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Scalar Curvature in Manifold Topology and Conformal Geometry
(1 November - 31 December 2014)

Organizing Committee · Visitors and Participants · Overview · Activities · Venue

 

 Organizing Committee

 

 

 Visitors and Participants

 

 

 Overview

 

This scalar curvature related program will concentrate on two sides: Analytical and topological.


Analytically our attention will focus on the scalar curvature related geometric differential equations. Such arising partial differential equations stimulate a lot of interest among many people.  Typical equations include elliptic and parabolic type, normally nonlinear on curved spaces. Hence curvature always plays an important role. Since the scalar curvature is just a real valued function, the conformal changes of the metrics naturally connect to scalar curvature or more generally the Q-curvature function. Recently, the parabolic type equations as well as the fully nonlinear equations also come to play, creating new possibilities for further research in this rich and interesting field.


The topological side is to consider the connections between various curvatures and index of metric related elliptic operators. The pioneering work of Lichnerowicz in the early sixties has shown that the scalar curvature is related to the index of Dirac operators on spin manifolds. The celebrated Atiyah-Singer index theorem tells us that the index of Dirac operator is a topological invariant of the manifold. This gives topological obstruction to the existence of positive scalar curvature metric on spin manifolds. Some driving forces for recent developments are the efforts of generalizing these known results about positive scalar curvature to infinite dimensional spaces and non-commutative spaces. A typical infinite dimensional space is the free loop space of a manifold. Stephan Stolz conjectured that if a string manifold admits Riemannian metric of positive Ricci curvature then the Witten genus (a modular form) of the manifold vanishes. This conjecture was heuristically argued to be true by formally viewing the Ricci curvature as scalar curvature on loop space and Witten genus as the index of Dirac operator on loop space. A typical example of non-commutative space comes from foliation. Alain Connes generalized the classical results for positive scalar curvature to foliated spaces concerning the existence of leaf-wise positive scalar curvature metrics. Recently Weiping Zhang gave a direct geometric proof of Conne’s vanishing theorem inspired by the analytic localization techniques developed by Bismut-Lebeau.


The purpose of the program is to bring together researchers working on the areas to communicate ideas and dig out the connections as well as stimulate possible research collaboration.



 Activities

 

1. Workshop on Positive Curvature and Index Theory: 17 - 21 Nov 2014

    Currently confirmed participants include:

  • Ulrich Bunke
  • Qingtao Chen
  • Xinyue Cheng
  • Anand Dessai
  • Xianzhe Dai
  • Fuquan Fang
  • Huitao Feng
  • Padram Hekmati
  • Peter Hochs
  • Wenchuan Hu
  • Ping Li
  • Joachim Michael
  • Xiaonan Ma
  • Shu Shen
  • Xiang Tang
  • Zihou Tang
  • Wilderich Tuschmann
  • Mathai Varghese
  • Bai-ling Wang
  • Hang Wang
  • Yong Wang
  • Siye Wu
  • Zhizhang Xie
  • Weiping Zhang
  • Jiuru Zhou


2. Workshop on Partial Differential Equation and its Applications: 8 - 12 Dec 2014


3. Winter School on Scalar Curvature and Related Problems: 16 - 19 Dec 2014
    There will be four mini courses.

4. Public Lecture


* Our office will be closed on Thursday, 25 Dec 2014 being Singapore public holiday.

 

Students and researchers who are interested in attending these activities are requested to complete the online registration form.

The following do not need to register:

  • Those invited to participate.

 

 Venue

 

Organizing Committee · Visitors and Participants · Overview · Activities · Venue

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