Activities, Summer 2016

A C T I V I T I E S , S U M M E R 2 0 1 6

DATE	SPEAKER	TITLE (click the title to see the abstract)
July 4 - 12	Mu-Tao Wang Columbia University	NCTS Summer Course on Aspects of Geometric Analysis Abstract: We plan to run this summer course from July 4 - July 12. The schedule will 10 - 12 am and 2- 3 pm. This summer course will consist of two parts. In the first part of this course, we focus on three geometric PDE's: the mean curvature flow, the isometric embedding equation, and the constraint equations (submanifold geometry). The topics are chosen so students who attend this summer course will be able to understand the basic materials covered in the research talks later on in follow-up activities. The follow-up activities will consist of student seminar on "stability of Schwarzschild spacetime", a special activity on "the isometric embedding equation" and several research talks by experts like Pengzi Miao, Lan-Hsuan Huang, Damin Wu and Richard Schoen. In order to make it accessible to undergraduate students, the first week will be dedicated to basic techniques which may include: Riemannian geometry: Riemannian curvature tensor, orthonormal frames, Ricci tensors and Einstein equations. This should include a lot of exercise problems related to the calculations of spherical symmetry and warped product. Geometric PDE: The introduction of the above three PDE's. Basic methods that include the implicit function theorem, the method of continuity, the Picard method, and the maximum principle. Submanifold geometry: Gauss--Codazzi equations, tubular neighborhood theorem in normal geodesic coordinates, second variation formula, the constraint equation and the static equation. The second part of the course will focus on "the geometry of black holes" in which we will discuss the basic geometry of black holes and their perturbations, with emphasis on the Schwarzschild spacetime.
Monday, July 4 10 - 12pm Tuesday, July 5 10 - 11am	Mu-Tao Wang Columbia University	Crash course on Riemannian geometry and geometric analysis
Tuesday, July 5 11 - 12pm Wednesday, July 6 10 - 11am	Ye-Kai Wang Michigan State University	The isometric embedding theorem of Weyl and Nirenberg
Thursday, July 7 10 - 12pm	Mao-Pei Tsui National Taiwan University	Introduction to Ricci flow, and Ricci flow on surfaces
Monday, July 11 10 - 12pm	Mu-Tao Wang Columbia University	Introduction to the Schwarzschild spacetime, and the Birkhoff uniqueness theorem
Tuesday, July 12 9:30 - 11am	Chung-Jun Tsai National Taiwan University	Poincaré uniformization theorem: an elliptic PDE approach
Tuesday, July 12 2 - 4pm	Mu-Tao Wang Columbia University	More about the Schwarzschild spacetime: global structure, Kruskal--Szekeres extension and Penrose diagram

NCTS webpage: link
organized by Mao-Pei Tsui (NTU)

Tuesday, August 2
10 - 12pm

Simon Blatt
Universität Salzburg

Introduction to Geometric Knot Theory

NCTS webpage: link
organized by Chun-Chi Lin (NTNU)

Wednesday, August 3
10 - 12pm

Simon Blatt
Universität Salzburg

Fractional Sobolev spaces and the Fractional Laplacian

NCTS webpage: link
organized by Chun-Chi Lin (NTNU)

Thursday, August 4 10 - 12pm	Simon Blatt Universität Salzburg	Regularity of Critical Points
Friday, August 5 10 - 12pm	Simon Blatt Universität Salzburg	The Heat Flow

NCTS webpage: link
organized by Chun-Chi Lin (NTNU)

Last modified: August 2, 2016.

