistable log Fano pairs with real coefficients (not just rational coefficients). In particular, we will construct a finite type Artin stack and show that it satisfies S-competeness and Theta-reductivity. This is based on an ongoing work with Yuchen Liu.
DINNER
- October 8 (Tuesday, 2PM), Feng Shao (Ewha Womans Univ) Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1
Lazarsfeld raised the problem that any surjective morphism f from a rational homogeneous space S of Picard number 1 to a projective manifold X must satisfy that either X is a projective space or f is an isomorphism. This problem was completely resolved in the affirmative by Hwang and Mok. In this talk, we will provide a partial extension of Lazarsfeld's problem to Fano manifolds of Picard number 1 that have big tangent bundles. As an application, we study the bigness of the tangent bundles of Fano manifolds with Picard number 1. This talk is based on my joint work with Guolei Zhong.
- October 16, Hyeonjun Park (KIAS) Cohomological Hall algebras for Calabi-Yau 3-folds at 과학관 B126 from 15:30
In this talk, I will explain how to construct cohomological Hall algebras for Calabi-Yau 3-folds, whose existence was conjectured by Kontsevich-Soibelman. This is achieved by proving the hyperbolic localization theorem for (-1)-shifted symplectic derived Artin stacks, which is a special case of the Joyce conjecture on functorial behavior of the perverse sheaves of vanishing cycles along Lagrangian correspondences. This is joint work with Tasuki Kinjo and Pavel Safronov.
- November 11 (Monday, 3PM), Naichung Conan Leung (The Chinese University of Hong Kong)
Special Lecture Series on 3d Mirror Symmetry at 과학관 255
Lecture 1 (15:00 - 16:00) What is 3d mirror symmetry?
3d mirror symmetry is a mysterious duality for certian pairs of hyperkahler manifolds, or more generally complex symplectic manifolds/stacks. In this talk, I introduce 3d mirror symmetry.
Lecture 2 (16:30 - 17:30) 3d mirror symmetry is mirror symmetry
3d mirror symmetry is a mysterious duality for certian pairs of hyperkahler manifolds, or more generally complex symplectic manifolds/stacks. In this talk, I introduce 3d mirror symmetry.
- November 20, Sukjoo Lee (IBS-CGP) Mirror P=W Phenomena at 과학관 255 from 15:30 14:30
The cohomology of a geometric space has long been a fundamental algebraic tool for studying geometric properties. For example, when X is a projective complex manifold or variety, the cohomology group H∗(X) carries a Hodge structure, a decreasing filtration that satisfies certain properties. A natural and intriguing question is what additional structures arise when X is non-compact. Two such structures are the perverse and weight filtrations. In algebraic geometry, a profound and mysterious relationship exists between these filtrations, known as the P=W phenomena, particularly in the contexts of non-abelian Hodge theory and mirror symmetry. In this talk, I will focus on the P=W phenomena in mirror symmetry and discuss an ongoing project with Andrew Harder.
- November 26 (Tuesday 2PM, 255) Special Geometry Workshop
Steven Dale Cutkosky (Univ of Missouri, Columbia)
Hema Srinivasan (Univ of Missouri,Columbia)
Steven Dale Cutkosky (Univ of Missouri) 14:00-15:00
The Proj of the section ring of a divisor above a singularity
We give a complete classification of the structure of the Proj of the section ring of a divisor over a two dimensional normal singularity, showing that it has a very simple structure, even when the section ring is not a finitely generated algebra over the singularity.
Gluing is a construction by which a semigroup A of embedding dimension p is joined with a semigroup B of embedding dimension q to produce a semigroup C of embedding dimension p + q with a strong requirement on the minimal number of generators for the three associated toric ideals. We show that any two homogeneous affine semigroups can be glued by embedding them suitably in a higher dimensional space. As a consequence, we show that the sum of their homogeneous toric ideals is again a homogeneous toric ideal, and that the minimal graded free resolution of the associated semigroup ring is the tensor product of the minimal resolutions of the two smaller parts. We also extend some of this to non homnogenous setting. We apply our results to toric ideals associated to graphs to determine when there is a splitting of a toric ideal associated to a graph or hypergraph. Specifically, suppose that a connected graph G splits in to G1 and G2 along an edge. Then the toric ideal IG splits as I_G=I_{G_1}+I_{G_2} if and only if at least one of the two graphs is bipartite. If neither G_1 nor G_2 are bipartite, there is indeed a 3-uniform hypergraph whose toric ideal splits as I_{G_1} + I_{G_2} .
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