Jaigyoung Choe

- Title: Some minimal submanifolds generalizing the Clifford torus

- Abstract: The Clifford torus is the simplest nontotally geodesic minimal surface in S^3. It is a product surface, it is helicoidal, and it is a solution obtained by separation of variables. We will see that there are more minimal submaniolfds with these properties in S^n.

Seungsu Hwang

- Title: The total scalar curvature and Bach tensor

- Abstract: In this talk, we discuss the behavior of the critical metric of the total scalar curvature on the space of constant scalar curvature under conditions which generalize Bach flatness of the metric. 

SunSook Jin

- Title: Riemann's minimal example

- Abstract: In this talk, we exam Riemann's original proof of the classification of minimal surfaces foliated by circles and lines in parallel planes and presents recent studies related by Riemann's minimal examples.

Hyunsuk Kang

- Title: Pinching estimate of curvature flows

- Abstract: We consider the curvature flows with concave and inverse concave speed in euclidean space.  With further restriction on the speed, one obtains a pinching estimate which is crucial to obtain convergence of the flow.  The main tool here is the Hamilton's maximum principle in matrix form.  We discuss the conditions to the speed and, if time permits, the extension of the known results to the anisotropic curvature flow.  This is a joint work with Ki-Ahm Lee.  

Daehwan Kim

- Title: Solitons for the mean curvature flow

- Abstract: The mean curvature flow (MCF) is the negative gradient flow of the area functional. MCF arises from the study of crystal growth, grain growth, image processing and other scientific fields. It is well known that any closed (hyper)surface flows in the direction of steepest descent for the area and occurs singularities in finite time under MCF. There are two types of singularities which are type 1 and type 2 represented by the self-similar solution and translating soliton, respectively. We construct new translating solitons invariant under the helicoidal motions, which seem to be single periodic translating solitons along the translating direction, by making analysis of an associated dynamical system. 

Jongsu Kim

- Title: A classification of 4-d gradient Ricci solitons with harmonic Weyl curvature 

- Abstract: I am going to present a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonic Weyl curvature. Roughly speaking, I prove that the soliton metric g  is locally isometric to  one of the following four types: an Einstein metric, the product R^2 X N of the Euclidean metric R^2 and a 2-d Riemannian manifold N of constant curvature, a certain singular metric and a locally conformally flat metric.

Seongtag Kim

- Title: A study of conformal scalar carvature rigidity

- Abstract: Let  $(M, bar g)$ be an $n$-dimensional complete Riemannian manifold. In this talk, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $Omega$ with $partial Omega$,  can one find a conformal metric $g$ whose scalar curvature $R[g]ge R[bar g]$ on $Omega$ and the mean curvature $H[g] ge H[ bar g]$ on $partial Omega$  with  $bar g = g$ on $partial Omega$? We review the developments of this problem and present a rigidity result showing that   $bar g = g$  on some smooth domains in a general Riemannian manifold using a conformal invariant.

Young Wook Kim

- Title: Machine learning and riemannian geometry

- Abstract: It has been long since differential geometry played major role in the theory of artificial intelligence (AI). Some geometric information theorists like Shun-ichi Amari had opened a new field of statistics using connection theories. But recently, from the opposite viewpoint, a new trend of machine learning is developed in engineering school and especially the deep learning method took the center stage of AI utilizing all possible mathematical theories. In this survey talk we will feel what deep learning is doing by looking at the software named Tensor Flow and try to see that some concepts of riemannian geometry finds their way into new neural network (NN) methods like GAN (Generative Adversarial Network).

Sung-Eun Koh

- Title: 3차원 Heisenberg Group에 들어있는 triply periodic compact minimal surface 만들기

- Abstract: 3차원 Heisenberg Group에는 직선으로 보이는 특별한 측지선들이 있는데, 더욱이 이 측지선에 대한 대칭이동은 거리보존 사상이 된다. 이 측지선과 이 측지선에 대한 대칭이동을 이용하여 triply periodic compact minimal surface를 만드는 간단한 방법 하나를 설명한다.