Up to the homepage of Chung-Jun Tsai

Friday, July 15 10 - 11am	Ye-Kai Wang Michigan State University	Isometric embedding into 3-dimensional hyperbolic space Abstract: We generalize Weyl and Nirenberg's work on isometric embedding. The main result says that given a metric g on a topogical sphere, if the Gauss curvature of g is greater than -1, then there exists an isometric embedding into the 3-dimensional hyperbolic space. Moreover, the embedding is unique up to isometries in hyperbolic space (due to Guan--Shen). This is a joint work with Chen-Yun Lin.
11:15 - 12:15pm	Zhizhang Wang Fudan University	The Weyl problem in warped product space (I) and (II) Abstract: In these two talks, we report the progress of the Weyl problem in warped product space. We will discuss the openness, non-rigidity and some applications. These results together with the a priori estimates obtained by Lu imply some existence results. Meanwhile we reprove the infinitesimal rigidity in the space forms. This is a joint work with Chunhe Li.
2 - 3pm	Zhizhang Wang Fudan University
3:30 - 4:30pm	Mu-Tao Wang Columbia University	Quasi-local mass and isometric embedding

Thursday, July 21 10 - 11am	Lan-Hsuan Huang University of Connecticut	Constraint manifolds with the dominant energy condition Abstract: The deformation results to the strict dominant energy condition are important analytic tools to study the space of initial data sets for the Einstein field equations. For non-vacuum initial data sets, there is a serious technical detail in deforming to the strict inequality from a weak inequality. In the joint work with Justin Corvino, we introduce a new modified Einstein constraint operator. By establishing local surjectivity of the operator, we can promote the dominant energy condition by compactly supported deformations and obtain new gluing results.
11:10 - 12:10pm	Damin Wu University of Connecticut	Holomorphic curvature and canonical metric Abstract: I will talk about a construction of Kähler--Einstein metric under the assumption of negative holomorphic sectional curvature, based on a joint work with Prof. Yau.

Monday, July 25 10 - 11am	Ye-Kai Wang Michigan State University	The linear stability of the Schwarzschild solution to gravitational perturbations (III): following Dafermos--Holzegel--Rodnianski Abstract: In their preprint (arXiv: 1601.06467v1), Dafermos--Holzegel--Rodnianski studies the linearized gravity on Schwarzschild spacetime in detail, obtaining new insights on both geometric and analytic aspects of the linearized vacuum Einstein equations. We plan to discuss the following: [1]. the equations in double null gauge; [2]. construction of the characteristic initial data; [3]. normalized solution and the characterization of the pure gauge, and the linearized Kerr solutions; [4]. wave equation on Schwarzschild background (arXiv: 0811.0354); [5]. the wave-type equations satisfied by the curvature components.
11 - 12pm	Jordan Keller Columbia University

Tuesday, July 26 10:30 - 12pm	Jongil Park Seoul National University	Symplectic fillings and Milnor fibers of quotient surface singularities (I) Backgrounds on singularities, symplectic fillings and Milnor fibers Abstract: One of the active research areas in 4-manifold theory is to classify symplectic fillings of certain 3-manifolds equipped with a natural contact structure. Among them, people have long studied symplectic fillings of the link of a normal complex surface singularity. Note that the link of a normal complex surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure. For example, P. Lisca classified symplectic fillings of cyclic quotient singularities whose corresponding link is a lens space, and A. Nemethi and P. Popescu-Pampu identified the correspondence between the symplectic fillings in Lisca's classification and the Milnor fibers for cyclic quotient singularities. Furthermore, M. Bhupal and K. Ono tried to extend these results, so that they classified all possible symplectic fillings of quotient surface singularities. In this series of lectures, I'd like to review basics on symplectic fillings and Milnor fibers of quotient surface singularities. Then I will explain some recent joint work (with Heesang Park, Dongsoo Shin, and Giancarlo Urzua) on identifying symplectic fillings with Milnor fibers of quotient surface singularities. Furthermore, if time allows, I'll also explain how we interpret symplectic fillings of quotient surface singularities as Lefschetz fibrations.
1:30 - 3:00pm	Jongil Park Seoul National University	Symplectic fillings and Milnor fibers of quotient surface singularities (II) A review of known results on symplectic fillings of quotient surface singularities
Wednesday, July 27 10:30 - 12pm	Jongil Park Seoul National University	Symplectic fillings and Milnor fibers of quotient surface singularities (III) A resolution of singularities and Milnor fibers
1:30 - 3:00pm	Jongil Park Seoul National University	Symplectic fillings and Milnor fibers of quotient surface singularities (IV) Identifying symplectic fillings with Milnor fibers
3:30 - 4:20pm	Cheuk Yu Mak University of Minnesota	Symplectic caps and application to symplectic fillings Abstract: We illustrate how a nice symplectic cap captures properties of symplectic fillings of a contact 3-manfiold. Three kinds of symplectic caps are introduced. Divisorial caps are motivated from compactifying divisors in algebraic geometry. Uniruled caps give strong restriction to symplectic fillings for a class of contact 3-manifolds strictly larger than the planar ones. Calabi--Yau caps, in particular, can be used to derive uniform Betti numbers bounds on Stein fillings of the standard unit cotangent bundle of any hyperbolic surface. This is a joint work with Tian-Jun Li and Kouichi Yasui.
Thursday, July 28 10:30 - 12pm	Jongil Park Seoul National University	Symplectic fillings and Milnor fibers of quotient surface singularities (V) A Lefschetz fibration viewpoint on symplectic fillings of quotient surface singularities
1:30 - 2:20pm	Cheuk Yu Mak University of Minnesota	Symplectic log Calabi-Yau surfaces Abstract: Recently, some mirror symmetric phenomena are observed for log Calabi--Yau surfaces. It is hence natural to ask for a classification of log Calabi-Yau surfaces from the symplectic perspective. In a joint work with Tian-Jun Li, we show that symplectic log Calabi-Yau surfaces are uniquely determined by the homological data of the boundary divisor up to symplectic deformation equivalence.

