Registration

Siao-Hao Guo 郭孝豪

Title: Extension of two-dimensional mean curvature flow with free boundary

Abstract: In this talk we will consider the mean curvature flow that meets a barrier surface orthogonally. The main result is that the flow can be extended provided that its mean curvature and perimeter stay uniformly bounded.

Lunch break

Jesse Madnick

Title: Associative Submanifolds of the Berger Space

Abstract: The Berger space B^7 = SO(5)/SO(3) is a rational homology 7-sphere that admits an (essentially unique) homogeneous metric of positive sectional curvature. In fact, this metric arises naturally in the theory of special holonomy: it is induced from a homogeneous nearly-parallel G2-structure on B, and its metric cone X^8 = Cone(B) was the first explicit example of an 8-manifold with Spin(7) holonomy. In this talk (joint work with Gavin Ball), we construct the first complete examples of associative 3-folds in B, a class of 3-dimensional minimal submanifolds that are expected to play a role in G2-geometry analogous to that of pseudo-holomorphic curves in symplectic geometry. Our examples are circle bundles over genus g surfaces for every g ≥ 0, thus providing infinitely many topological types of compact associative 3-folds in B.

Tea break

I-Hsun Tsai 蔡宜洵

Title: Index theorems of transversal type

Abstract: This is based on a joint work with J.-H.Cheng (鄭日新) and C.-Y. Hsiao (蕭欽玉). I will start out by describing M. Atiyah's formulation of index theorems for transversally elliptic operators. One trouble here is that both kernel and cokernel are infinite dimensional. Then I will go to our m-th index version for CR manifolds, which does not seem to appear in Atiyah's discussions. Finally, a comparison with Atiyah's work is made. If time remains, I may mention the recent complex manifold case, done also jointly with both authors above.

Break

forum (and MOST orientation)

Symposium Banquet (The expense will be covered by NCTS)

Place: 豐盛食堂

Address: 台北市大安區麗水街1之3號

Wei-Ting Kao 高尉庭

Title: The variation formula of the p-mass on three dimensional compact CR manifolds embedded in C^{2}

Abstract: In recent years, Cheng, Malchiodi and Yang defined the pseudohermitian mass (abbreviated p-mass) for three dimensional asymptotically flat CR manifolds and proved a positive mass theorem for such CR manifolds. Similarly as in the Riemannian case, the p-mass of a certain blow-up of a compact CR 3-manifold is the first nontrivial coefficient in the expansion of the Green function of its CR conformal Laplacian. We study a family of compact CR 3-manifolds embedded in C^{2} , described by the level sets of a defining function, and derive the variation formula of the p-mass for this deformation. To obtain this result, we generalize the ambient Graham-Lee connection to the one for arbitrary contact form and obtain the conformal transformation law in this extended theory.

Tea break

Trần Thế Dũng 陳世勇

Title: Evolution of elasticae with free boundaries.

Abstract: In this report I investigate the $L^2$-flow of elastic curves in the case the angles between unit tangential and $x$-axis are fixed but non equal to zero, fixed area and free boundary conditions in 2-dimensional Euclidean spaces. Global existence of $C^{\infty}$-smooth solutions together with subconvergence to critical points is proved.

Discussion and Lunch

Students sessions and hiking

台北大縱走第七段 which is the trail in 貓空. The trail be like

Start from 動物園站 =>指南宮 => stop by an organic tea shop => 茶展中心步道 => 貓空 =>樟樹步道=> 動物園站 [map]

The lunch and discussion will take place in 紅木屋休閒茶館.

Speakers of student sessions: 高尉庭 (NTU) and 陳世勇 (NTNU), each will give a 30-minutes talk.