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

Title. How Floer homology sees topological entropy?

Abstract. The topological entropy of a self-diffeomorphism of a smooth manifold measures the complexity of the map in terms of volume growth. It is then a fundamental question whether the topological entropy is positive. If we consider a symplectomorphism, that is, an automorphism of a symplectic manifold, then we can introduce the Floer theoretic entropy. We will explain how the Floer theoretic entropy gives a lower bound of the topological entropy. This is joint work in progress with Myeonggi Kwon.

Title: Ways of counting geometric objects

Abstract: One natural problem, counting geometric objects has attracted interests once its connection with quantum field theories has discovered since 80-90's. Gromov-Witten theory and Donaldson-Thomas theory count complex curves in a manifold and holomorphic vector bundles on a low dimensional manifold respectively. In this talk, I would like to explain a development of the theory of a counting since Donaldson's idea in 80's and some new progresses.

Title. Deformation of Pythagorean hodograph curves using rectifying control polygon

Abstract. Pythagorean hodograph (PH) curves are a special type of polynomial curves that their speed functions are also polynomials. We introduce the concept of a rectifying control polygon for PH curves, which has (i) the end point interpolation property, (ii) the rectifying property, in the sense that the length of the polygon is the same as the arc length of the PH curve, and (iii) the same degree of freedom as the PH curve has. This is a joint work with Hwan Pyo Moon.

Title. Sasaki-Einstein 5-manifolds

Abstract. The theory of valuative criterions for K-stability of Fano varieties has been developed for the last ten years. In this talk, using this theory we prove that closed simply connected 5-manifolds $2(S^2 \times S^3) \# nM_2$ allow Sasaki-Einstein structure.

Title. Some uniqueness results for rotationally symmetric cmc-H hypersurfaces

Abstract. We introduce some uniqueness results for rotationally symmetric cmc-H hypersurfaces in a space form.

Title. On toric Schubert varieties

Abstract. Let $G$ be a simple Lie group and let $B$ be a Borel subgroup. The homogeneous space $G/B$ becomes a smooth projective variety, called the flag variety. A maximal (complex) torus $T$ acts on the flag variety and Schubert varieties are some of the most interesting $T$-invariant subvarieties of the flag variety. In this talk, we consider \emph{toric} Schubert varieties (with respect to the action of $T$) and their isomorphism classes. This talk is based on joint work with Mikiya Masuda and Seonjeong Park.

Title. On determinantal equations of higher secant varieties

Abstract. In this talk we discuss higher secant varieties and their equations that come from minors of matrices whose entries are linear forms. We begin with higher secant varieties to Veronese varieties and mention three topics: (1) determinantal presentation of higher secant varieties of minimal degree, (2) a generalization of the gonality conjecture and (3) a vector bundle method for equations of higher secant varieties. This talk is based on joint works with Prof. Sijong Kwak.