Registration

Rung-Tzung Huang 黃榮宗 (National Central University)

Title: Geometric quantization on complex manifolds with boundary

Abstract: In this talk I will first review the principle that “quantization commutes with reduction” ([Q, R]=0) for symplectic manifolds. Then I will discuss the [Q, R]=0 principle for CR manifolds, introduced recently by Chin-Yu Hsiao, Xiaonan Ma and George Marinescu, and for complex manifolds with boundary, jointly with Chin-Yu Hsiao, Xiaoshan Li and Guokuan Shao. An important difference between the CR manifolds and the complex manifolds with boundary setting and the symplectic setting is that the quantum spaces in the case of compact symplectic manifolds are finite dimensional, whereas the quantum spaces consisting of CR functions for the compact CR manifolds and holomorphic functions smooth up to the boundary for the compact complex manifolds with boundary are infinite dimensional. We will present that under natural pseudoconvexity assumptions that the Guillemin-Sternberg map is Fredholm at the level of Sobolev spaces of CR functions and of holomorphic functions smooth up the boundary.

Lunch break

Simon-Raphaël Fischer (National Center for Theoretical Sciences)

Title: Curved Yang-Mills-Higgs gauge theories

Abstract: We will introduce and discuss Curved Yang-Mills-Higgs gauge theories (CYMH); these are infinitesimal gauge theories originally developed by Thomas Strobl and Alexei Kotov. The main idea is to replace the structural Lie algebra with a Lie algebra bundle (Yang-Mills) or a Lie algebroid (Yang-Mills-Higgs) $E \to N$. This Lie algebroid is equipped with a connection, a metric on the base, a fibre metric on $E$, and, last but not least, a 2-form $\zeta$ on $N$ with values in $E$ which contributes to the field strength. Gauge invariance of the Yang-Mills type functional leads to four compatibility conditions to be satisfied between these structures. If the connection on $E$ is flat, the compatibilities imply that the Lie algebroid is locally what we call an action Lie algebroid, and one gets back to the standard Yang-Mills-Higgs gauge theory if additionally $\zeta \equiv 0$. Thus, the theory represents a covariantized version of gauge theory equipped with an additional 2-form $\zeta$. The 2-form $\zeta$ is needed in order to have curved connections on $E$. We will mainly focus on the simpler Yang-Mills infinitesimal gauge theory, and if time allows it, we will discuss the most recent developments of my research about integrating such gauge theories.

Tea break

Brian Harvie (National Center for Theoretical Sciences)

Title: Dynamical Stability in the Inverse Mean Curvature Flow

Abstract: My presentation will focus on the problem of dynamical stability of the round sphere in the Inverse Mean Curvature Flow (IMCF), an expanding extrinsic geometric flow with numerous applications to geometric inequalities and to general relativity.

I will first present a sufficient condition on a solution $\{N_t\}_{0\leq t< T}\subset\mathbb{R}^{n+1}$ of IMCF to guarantee global existence and homothetic convergence to a round sphere at large times that arises from studying variational weak solutions. Then, I will discuss a proof of dynamical stability of the round sphere in $\mathbb{R}^{n+1}$ under axially symmetric perturbations. This proof uses the first result on global existence and a localized version of the parabolic maximum principle. Finally, I will explain a connection between dynamical stability in IMCF and previous work on minimal surfaces by Meeks and Yau. In particular, I prove that certain global area-minimizers in $\mathbb{R}^{3}$ are embedded via embedded, global solutions of IMCF.

Forum discussion

Symposium Banquet

Cancelled. Meal box will provided for dinner.