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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
- Fei Han (National University of Singapore)
- Wei-ming Ni (University of Minnesota and East China Normal University)
- Xingwang Xu (National University of Singapore)
- Weiping Zhang (Nankai University)
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.
1. Workshop on Positive Curvature and Index Theory: 17 - 21 Nov 2014
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.
* 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.
Organizing Committee · Visitors and Participants · Overview · Activities · Venue