이 방법은 고려대의 김영욱 교수, 양성덕 교수, 이형용 박사, 중앙대의 신해용 교수와 함께 고안해낸 것이다.

Eunjoo Lee

-Title: New construction techniques for minimal surfaces

-Abstract: While the study on minimal surfaces has been conducted for more than 250 years, little is known about minimal submanifolds in higher dimensional Euclidean space. Not many examples are known explicitly and no general technique was developed for solving the nonlinear minimal surface equations. In this talk, we investigate J. Hoppe and V. Tkachev's recent paper on the construction techniques for minimal surfaces. It will be pointed out that their method generates new minimal surfaces from known ones despite the non-linearity of the underlying equations. 

Jihyeon Lee

- Title: The Common Constant Mean Curavature Surfaces both in R3 and H3

- Abstract: A surface S in the upper halfspace of R3 may be regarded as the surface S in R3 endowed with the Euclidean metric as well as the surface S in H3 induced with the upper halfspace metric. Since constant mean curvature surfaces including minimal are of interest, I proved that the common con-stant mean curvature surfaces in both R3 and H3 are (a piece of) planes and (a piece of) spheres. To prove the theorem, I referred [1], which contains the relation of mean curvatures of a surface in Rn+1 and Hn+1.

Xin Nie

- Title: The coupled vortex equation and geometry

- Abstract: Given a Riemann surface equipped with a holomorphic quadratic or cubic differential $U$, a conformal Riemannian metric $g$ is said to satisfy the coupled vortex equation if $k=-1+|U|^2$, where $k$ is the curvature of $g$ and $|U|$ is the norm of $U$ with respect to $g$. We will first explain how this equation arises in geometry: when $U$ is a quadratic differential, this is essentially the Gauss-Codazzi equation for certain surfaces in 3-manifolds, namely, minimal surfaces in $mathbb{H}^2timesmathbb{R}$, maximal surfaces in AdS3, CMC surfaces in the Minkowski 3-space, etc.; when $U$ is a cubic differential, the equation is the structural equation for hyperbolic affine spheres. We then show some recent asymptotic results on this equation and their geometric significance.

Juncheol Pyo

- Title: Rigidity results of capillary surfaces

- Abstract: We introduce some recent progress on capillary surfaces to geometric domains, in particular the rigidity of stable capillary surfaces in a ball proved by Wang and Xia. We prove that the rigidity of free boundary surface in a wedge in a space form.  And we give the rigidity results of capillary surfaces to a domain which is bounded by two concentric spheres proved with Sung-ho Park.

Leobardo Rosales

- Title: Generalizing Hopf's boundary point lemma}

- Abstract: We give a Hopf boundary point lemma for weak solutions of linear divergence form uniformly elliptic equations, with Holder continuous top-order coefficients and lower-order coefficients in a Morrey space. We apply this to show a regularity of the boundary result for co-dimension one area minimizing currents with $C^{1,alpha}$ tangentially immersed boundary, with boundary having Lipschitz co-oriented mean curvature.

Keomkyo Seo

- Title: Necessary conditions for minimal submanifolds to be connected

- Abstract: In general, a connected solution to the Plateau problem does not exist. We give a quantitative description of necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L^2 norm of the mean curvature vector in a Riemannian manifold.

Heayong Shin

- Title: Schwarz D-surfaces in the Heigenberg group

- Abstract:  There are many examples of periodic minimal surfaces in Euclidean space. Some of their constructions can be successfully applied to the homogeneous spaces. As an example, we will consider the construction of Schwarz D-Surface in Heigenberg group.

Chung-Jun Tsai

- Title: Noncollapsing of curve-shortening flow in surfaces

- Abstract: Grayson's original work on the curve shortening flow on the plane has been refined multiple times, especially by considering the ratio between the extrinsic and intrinsic distance.  Recently, Nick Edelen showed the noncollapsing of curve-shortening flow in general surfaces.  We will first explain some background of curve shortening flow, and then the work of Edelen.