International Conference on Differential Geometry

Transcription

1 International Conference on Differential Geometry (for Beijing Section) June 2018 Capital Normal University & East China Normal University

2 International Conference on Differential Geometry In Honor of Jeff Cheeger s 75th Birthday June at Capital Normal University in Beijing June at East China Normal University in Shanghai The focus of the conference will be on recent developments in geometric partial differential equations, Riemannian geometry, spectral theory, metric geometry, and analysis on metric measure spaces. Invited Speakers: Jean-Pierre Bourguignon (CNRS-IHES) Xianzhe Dai (UC, Santa Barbara) Sylvester Eriksson-Bique (UCLA) Fuquan Fang (Capital Normal University) Wenshuai Jiang (Zhejiang University) Ilkka Holopainen (University of Helsinki) Chi Li (Purdue University) Werner Müller (University of Bonn) Aaron Naber (Northwestern University) Tony Phillips (Stony Brook University) Eero Saksman (University of Helsinki) Gang Tian (Peking University) Weiping Zhang (Nankai University) Yuguang Zhang (Bath University) Zhenlei Zhang (Capital Normal University) Conference Organizing Committee: Xianzhe Dai Fuquan Fang Xiaochun Rong Local Organizing Committee: CNU: Yanyan Niu, Ming Xu, Shicheng Xu, Zhenlei Zhang ECNU: Guofang Wei, Hongyan Zhang, Yu Zheng, Linfeng Zhou

3 Conference Placement in Beijing Beijing Kuangou Conference Center is located in Huai-Rou District, to the north of Beijing City, 60 kilometers away from Capital Normal University. (1) Registration Time: June 24 (Sunday) 2-7pm (at Kuangou Conference Center) June 25 (Monday) 8:30am (at Kuangou Conference Center) (2) Registration Location: Beijing Kuangou Conference Center No.3 Building, Lobby 北京宽沟会议中心 3 号楼, 一层大厅 (3) Conference Location: Beijing Kuangou Conference Center No.3 Building, No.1 Conference Room 北京宽沟会议中心 3 号楼, 第一会议室 (4) Meals: Beijing Kuangou Conference Center No.3 Building, Cafeteria 北京宽沟会议中心 3 号楼, 餐厅 (3) Transportation You can go by yourself or take a shuttle bus to the Beijing Kuangou Conference Center.

4 Bus Schedule in Beijing Shuttle Time Route Passengers June 24,3:00pm June 24, 7:00pm June 25, 6:40am June 25, 7:00pm from East Gate of CNU to Kuangou Conference Center from East Gate of CNU to Kuangou Conference Center from East Gate of CNU to Kuangou Conference Center from Kuangou Conference Center to East Gate of CNU For whom will live at Kuangou Conference Center For whom lives near CNU June 26, 6:40am June 26, about 8:00pm (TBA) from East Gate of CNU to Kuangou Conference Center from Kuangou Conference Center to East Gate of CNU For whom lives near CNU June 27, 9:30am from Kuangou Conference Center to Beijing South Station For whom will take train to Shanghai June 27, 10:00am from Kuangou Conference Center to East Gate of CNU For whom back to CNU or Beijing City Tips: (1) Room 508 in New 2nd Academic Building, Capital Normal University is for rest and waiting for the bus at 3:00pm and 7:00pm, June 24 (Sunday). (2) The address of Capital Normal University: 105 Xisanhuan North Road, Haidian District, Beijing 西三环北路 105 号, 首都师范大学 (3) The address of Kuangou Conference Center: No.1 Kuangou Road, Huai-Rou District, Beijing 宽沟路 1 号市政府宽沟招待所

5 Meals Service in Beijing Place: Beijing Kuangou Conference Center No.3 Building, Cafeteria 北京宽沟会议中心 3 号楼, 餐厅 Breakfast: 7:30am-9:00am Tips: Breakfast is also served for people who takes shuttle bus to Kuangou in the morning. Buffet lunch: 12:00am-1:30pm Dinner: 5:30pm-7:00pm

