## 談話会　Young-Jun Choi「Fiberwise Ricci-flat metrics of a family of Calabi-Yau manifolds」 、Ji-Hun Yoon「Pricing vulnerable path-dependent options using integral transforms」、Juncheol Pyo「Rigidity theorems of capillary hypersurfaces 」

2017年 　 12月 19日 14時 45分 ～ 2017年 　 12月 19日 17時 50分

Young-Jun Choi 氏(Pusan National University) 、Ji-Hun Yoon 氏(Pusan National University)、Juncheol Pyo 氏(Pusan National University)

スケジュール：
14：45-15：35　Young-Jun Choi氏(Pusan National University)
15：35-16：00　teatime（場所：5階コモンスペース）
16：00-16：50　Ji-Hun Yoon氏 (Pusan National University)
16：50-17：00　teatime（場所：5階コモンスペース）
17：00-17：50　Juncheol Pyo氏 (Pusan National University)

Young-Jun Choi 氏(Pusan National University)
title:Fiberwise Ricci-flat metrics of a family of Calabi-Yau manifolds
abstract:
Let $p:X\rightarrow Y$ be a surjective holomorphic submersion between complex manifolds such that every fiber $X_y:=p^{-1}(y)$ for $y\in Y$, is a Calabi-Yau manifold, i.e., a compact K\"ahler manifold with trivial canonical line bundle. This is called a \emph{family of Calabi-Yau manifolds} or a \emph{Calabi-Yau fibration}. If $(X,\omega)$ is a K\"ahler manifold, then every fiber $X_y$ equpped has a unique Ricci-flat K\"ahler metric whose associated K\"ahler form belongs to the fixed K\"ahler class $[\omega\vert_{X_y}]$ by Calabi-Yau theorem. This family of Ricci-flat metrics induces the fiberwise Ricci-flat metric on a Calabi-Yau fibration.
In this talk, we discuss the positivity of direct images of fiberwise Ricci-flat metrics on the base $Y$. We also discuss the extension of the Weil-Petersson metric on $Y$.

Ji-Hun Yoon 氏(Pusan National University)
title:Pricing vulnerable path-dependent options using integral transforms
abstract:
In the over-the-counter (OTC) markets, the holders of many contracts are vulnerable to counterparty credit risk.
Because of this issue, vulnerable options must be considered. In addition, in a financial environment, the pricing of path-dependent options yields many interesting mathematical challenges. In this paper, we study the pricing of vulnerable path-dependent options using double Mellin transforms to investigate an explicit (closed) form pricing formula or semi-analytic formula in each path-dependent option.

Keywords: Vulnerable barrier option, Vulnerable double barrier option, Vulnerable lookback option, Method of image, Double Mellin transform

References
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Juncheol Pyo 氏(Pusan National University)
title:Rigidity theorems of capillary hypersurfaces
abstract:
In this talk, we give some rigidity theorems of capillary hypersurfaces to geometric domains. First, we prove that every compact immersed stable constant mean curvature hypersurface with free boundary in a wedge is part of a geodesic sphere centered at a point of the edge of the wedge. Second, we show that the same rigidity result holds for a compact embedded constant higher order mean curvature hypersurface with free boundary in a wedge. Finally, we give rigidity results for capillary hypersurfaces to a domain which is bounded by two concentric spheres