김재현 (이화여자대학교)

A Conjecture on K-stability and Cylindricity

Jaehyun Kim

Postdoctoral researcher, Ewha Womans University

 

Abstract
There is a conjecture regarding the relationship between K-stability and cylindricity by I. Cheltsov, J. Park, Y. Prokhorov and M. Zaidenberg. They expected that Fano varieties with at worst klt singularities without any anticanonical polar cylinder are K-polystable. In this talk, we answer to the conjecture by considering certain singular del Pezzo hypersurfaces in three dimensional weighted projective spaces. This is joint work with In-Kyun Kim and Joonyeong Won.

 

 

 

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김정섭 (KIAS) 

Stability of symmetric powers of vector bundles

Jeong-Seop Kim

Project Research Fellow, Korea Institute for Advanced Study (KIAS)

 

Abstract
The symmetric powers of a μ-semistable vector bundle remain μ-semistable, although μ-stability may not be preserved. However, to confirm the μ-stability of all symmetric powers, it is known to verify up to the 6th symmetric power. In this talk, I will illustrate how this process works for rank two vector bundles over curves, and then review a general result by Balaji and Kollár. Additionally, I will introduce specific examples where alternative methods can be employed.

 

 

 

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박현준 (KIAS) 

Shifted symplectic Weil restrictions

Hyeonjun Park

June E Huh Fellow, Korea Institute for Advanced Study (KIAS)

 

Abstract
Shifted symplectic structures are generalizations of symplectic structures in the context of derived algebraic geometry. Their local structure theorem provides important applications to enumerative geometry of counting sheaves on Calabi-Yau varieties.

In this talk, we discuss the shifted symplectic geometry in the relative setting. We explain how to pushforward symplectic fibrations along base changes. Examples include symplectic quotients and symplectic zero loci. We use these symplectic pushforwards to provide local structure theorems for symplectic fibrations of derived Artin stacks.

As an application, we resolve the deformation invariance issue in Donaldson-Thomas theory of Calabi-Yau 4-folds and ensure that the reduced virtual cycles for counting surfaces detect the variational Hodge conjecture. We also conjecture the existence of refined DT4 invariants via the vanishing cycle cohomology of the Hodge loci.

 

 

 

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윤영호 (충북대학교)

Spectrum of Non-degenerate Functions with Simplicial Newton Polytopes

Youngho Yoon

Assistant Professor, Chungbuk National University

 

Abstract
The weighted spectrum is a crucial analytic invariant of singularities, encoding information about monodromy and Hodge structure on vanishing cycles. Calculations are not easy, particularly for non-isolated singularities. In collaboration with Seung-Jo Jung, In-Kyun Kim, and Morihiko Saito, we introduce a Gamma-spectrum as a first approximation and derive formulas for the weighted spectrum in the cases of 3 or 4 variables.

 

 

 

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이경석 (POSTECH) 

Floer homology of 3-manifolds via algebraic geometry

Kyoung-Seog Lee

Assistant Professor, Pohang University of Science and Technology (POSTECH)

 

Abstract
Recently there have been increasing interactions between low dimensional topology and algebraic geometry. In this talk, I will discuss some of these interactions focusing on certain Floer homology of some well-known 3-manifolds. The last part of this talk is based on joint works (some in progress) with Anatoly Libgober and Nikolai Saveliev.

 

 

 

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이대원 (이화여자대학교)

Characterizations of algebraic varieties and anticanonical minimal models

Dae-Won Lee

Research fellow at Ewha Womans University

 

Abstract
In this talk, we introduce the concept of absolute complexity and give various characterizations of algebraic varieties via absolute complexity. Additionally, we provide a criterion for the non-existence of log canonical anticanonical minimal models in terms of absolute complexity. Moreover, we present a necessary and sufficient condition for a log pair to have klt good anticanonical minimal models in terms of complement.

 

 

 

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이동건 (IBS-CCG)

Automorphisms and deformations of Hessenberg varieties

Donggun Lee

Postdoctoral Research Fellow, Center of Complex Geometry at the Institute for Basic Science (IBS-CCG)

 

Abstract
Hessenberg varieties are subvarieties in flag varieties which have interesting nontrivial symmetric group actions on their cohomology. The positivity of induced representations in the permutation module basis expansion is known to be equivalent to a long-standing conjecture proposed by Stanley-Stembridge in combinatorics.

To enhance our understanding of these representations, one might hope to identify a lift of the action on the cohomology to Hessenberg varieties themselves or to discover useful deformations of them. In this talk, we discuss automorphisms and deformations of Hessenberg varieties when they are hypersurfaces in flag varieties. Especially in type A, we provide a complete classification along with an interpretation in terms of moduli of pointed rational curves. This is a joint work in progress with P. Brosnan, L. Escobar, J. Hong, E. Lee, A. Mellit and E. Sommers.
 

 

 

 

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전재관 (IBS-CGP)

Deformations of rational surface singularities

Jaekwan Jeon

Research Fellow, Center for Geometry and Physics, Institute for Basic Science (IBS)

 

Abstract
In general, an isolated singularity has the semi-universal deformation, from which any deformation is induced by a base change. Therefore, we want to study the semi-universal base space, such as its irreducible components. But defining equation approach is very complicated. 
A convenient approach is the Koll'{a}r conjecture. J'{a}nos Koll'{a}r conjectured that every irreducible component of the deformation space of a rational surface singularity can be induced from a certain partial resolution of the singularity. By the partial resolution, we can calculate some invariants. But the conjecture is not much advanced and only some results are known.
In this talk I will introduce the conjecture, known result, recent progress and future work.

 

 

 

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정재우 (IBS-CCG)

Some Invariants of Sums of Squares Cones on Varieties

Jaewoo Jung

Postdoctoral Research Fellow, Center of Complex Geometry at the Institute for Basic Science (IBS-CCG)

 

Abstract
Exploring the structure of the sums of squares cone (SoS cone) and comparing it with the cone of positive semidefinite polynomials (PSD cone) on varieties is a crucial and captivating challenge in real algebraic geometry. Since Hilbert's groundbreaking result in 1888, which categorized polynomials allowing the representation of positive semidefinite polynomials as sums of squares, contemporary mathematicians such as Blekherman, Sinn, Smith, and Velasco have achieved a recent breakthrough. They broadened the investigation from polynomials over real numbers to forms on varieties with dense real loci. An important outcome of their works is determining the class of varieties on which any nonnegative quadratic polynomials can be expressed as a sum of squares.

In this talk, we introduce some semi-algebraic invariants on the SoS cones over varieties and discuss algebraic invariants that bound these semi-algebraic measures. The talk includes a collaborative result with Grigoriy Blekherman and Justin Chen.

 

 

 

 

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조창연 (서강대학교)

Twisted equivalences in spectral algebraic geometry

Chang-Yeon Chough

Assistant Professor, Department of Mathematics, Sogang University

 

Abstract
We investigate twisted derived equivalence for schemes in the setting of spectral algebraic geometry. By introducing the notion of a twisted equivalence in the spectral setting, I'll show that a twisted equivalence for perfect spectral algebraic stacks admitting a quasi-finite presentation gives rise to an equivalence of the stacks, which compensates for the failure of twisted derived equivalences for non-affine schemes to provide an isomorphism of the schemes. If time permits, I'll also talk about a spectral analogue of Rickard's theorem. This talk will be mainly devoted to giving brief expository accounts of some background materials needed to understand the notion of twisted equivalences, and no prior knowledge on derived/spectral algebraic geometry is required.