6 Conference Schedule MONDAY, JUNE 25 TUESDAY, JUNE 26 WEDNESDAY, JUNE 27 THURSDAY, JUNE 28 FRIDAY, JUNE 29 9:00-9:15 Openning 8:50-9:50 Gang Tian 9:15-10:15 Jean-Pierre Bourguignon 10:15-10:45 Break 10:10-11:10 10:45-11:45 Eero Saksman 9:50-10:10 Group Photo Sylvester Eriksson-Bique 9:00-12:00 Discussion/ Shuttle Bus 9:00-10:00 11:10-11:20 Break 10:30-11:30 11:20-12:20 Zhenglei Zhang Werner Mueller 9:00-10:00 Ilkka Holopainen 10:00-10:30 Break 10:00-10:30 Break Xianzhe Dai 10:30-11:30 Chi Li 12:00-14:00 Lunch 12:20-14:00 Lunch 12:00-14:00 Lunch 12:00-14:00 Lunch Weiping Wenshuai 14:00-15:00 Fuquan Fang 14:00-15:00 14:00-15:00 Zhang Jiang Travel to 15:00-15:30 Break 12:00-17:00 15:00-15:30 Break 15:00-15:30 Break Shanghai 14:00-17:30 Discussion Yuguang Tony 15:30-16:30 Aaron Naber 15:30-16:30 15:30-16:30 Zhang Phillips 16:30-17:30 Discussion 16:30-17:30 Discussion 16:30-17:30 Discussion

7 25 June, Monday Morning Section: Speaker: Jean-Pierre Bourguignon (CNRS-IHES) Title: Revisiting the Bel-Robinson Tensor Speaker: Eero Saksman (University of Helsinki) Title: On random geometry and its challenges to analysis Abstract: We consider a couple of problems belonging to random geometry, and describe some new analytical challenges they pose for analysis, especially in the context of conformal geometry and Beltrami equations. Afternoon Section: Speaker: Fuquan Fang (Capital Normal University) Title: Polar actions and non-negative curvature Abstract. This is a brief survey on the geometry and topology of non-negatively curved manifolds with polar actions. Speaker: Aaron Naber (Northwestern University) Title: Recent Progress on Manifolds with Ricci Curvature Bounds. Abstract: In this talk we survey some of the recent advances made in the understanding of manifolds with lower and two sided Ricci curvature bounds. We will discuss the structure theory of such spaces and overview the relevant ideas. We will point out potential applications of these ideas to other geometric equations.

8 Morning Section: 26 June, Tuesday Speaker: Gang Tian (Peking University) Title: Kahler-Ricci flow on Fano G-maniolds Abstract. In this talk, I will show examples of Fano manifolds on which Kahler-Ricci flow develop type II singularities. This against an old speculation on limiting behavior of Kahler-Ricci flows. Speaker: Sylvester Eriksson-Bique (University of California, Los Angeles) Title: Structure of spaces permitting differentiation of Lipschitz functions Abstract. Rough metric spaces arise in a variety of contexts, such as limits of Ricci bounded manifolds or as asymptotic boundaries of hyperbolic spaces and Gromov- Hyperbolic-groups. Differentiation provides a powerful tool to study the structure of such spaces. In the context of smooth spaces these concepts have clear meaning, but for metric spaces even the sensibility of such notions is questionable. Jeff Cheeger in a pioneering paper introduced a notion of differentiation that held for large classes of spaces. With relatively simple assumptions of doubling and a Poincare inequality, he developed a comprehensive theory of differentiation, measurable co-tangent bundles, harmonic functions and Sobolev spaces for metric spaces. However, much remained unclear about the precise nature of these assumptions, as well as if they were necessary for defining these associated structures. In this talk we will survey some of the exciting developments by the speaker and others in this field from the past few years, including partial characterizations, counter-examples and examples of exotic spaces satisfying Poincare inequalities. Speaker: Zhenglei Zhang (Capital Normal University) Title: Relative volume volume comparison of Ricci flow and its application Abstract. In the talk I will present a relative volume comparison property of Ricci flow. It is a refinement of Perelman's no local collapsing theorem of Ricci flow. One application to the convergence of Ricci flow on minimal manifolds will also be discussed. It is a joint work with professor Tian. 27 June, Wednesday Have a good trip to Shanghai!

