홍한솔 연세대학교

월1

Tropical method in enumerative geometry

Following Gromov’s seminal work, lots of symplectic invariants have been developed through the enumeration of various types of (pseudo-)holomorphic curves. Despite their powerfulness, the explicit computation of these invariants mostly remains challenging. In this lecture, I will introduce one possible approach to this computational challenge utilizing tropical geometry. Its basic idea is to degenerate holomorphic curves into piecewise linear objects thereby transforming the enumeration problem into a relatively simple combinatorics.

화2

Mirror symmetry for log Calabi-Yau surfaces

We discuss the count of holomorphic disks in log Calabi-Yau surfaces and its implications for associated questions on their mirror symmetry. Establishing the correspondence between holomorphic and tropical objects, it amounts to finding linear graphs on a certain planar diagram. Moreover, the mirror geometry can be read off from a tropical degeneration of the generating function associated with the disk counting invariants.

 

 

 

신진우 고등과학원

월2

Blow-up phenomena for the Yamabe equation

  Let be a compact Riemannian manifold of dimension . As a generalization of the uniformization theorem for surfaces, the Yamabe problem is to find a metric conformal to such that its scalar curvature  is constant. This problem can be reduced to finding a positive smooth solution of the follwoing semi-linear elliptic PDE:

(1) .

  It is well known that the PDE (1) has at least one positive solution for anychoice of . Solutions to (1) are usually not unique. Thus it is interesting question whether the set of all solutions to (1) is compact in the -topology. In this talk, we will discuss the compactness and non-compactness results of the Yamabe equation (1). If time allows, we present our results (joint work with P. T. Ho) on the non-compactness of the constant scalar curvature and constant mean curvature equation, a generalization of the Yamabe equation.

 

 

 

조다혜 연세대학교

월3

사교적인 코호몰로지에 대한 소개 (Introduction to Symplectic Cohomology)

해밀토니안 플루어 이론은 특정 사교 공간 위에 있는 끈들의 공간에 모스 이론을 적용하여 사교 공간을 이해하는 이론입니다. 사교 공간의 구조 중에서 Completion of Liouville Domain에 대해 설명해 드리고, 어떻게 Symplectic cohomology를 정의하는지에 대해 설명해 드리겠습니다. 이번 시간에는 특별히 볼록함과 최대원리가 쓰이는 곳에 대해 간략하게 소개해 드리겠습니다.

 

 

 

김준태 서강대학교

월4

Yomdin theorem and Morse homology

Given a diffeomorphism of a smooth manifold, its topological entropy is a non-negative quantity which measures dynamical complexity of the map. The classical Yomdin theorem asserts that this quantity has a lower bound determined by the induced map on homology.

We provide an overview of this story and (its connection to) Morse homology.

화1

What can Floer homology say about entropy?

In the previous talk, we focused on the classical side of topological entropy. If we look at smooth manifolds equipped with symplectic forms, i.e., symplectic manifolds, then we can say more about entropies of diffeomorphisms preserving symplectic forms. To this end, we give an overview of Floer homology and its applications to dynamical complexity.

 

 

 

이상훈 고등과학원

수1

Liouville-Type Theorems on the Hyperbolic Space

In this talk, we establish Liouville-type theorems for a one-parameter family of elliptic PDEs on the standard upper half-plane model of the hyperbolic space, under specific geometric assumptions. Our results indicate that the Euclidean half-plane is the only compactification of the hyperbolic space when the scalar curvature of the compactified metric has a designated sign.

금2

Improved energy decay estimate for Dir-stationary Q-valued functions and its applications

In this talk, we prove an improved decay estimate for the Dirichlet energy of Dir-stationary Q-valued functions. As a direct application of this estimate, we derive a Liouville-type theorem for bounded Dir-stationary Q-valued functions defined on . Additionally, in an attempt to establish the continuity of Dir-stationary Q-valued functions, we confirm that such functions exhibit the Lebesgue property at every point within their domain.

 

 

 

이상진 고등과학원

수2목2

A construction of diffeomorphic, but not symplectically

isomorphic manifolds

A smooth manifold is a topological manifold having a differential structure, and a symplectic manifold is a smooth manifold equipped with an "extra" structure, called "symplectic" structure. The following question asks the difference between smooth and symplectic manifolds: Is it possible to put two different symplectic structures on a smooth manifold? In this talk, we discuss an answer to the question.

The first talk will briefly introduce basic concepts in symplectic topology, including the notion of symplectic handle decomposition. Using the concept of symplectic handle decomposition, we will construct a pair of diffeomorphic, but not symplectically isomorphic manifolds in the second talk.

 

 

 

서동균 서울대학교

수3

Gentle introduction to word-hyperbolic groups

We define a geodesic metric space as Gromov-hyperbolic if every geodesic triangle is thin. Gromov-hyperbolic spaces have been extensively studied in geometric group theory and topological dynamics for several decades. This talk will specifically focus on the properties of isometric group actions on Gromov-hyperbolic graphs. T his talk will provide a gentle introduction to word-hyperbolic groups. Gromov originally studied word-hyperbolic groups to observe their asymptotic behavior.

목3

Groups acting on Gromov-hyperbolic spaces

We will explore various generalizations of word-hyperbolic groups. While these groups may not be word-hyperbolic themselves, their algebraic properties can be derived from well-defined isometric actions on Gromov-hyperbolic graphs. This research is a collaborative effort with Hyungryul Baik and Hyunshik Shin.

 

 

 

이태훈 고등과학원

수4목1

On Morse index of minimal surfaces

이 강연에서는 최소곡면의 인덱스에 대해 다룹니다. 최소곡면은 면적 범함수의 임계점으로, 이를 연구하는 데 있어서 이 변동의 안정성과 인덱스가 중요한 역할을 합니다. 본 강연의 중점은 최소곡면의 인덱스와 이것이 곡면의 위상적 특성과 어떠한 관련성을 가지는지 알아보는 것입니다. 시간이 된다면, 최소곡면의 인덱스가 최소곡면에서 발생하는 고대 평균 곡률 흐름(ancient mean curvature flow)의 풍부한 해들을 어떻게 형성하는지에 대해서도 살펴볼 예정입니다.

 

 

 

박지원 KAIST

목4

Monotonicity formulas in geometric analysis

Monotonicity formulas are one of the most powerful tools in many problems in geometric analysis, including the structure of manifolds with curvature bounded below and various curvature flows. In the first talk, we will introduce some of these formulas and look at how some of them can be derived from Li-Yau-Hamilton type estimates for elliptic or parabolic equations. We will also introduce a novel monotonicity formula based on this principle.

 

금1

Geometric applications of various monotonicity formulas

In the second talk, we will explain how monotonicity formulas govern the convergence of geometric objects. We will focus on two illuminating examples, namely manifolds with Ricci curvature bounds, and the mean curvature flow. For Ricci curvature, certain monotonicity formulas for the Laplacian and the p-Laplacian can be used to show geometric inequalities such as Willmore and isoperimetric inequalities and Penrose inequality. For mean curvature flow, Huisken’s monotonicity of the F-functional implies the self-similarity of backward time limits. Moreover, there are quantitative estimates associated to this F-functional. We will discuss some forward-time counterparts of these results.