제 14회 고등과학원 기하학 겨울학교

(2018년 12월 18일-12월 22일)

김 성 운 (제주대학교) (화 19:00–19:50, 수 9:00–9:50)

Sullivan's structural stability and Anosov subgroups

Sullivan proved a structural stability theorem for group actions satisfying certain expansion-hyperbolicity axioms.
We add more details on his somewhat sketchy proof. After this, we show that Anosov subgroups of semi-simple Lie groups satisfy these axioms, thereby obtaining the stability of Anosov subgroups in a broader setting of differential dynamics. This is a joint work with M. Kapovich and J. Lee.

서 애 령 (고등과학원) (화 20:00–20:50, 수 10:05-10:55)

Kobayashi hyperbolicity of flag domains

In this talk, we generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow's theorem in sub-Riemannian geometry. Let be a linear semisimple Lie group. For a complex -homogeneous manifold with a -invariant holomorphic bracket generating distribution , we prove that is Kobayashi hyperbolic if and only if the universal covering of M is a canonical flag domain and the induced distribution is the superhorizontal distribution.

안 병 희 (IBS-CGP) (수 11:10–12:00, 목 10:05-10:55)

Braid groups and configuration spaces of graphs

The (classical) braid groups have many aspects in topology and geometry, group theory, cryptography and so on, and since 1920's when Artin firstly introduced braids as mathematical objects, they have been generalized in many different ways for their own purpose such as braid groups on surfaces and higher dimensional manifolds, and singular braid groups by allowing double points. Most of the studies of these generalizations have been done by using the configuration spaces -- the collection of some number of points in the given space---introduced by Fadell and Neuwirth in 1960's.
Rather recently, in the end of 20's century, Ghrist published a pioneering paper about braid groups on graphs, which are non-collision motions of points on the graphs. Typically it can describe the motions of robots in a factory, trains on rails, or any number of particles on a given trajectory in general.
In these two talks, we will focus on the homotopy invariant for braid groups on graphs. In particular, for configuration spaces of graphs, we will see chain-level module structures coming from certain stabilization processes, and define a sequence of topological invariants of graphs related with the growth of betti numbers.
If time permits, we will also discuss further algebraic structures on the chain complexes of configuration spaces of graphs.
This is a joint work with Gabriel C. Drummond-Cole (IBS-CGP) and Ben Knudsen (Harvard University).

강 현 석 (광주과학기술원) (수 19:00–19:50)

Introduction to mean curvature flow for convex hypersurfaces

We introduce the basics of mean curvature flows for convex hypersurfaces. The main breakthrough was made by Huisken using backward heat kernel to obtain monotonicity formula and also applying De Giorgi method, well known in PDE theory, which was a turning point in the study of extrinsic curvature flow. We start with definitions of extrinsic curvature and build up from evolution equations to the aimed results.

김 종 수 (서강대학교) (수 20:00–20:50)

Generalization of Einstein metrics

I will discuss various classes of Riemannian metrics which can be considered as generalizations of EInstein metrics. The common part in my lecture will be the so-called Codazzi tensors. I will talk about RIemannian metrics with Codazzi tensors, harmoinic Weyl curvature or harmonic curvature. Then I treat gradient Ricci solitons and quasi—Einstein metrics.

김 성 택 (인하대학교) (목 9:00–9:50, 11:10-12:00, 금 10:05–10:55)

Introduction to analytic methods in differential geometry

In this lectures, I will review the basic analytic methods used in differential geometry. Main goal is to provide the basic materials needed to understand the short time existence of Ricci curvature flow and mean curvature flow. I will cover the materials in heat equations and nonlinear parabolic system. Lectures will be friendly to geometry community.

고 성 은 (건국대학교) (목 20:00–20:50)

Legendre Transformation 소개

초보적인 수준에서 Legendre 변환을 소개한다.

서 검 교 (숙명여자대학교) (금 9:00–9:50, 11:10-12:00)

Properties of minimal surfaces from a variational viewpoint

We discuss basic properties of minimal surfaces using 1st and 2nd variation of area functional.

박 경 동 (IBS-CGP) (금 19:00–19:50 토 9:00–9:50)

The Cayley Grassmannian and smooth projective symmetric varieties

Four-dimensional subalgebras of the complexified algebra of octonions are parametrized by a closed subvariety of the Grassmannian , which we call the Cayley Grassmannian. It is a smooth irreducible projective variety of dimension 8, and a spherical variety with 3 orbits under the action of the complex Lie group . Since the (unique) open orbit in the Cayley Grassmannian is a symmetric homogeneous space given by the (nontrivial) involution of , it is one example of smooth projective symmetric varieties with Picard number 1. Interestingly, these symmetric varieties are closely related to isoparametric hypersurfaces in spheres having 3 or 6 principal curvatures.

이 재 혁 (이화여자대학교) (금 20:00–20:50)

Isoparametric hypersurfaces and minimal Lagrangians

Recently, the classification of isoparametric hypersurfaces in spheres has been completed. Therefrom, new research projects have been initiated from various branch of geometry. The study of minimal lagrangian submanifolds via isoparametric hypersurfaces is one of the most active projects a la mode. In this talk, we have an introduction to Isoparametric hypersurfaces and minimal Lagrangian submanfolds, and discuss the relationship between them.

김 대 환 (고등과학원) (토 10:05–10:55)

Solitons for the inverse mean curvature flow and their properties

Inverse mean curvature flow have been extensively studied as a geometric flow and for applications to prove several geometric inequalities. Analyzing special solutions of types of geometric flow is helpful to understand and apply solutions of those, and so does the inverse mean curvature flow. In this talk, we concern the homothetic and translating solitons for the inverse mean curvature flow that are self-similar solutions deformed by only homothety and translation under the flow, respectively, and introduce several examples of the solitons. To be specific, the incompleteness of the homothetic soliton for and the translating soliton, and their area growth are proved.