9 Morning Section: 28 June, Thursday Speaker: Werner Mueller (University of Bonn) Title: Analytic torsion of locally symmetric spaces and torsion in the cohomology of arithmetic groups Abstract. Analytic torsion is a spectral invariant of a compact Riemannian manifold defined in terms of regularized determinants of Laplace operators on forms twisted by a flat bundle. By a theorem, which is due to Jeff Cheeger and myself, the analytic torsion equals the Reidemeister torsion. Recently, this result has found interesting applications in the study of the growth of torsion in the cohomology of cocompact arithmetic groups. Since many arithmetic groups are not cocompact, it is interesting to extend these results to the non-cocompact case. In this talk I will report on recent progress concerning this problem. Speaker: Xianzhe Dai (UC, Santa Barbara) Title: Holomorphic Torsion and Conical Singularity Abstract. Holomorphic torsion is the complex analog of the Ray-Singer analytic torsion. Unlike the analytic torsion, it has no topological interpretation. Nevertheless it still plays crucial roles in diverse fields. We study the holomorphic torsion for complex manifolds with conical singularity, showing that it is well defined, and then study its basic properties and applications. Afternoon Section: Speaker: Weiping Zhang (Nankai University) Title: Positive scalar curvature on foliations. Abstract. We present a generalization of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations. An a consequence, there is no foliation of positive leafwise scalar curvature on any torus. This later result extends the corresponding result of Schoen-Yau and Gromov-Lawson, which states that there is no metric of positive scalar curvature on any torus, to the case of foliations. Speaker: Yuguang Zhang (Bath University) Title: Equivalence of degenerations of Calabi-Yau manifolds Abstract. In this talk, we show the equivalence of degenerations of Calabi-Yau manifolds among the convergence of Ricci-flat Kahler-Einstein metrics, cohomological classes of calibration forms and the finiteness of Weil-Peterson metric.

10 Morning Section: 29 June, Friday Speaker: Ilkka Holopainen (University of Helsinki) Title: Asymptotic Dirichlet problem for the mean curvature operator in warped products Abstract Speaker: Chi Li (Purdue University) Title: On the Yau-Tian-Donaldson conjecture for singular Fano varieties Abstract. I will talk about an existence result for the Yau-Tian-Donaldson conjecture on any Q-factorial Fano variety that has a log smooth resolution of singularities such that discrepancies of all exceptional divisors are non-positive. We will show that if such a Fano variety is K-polystable, then it admits a K\ ahler-einstein metric. This extends the the previous result for Fano manifolds to this class of singular Fano varieties. The proof uses various techniques from differential complex geometry. This is a joint work with Gang Tian and Feng Wang. Afternoon Section: Speaker: Wenshuai Jiang (Zhejiang University) Title: The Singular set of Noncollapsing Ricci Limit Spaces Abstract: Let us consider (M^n_i,g_i,p_i)-> (X,d,p) in Gromov-Hausdorff sense with Vol(B_1(p_i))>v>0 and Ric>=-(n-1). It is known that X is a metric space and X has a

11 regular-singular decomposition X=R\cup S with dim S<= n-2 by Cheeger-Colding. In this talk, we will show that the singular set S of X is (n-2)-rectifiable. More generally, for 0<=k<n, we will shot that the k-stratum S^k={x\in X: no tangent cone at x splits off a R^{k+1} factor} is k-rectifiable. We will also discuss the quantitative estimate of S^k. This is joint work with Professor Jeff Cheeger and Aaron Naber. Speaker: Tony Phillips (Stony Brook University) Title: Fifty-odd years of sphere eversion Abstract. Stephen Smale proved in 1959 that any two immersions of the 2-sphere in 3-space are regularly homotopic. In particular, the sphere can be turned inside-out (everted) along a path of immersions with continuously varying tangent map. This lecture will survey the attempts, which started then and continue to this day, to make that process intelligible.