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김승혁 한양대학교 (1, 2, 3 강의)

 

라플라스 작용소, 소볼레프 부등식과 공형기하학

 

     공형기하학은 미분기하학의 세부학문 중 하나로서 주어진 리만 다양체에서 정의된 등각사상(각도를 보전하는 사상)을 연구하는 학문이다. 이 강연에서는 미분방정식론의 기본 개념인 라플라스 작용소와 소볼레프 부등식이 공형기하학에서 어떠한 의미를 갖는지를 살펴본다.

강연 1
 소볼레프 부등식, 연관 성질(대칭성 포함) 및 정리, 유클리드 공간 위에서 정의된 라플라스 작용소의 공형 불변성

강연 2
리만 다양체 위의 공형 라플라스 작용소 정의, 야마베 문제의 개략적 소개, 소볼레프 부등식과의 관련성 탐구

강연 3
야마베 문제의 해법, 양수 질량 정리, (시간이 허락하는 경우) 최근 연구 주제에 대한 간략한 소개

 

 

 

최영준 부산대학교

 

Yau’s answer for Calabi’s conjecture

 

     In the study of compact Kähler manifolds, the first Chern class is represented by the Ricci curvature of any Kähler metric. In 1954, E. Calabi raised the following question:

     In 1978, S.-T. Yau solved this conjecture by solving complex Monge-Ampère equations. A direct consequence is the existence of the Kähler-Einstein metrics on compact Kähler manifold with nonpositive first Chern class.
     In this talk, we will discuss Calabi’s conjecture and Kähler-Einstein metric. We also talk about the sketch of Yau’s solution. 

 

 

 

김준태 고등과학원

 

Real Lagrangian tori in symplectic four-manifolds

 

     The study and classification of Lagrangian submanifolds in symplectic manifolds has been a central topic of modern symplectic topology. In this talk, we address its historical results, and explore the topology of real Lagrangian tori in monotone symplectic four-manifolds. In particular, the existence and exoticness of real Lagrangian tori in del Pezzo surfaces are discussed.

 

 

강정수 서울대학교

 

Symplectic geometry and Hamiltonian dynamics

 

     The aim of this series of lectures is to give an (incomplete) overview on the interplay between symplectic geometry and Hamiltonian dynamics. Starting from very basics on symplectic geometry, I will introduce several symplectic measurements for symplectic manifolds and applications thereof. If time permits, I will also mention recent studies on the systolic-type inequality for Hamiltonian periodic solutions.

 

 

 

김호성 IBS-복소기하학 연구단

 

Cartan-Fubini type extension problem

   

     In the first talk I will define the VMRT associated to a family of minimal rational curves on a Fano manifold of Picard number 1 and describe some properties of it. I will also introduce the Cartan-Fubini type extension problem and J.-M. Hwang and N. Mok’s result on the cases with positive dimensional VMRT.

     The aim of the second talk is to prove the Cartan-Fubini type extension property for a double cover of projective space appearing in the joint work with J.-M. Hwang. To do this we used a property of the flat projective connection on a projective space and the explicit description of the projective geometry of VMRT.

 

 

 

이재혁 이화여자대학교

 

Calibrated geometry and Vector cross products

 

     Calibrated geometry is a study of submanifolds characterized by closed forms with volume related conditions. The interesting cases of these forms appear in manifolds with special holonomies such as Calabi-Yau manifolds and -manifolds, and the related calibrated geometry derives more attention thanks to mirror symmetry. In the first talk, we have a brief introduction to calibrated geometry of manifolds with special holonomies and give a uniform approach to these submanifolds along the vector cross products. In the second talk, we discuss the comparison between calibrated geometry of Calabi-Yau 3-folds and -manifolds induced by holonomy reduction and mirror symmetry. This is a joint work with N. C. Leung.

 

 

 

박웅배 University of Pittsburgh
- 박사학위 : 2021년 Michigan State University, USA, (지도교수:  Thomas H. Parker)
- 연구경력 : 2021년-현재 University of Pittsburgh

 

Harmonic maps over nodal domain using Deligne-Mumford moduli space

 

     This talk consists of three parts. In the first part I will review basic facts about harmonic maps in Riemann surfaces and consider the situation when a sequence of harmonic maps converges away from finitely many points, called bubble points. Next, I will introduce the brief notion of Deligne-Mumford moduli space. Finally, I will describe how we can capture the bubbles using Deligne-Mumford moduli space.

 

 

 

김민현 Uniersitat Bielefeld

- 박사학위 : 2020년 서울대학교 (지도교수: 이기암)

- 연구경력 : 2020년 중앙대학교

                  2021년-현재 Universität Bielefeld, Bielefeld, Germany

 

 

 

김성찬 고등과학원
- 박사학위 : 2018년 Universit¨at Augsburg, Germany (지도교수: Urs Frauenfelder)
- 연구경력 : 2019년-2020년 Université de Neuchâtel, 2021년 Seoul National University
                  2021년-현재 고등과학원

 

Transverse foliations in Hamiltonian systems

 

The modern dynamical systems theory and symplectic geometry have their origin in Poincare's work in Hamiltonian systems. Since then, both research fields have successfully developed independently. Symplectic Dynamics is a new research field trying to bring them closer together again. Namely, it studies Hamiltonian systems utilising highly integrated ideas from both research fields. In this talk, I will give a mild introduction to the field of symplectic dynamics. In particular, I would like to talk about a transverse foliation, which is a powerful tool to study the orbit structure of a given Hamiltonian system, and some consequences of its existence.