鉴于雷-辛格猜想/齐格-莫勒定理，伯格曼核和瑞奇流在几何，拓扑及数学物理等多个领域中的重要作用，本项目将在已完成的项目研究基础上，主要围绕带锥形奇异点空间的雷-辛格猜想，伯格曼核以及非紧空间上的瑞奇流，开展如下几方面问题的研究与探索：.在关于带锥形奇异点空间的雷-辛格猜想方面，将着重在解析绕率与 R-绕率之间的关系，解析绕率在奇异形变下的性状研究及相关的包括解析绕率与模形式关系等问题的深入研究。.在关于伯格曼核方面，将着重在关于奇异空间上伯格曼核的研究和伯格曼核在半正性下的渐近展开及应用的深入研究。最终将与最近在所谓田-丘-唐纳生猜想的突破性发展中扮演极其重要脚色的部分C零估计挂钩。.在瑞奇流方面，将着重于一类重要的非紧空间即渐进平坦空间上瑞奇流的长时间性状和收敛性。这类空间具有物理意义，且和紧致流形相差不大， 因此有希望获得比较完全的了解。最终目标是希望能给出正质量定定理的一个瑞奇流证明

This proposal concerns problems in Atiyah-Singer index theory and several problems in geometry that are related to Dirac operators and conical singularity. These include the study of analytic torsion and intersection R-torsion for manifolds with conical singularity, the study of the heat kernel and the Bergman kernel using the local index theory technique and the study of their relation with canonical metrics, and the study of Ricci flow on manifolds with conical singularity and a class of noncompact manifolds, the ALE spaces (which are manifolds with conical ends)...Analytic torsion has found many interesting connections and significant applications, in Seiberg-Witten theory, moduli spaces of Riemann surfaces, hyperbolic geometry, and mirror symmetry. The Cheeger-Mueller theorem has played an important role in all these. The PI, along with his collaborator and graduate students, would like to prove the Ray-Singer conjecture/Cheeger-Mueller theorem for manifolds with conical singularities which should help us understand more complicated singularities. One of the most important and extensively.studied area of geometry is the study of canonical metrics. Bergman kernels and conical singularity have been esential ingredients in the recent spectacular solution of the Yau-Tian-Donaldson conjecture. This proposal aims for the solution of the Ray-Singer conjecture for manifolds with conical singularity, better understanding of the Bergman kernel, and the Ricci flow on noncompact manifolds. It also explores the connection with positive mass theorems.