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서검교 (숙명여자대학교) / 표준철 (부산대학교)

 

극소곡면론 소개

 

유클리드 공간에 놓여있는 곡면들 중 평균곡률이 영인 곡면을 극소곡면이라고 한다. 공간에 놓인 Jordan 곡선에 대해 극소곡면(disk-type)이 항상 존재하고, catenoid, helicoid 등 경계가 없는 완비된 극소곡면들도 존재한다. 유클리드 공간에 풍부한 예들이 있고, 주어진 경계에 대해 넓이가 최소가 된다는 성질로 부터 흥미로운 성질들이 유도된다. 수학의 여러 분야의 이론을 통하여 극소곡면의 구성방법과 중요한 성질을 소개하고자 한다.

 

목차

1. Definition of minimal surfaces, 1st variational formula - 서검교

2. Examples and finding properties of minimal surfaces - 표준철

3. minimal surfaces theory based on PDEs (maximum principle, convex hull property, harmonicity of the coordinate functions, Bernstein theorem)- 서검교

4. minimal surface theory based on complex analysis (Enneper-Weierstrass representation theorem, reflection principles) - 표준철

5. 2nd variational formula and stability- 서검교

6. minimal surfaces in the 3-dim sphere - 표준철

 

 

 

이재혁 (이화여자대학교)

 

삼차원 공간의 곡면에 대한 모스함수의 풍부성

 

모스 함수의 중요성은 기하학과 위상수학의 다양한 분야에서 나타난다. 우리는 모스함수의 기본적인 성질을 소개하고 다양체의 위상에 대한 활용을 살펴본다. 그리고 유클리드 공간의 부분다양체에 대한 모스함수의 풍부성에 대한 결론을 삼차원 곡면에 관한 미분기하학에서 사용하는 용어로 소개한다. 이 강연은 곡선과 곡면 관한 미분기하학과 Milnor의 Morse theory의 제1장을 배경으로 한다.

 

 

 

김영주 (건국대학교)

 

쌍곡기하 산책

 

본 강연에서는 쌍곡기하공간의 정의와 모델, 등장사상 등의 기초적인 정의에서 시작하여 쌍곡 구조를 갖는 다양체가 갖는 특징을 살펴본다.

 

 

 

박경동 (IBS-CGP)

 

Holonomy groups and hyperkaehler manifolds in homogeneous varieties

 

The holonomy group of the Levi-Civita connection on a Riemannian manifold is a Lie group given by parallel transport maps along loops based at a fixed point, and is a global invariant which measures constant tensors on the manifold. Interesting in various aspects are Riemannian manifolds with special holonomy groups, which include Calabi-Yau, hyperkaehler, and quaternionic Kaehler manifolds. However, it is extremely hard to construct manifolds with a specific holonomy. For instance, there are no known examples of compact quaternionic Kaehler manifold that are not symmetric spaces. I will explain Berger's classification of holonomy groups and explore attractive problems involved in special holonomy groups. The latter half of my talk will be devoted to hyperkaehler manifolds given by zero loci of global sections of equivariant vector bundles on homogeneous varieties.

 

 

 

홍한솔 (연세대학교)

 

Toric mirror symmetry and beyond

 

Mirror symmetry of a toric manifold is now a well-studied subject, in which one can see explicit constructions of the mirror space and mirror functors. I will give a short survey on the toric mirror symmetry, mainly using simple examples, and discuss more recent progresses if time permits.

 

 

 

김현규 (이화여자대학교)

 

Goldman symplectic structure on space of surface group representations, and skein algebra

 

The moduli space of group homomorphisms from the fundamental group of a surface into a Lie group G appears in many areas of mathematics and physics, like Teichmuller theory and gauge theory. In 1980's Goldman constructed a natural symplectic structure on this space, which I will review in this talk. On the other hand, I will review the skein algebra of a surface, which appeared as an approach to link invariants in 3d spaces. I will explain how Turaev showed that the skein algebra can be viewed as a quantization of the above symplectic moduli space, in case when G is SL(2,C). If time allows, I will discuss recent research problems.

 

 

 

안병희 (IBS-CGP)

 

Legendrians and constructible sheaves

 

This series of two lectures will introduce the microlocal study of constructible sheaves and their interactions with Legendrian graphs. More precisely, we will briefly review microlocal sheaf theory firstly introduced by Kashiwara and Schapira in 1980's, and define an invariant for Legendrian graphs by using the microlocal sheaf theory which is related with several questions in symplectic geometry.

 

 

 

최학호 (고등과학원)

 

On symplectic fillings of small Seifert 3-manifolds

 

One of the fundamental problems in symplectic 4-manifold topology is in classifying symplectic fillings of certain 3-manifolds equipped with a natural contact structure. If we get the classification result, then it is natural to ask that Is there any surgery description of those fillings.
In this talk, we discuss classification of minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we demonstrate that every minimal symplectic filling of small Seifert 3-manifolds satisfying the conditions can be obtained by a sequence of rational blowdowns from the minimal resolution. This is joint work with Jongil Park.

 

 

 

한지영 (서울대학교)

 

Homogeneous dynamics and application to the lattice counting problem

 

Homogeneous dynamics is to study dynamical properties-e.g. ergodicity, equidistribution property-of orbits of a flow defined on the quotient space of , where  is a Lie gorup and  is a discrete subgroup, for which  is finite. The most famous example is , which can be identified with the space of unit-covolume lattices of . Using this relation, we can apply homogeneous dynamics to a problems in the geometry of numbers.

 

 

 

서동휘 (KAIST)

 

Shape optimization problems for Steklov eigenvalue

 

1954년에 Weinstock은 주어진 둘레를 가진 planar simply connected set중 first Steklov eigenvalue가 최대가 되는 경우는 disk임을 보였다. 이 후에 관련 결과들이 다양한 기하학적 방법론으로 증명되었다. 이 강연에서는 최근 연구되고 있는 반지름이 주어진 두 개의 ball로 bound되는 annular domain에서의 first eigenvalue에 대한 shape optimization problem들을 소개하고 이 결과들이 two-point homogeneous space로까지 확장될 수 있음을 보이려고 한다.