Registration

Camille Laurent-Gengoux

Title: The geometry of singular points and leaves

Abstract: I will give an overview on recent research about singular foliations, i.e. partitions of a manifold by points which reachable one from the other by following flows of a finite number of differential equations.

I will explain how "hidden" algebraic structures are attached to singular points or leaves, and how they are relevant for understanding the geometry of those. I will also explain how those are used in non-commutative geometry, and, last, present some conjectures and recent results about deformations of those.

Lunch break

Yen-Chang Huang 黃彥彰

Title: Relation between projections and surface areas in the flat pseudo-hermitian manifolds

Abstract: In the talk I will first review some geometric invariants (as shown in the title) in the Heisenberg groups regarded as the flat pseudo-hermitian manifolds. Of particular interest is the establishment of the notion of p-areas defined by Cheng-Hwang-Malchiodi-Yang and Cheng-Hwang-Yang by a natural variational approach for volumes. Recently I gave an equivalent definition of p-areas by Lie group theory and the viewpoint of Integral Geometry and showed the Cauchy’s surface area formula. The formula states that the p-area of the boundary of any domain in can be obtained by finding the average of projected p-areas along all directions on the Pansu sphere, a special rotationally symmetric surface in which is believed as the sharp case for the isoperimetric inequality. In contrast to the Cauchy’s formula in the Euclidean spaces , the new definition helps to remove the convex assumption for domains and the formula provides a support that the Pansu spheres are the better objects compatible with the geometry of rather than the standard spheres or other rotationally symmetric surfaces. The similar definition can be applied back to so that we can generalize the original Cauchy’s formula to arbitrary domain with boundary. We are also developing the formulas for the volumes of arbitrary k-submanifolds.

Tea break

Albert Wood

Title: Cohomogeneity-One Lagrangian Mean Curvature Flow

Abstract: Lagrangian mean curvature flow (LMCF) is the name given to the observation that Lagrangian submanifolds of Calabi-Yau manifolds are preserved under mean curvature flow. This observation eventually gave rise to the conjecture of Thomas-Yau, which suggests that assuming a ‘stability’ condition on the Lagrangian submanifold, the flow should converge to a unique volume-minimising representative: a special Lagrangian. Since LMCF typically forms finite-time singularities, a surgery procedure must be defined to resolve this conjecture, and an understanding of the possible singularity models is therefore a vital first step.

In this exposition of recent joint work with Jesse Madnick, we specialise to highly symmetric Lagrangians; Lagrangians in complex n-space that are cohomogeneity-one with respect to a group action respecting the Calabi-Yau structure. Such Lagrangians must be contained in an (n+1)-dimensional submanifold (a level set of the moment map μ), and taking a quotient produces a curve in a 2-manifold. LMCF may therefore be studied via a related flow of curves, which we show does not depend on the group action. By this method, we are able to classify cohomogeneity-one shrinking and expanding solitons, as well as fully classify singularities in a restricted setting.

Break

forum

Symposium Banquet (The expense will be covered by NCTS)

Place: 易牙居 (Yi Ya Ju Restaurant) which is very close to the NTU campus. website: https://www.eyaji.com.tw/

Address: 台北市羅斯福路三段286巷16號

16, Lane 286, Section 3, Roosevelt Rd, Zhongzheng District, Taipei City, 100