Thursday, July 28 3 - 4pm	Gerhardt Huisken Universität Tübingen/MFO(Oberwolfach)	Meet with Professor Gerhardt Huisken and Professor Richard Schoen In this rare opportunity, Professor Huisken and Professor Schoen will share their perspectives and answer questions related to their own research experience in geometric analysis.
	Richard Schoen Stanford University/UC Irvine

Tuesday, August 2 2:30 - 3:30pm	Pengzi Miao University of Miami	On the total mean curvature of boundary of compact manifolds with nonnegative scalar curvature Abstract: In these two talks, I will discuss recent work with Christos Mantoulidis on the supremum of the total boundary mean curvature over compact 3-manifolds with nonnegative scalar curvature, with a prescribed boundary metric. When the manifold boundary consists of topological 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Wang-Yau and Shi-Tam on the quasi-local mass problem in general relativity. When the manifold boundary consists of arbitrary 2-surfaces, we establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. In the proof of the additivity property, we make use of the recent method of Mantoulidis-Schoen on computing the Bartnik mass of apparent horizons. A main feature of the results discussed here is that they do not require the Gauss curvature of the boundary to be positive. I plan to provide complete statement and background of these results in the first lecture and then provide more details of the proof in the second lecture.
4 - 5pm

NCTS Summer Course on Aspects of Geometric Analysis (at Astro-Math 440)

NCTS Special Day on Isometric Embeddings (at Astro-Math 440)

Seminar on the linear stability of the Schwarzschild metric (at Astro-Math 440)

NCTS/NTU Differential Geometry Seminar (at Astro-Math 440)

Seminar on the linear stability of the Schwarzschild metric, continued (at Astro-Math 440)

Seminar on the linear stability of the Schwarzschild metric, continued (at Astro-Math 440)

NCTS Seminar in Algebraic Geometry (at Astro-Math 201)

Mini Course on Symplectic Fillings and More (at Astro-Math 440)

Special Event of Academia Sinica/NCTS/NTU (at Astro-Math 202)

Summer Course on Fractional Sobolev Spaces in Geometric Knot Theory (at Astro-Math 440)

NCTS/NTU Differential Geometry Seminar (at Astro-Math 440)

Summer Course on Fractional Sobolev Spaces in Geometric Knot Theory (at Astro-Math 440)

Academia--NCTS Distinguished Lecture (at Astro-Math 202)

Summer Course on Fractional Sobolev Spaces in Geometric Knot Theory (at Astro-Math 440)

TIMS Seminar in Algebraic Geometry (at TIMS 103)