부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

• 박경동 (경상국립대학교) - Lie groups, Lie algebras, and their representations

• 이재혁 (이화여자대학교) - Geometry with Calibrations

• 조윤형 (성균관대학교) - Invitation to toric geometry

Title. Kodaira-Spencer maps for punctured Riemann surfaces

Abstract. Mirror symmetry relates two seemingly apart areas which are symplectic geometry and algebraic geometry. There are many versions of mirror symmetry statements, and we will focus on the statement about the comparison of quantum/symplectic cohomology and Jacobian ring. We will briefly review the preliminary definitions at first, and we will see some new features if we consider group actions in addition. More specifically, for the 3-punctured sphere we will construct a version of equivariant symplectic cohomology and compare it to the orbifold Jacobian ring of mirror potential function. (This talk is based on the joint work with H. Hong and H. Jin.)

Title. On presentations of immersed surface-links

Abstract. An immersed surface-link is a closed surface immersed in $\Bbb R^4$ such that the multiple points are transverse double points. The author introduced immersed surface-links can be presented by diagrams on the plane of 4-valent spatial graphs with markers on the vertices, called marked graph diagrams. Jabłonowski defined singular marked graph diagrams, which are diagrams on the plane of 4-valent spatial graphs with marked vertices and singular vertices. In this talk, we will compare two methods of presenting immersed surface-links. Also, we introduce how to convert marked graph diagrams to singular marked graph diagrams.

Title. Holomorphic ball bundles over complex manifolds

Abstract. A holomorphic ball bundle over a compact manifold is a locally trivial holomorphic frame bundle with the complex unit ball as its frame. This bundle can be regarded as a relatively compact open set in the associated bundle whose fiber is the complex projective space. Thus, we can investigate the Levi problem--including the Steiness of such bundles--i.e., determining how many (suitable) holomorphic functions exist. It truns out that the Levi geometry such a bundle depends on Kählerness of the base manifold or a given representation of the holomorphic automorphisms on the universal cover of the bundle onto the complex unit ball. 

In this context, we will explore several relations between the theory of holomorphic functions and complex analytic geometry of such a bundle. This presentation is based on joint work with Aeryeong Seo of Kyungpook National University.

Title. Mirror Symmetry and Blowups

Abstract. Through the framework of tropical geometry, we analyze the critical behavior of the Landau-Ginzburg mirror of toric/non-toric blowups of possibly non-Fano toric surfaces. After a brief review on the mirror construction for log Calabi-Yau surfaces, we introduce a method for identifying the precise location of critical points of the superpotential (or equivalently, non-displaceable fibers). We further show their non-degeneracy for generic parameters, proving closed and open string mirror symmetry.

Title. On the first Steklov-Dirichlet eigenvalue for eccentric annuli

Abstract. The Steklov eigenvalue problem is an eigenvalue problem for an operator which is defined in the boundary of a domain. Since the operator is nonlocal, the eigenvalues depend on both the geometries of the interior and the boundary of the domain. In this talk, we consider the Steklov-Dirichlet eigenvalue problem in eccentric annuli and related problems. We obtain a lower bound of the first Steklov-Dirichlet eigenvalues of the eccentric annuli by analyzing the first eigenvalues if the distance between the boundary components are sufficiently close. This is based on joint work with Jiho Hong and Mikyoung Lim.

Title. Complete minimal surfaces of finite topology in the doubled Schwarzschild 3-manifold

Abstract. The Schwarzschild manifold has been a fundamental example of asymptotically flat space in mathematical general relativity. However, despite its asymptotic closeness to the Euclidean metric, examples of included minimal surfaces are not well-known. In this talk, we will discuss the differences between the Euclidean metric and the Schwarzschild metric in the context of minimal surfaces and address newly discovered minimal surfaces in the doubled Schwarzschild 3-manifold. This talk is based on joint work with Jaigyoung Choe and Eungbeom Yeon.

Title: Structural stability of group action
Abstract: We review the history and recent results in the study of structural stability of group action. Then we introduce a new structurally stable notion of meandering hyperbolicity for group actions and show that this generalization is substantial enough to encompass actions of uniform lattices in semisimple Lie groups on flag manifolds. This is a joint work with Misha Kapovich and Jaejeong Lee.

Title: Projective manifolds with big tangent bundles

Abstract: After Mori's solution to Hartshorne's conjecture on ample tangent bundles, there are similar questions to characterize a projective manifold with certain positivity of its tangent bundle, including Campana-Peternell's conjecture on nef tangent bundles. In this talk, I will review recent progress on the question about big tangent bundles, and introduce some criteria and examples. This talk is based on joint work with Hosung Kim and Yongnam Lee.