Principal Investigator: Xianzhe Dai
This proposal concerns several problems in geometry that are related to Dirac operators and Atiyah-Singer index theory. It includes the use of Dirac operator and spinors in the study of stability problem for Einstein metrics. The PI will seek better understanding of the stability of Einstein metrics with positive scalar curvature by exploring the connection with Killing spinors and Sasakian-Einstein metrics. Another problem involves the study of heat kernel and Bergman kernel using local index theory technique and the study of their relation with canonical metrics. The PI would like to study the spectral gap of Dirac operators and the asymptotic expansion of Bergman kernel in the semipostive case. The PI will also study Ricci flows on a class of noncompact manifolds, the ALE spaces. The question of long time convergence will be the main focus. Finally, the PI will investigate the behavior of geometric invariants under metric degeneration, including adiabatic limit and conical degeneration. In particular, one of the applications will be the Ray-Singer conjecture for manifolds with conical singularities.
Einstein's General relativity geometrizes gravity, one of the four fundamental forces in nature and the dominating one in shaping our universe. Einstein manifolds play essential role in mathematics and physics. It is important to understand the stability of Einstein manifolds. Stability issue is also important in the study of geometric evolution equations such as the Ricci flow. Recent development shows the extraordinary power of the Ricci flow. Dirac operators and related geometric invariants are playing significant and important role in diverse fields of mathematics and physics. This proposal aims for better understanding of the stability of Einstein manifolds, of the Ricci flow on noncompact manifolds, and of geometric invariants.
Singuarities occur very naturally in mathematics (and other sciences). The results and techniques we developed lead to better and further understanding of the effects of singularities in mathematics. The Ray-Singer conjecture is about the relation between a topological invariant, the Reidemeister torsion and an analytic invariant, the Ray-Singer analytic torsion. For closed manifolds (smooth spaces), the Ray-Singer conjecture has been proven independently by Jeff Cheeger and Werner Mueller, and has found many remarkable applications in topology, number theory, mathematical physics and geometry. It is thus highly interesting to generalize it to singular spaces. We investigated a natural class of singular spaces, manifolds with conical singularity and found that the singularity will contribute an extra geometric correction term to the Ray-Singer conjecture. We are currently in the process of identifying this extra geometric correction term. Another analytic invariant that we introduced earlier, the even dimensional eta invariant, plays the role of the eta invariant of Atiyah-Patodi-Singer which is for odd dimensional manifolds. We succeeded in finding an intrinsic formula for the even dimensional eta invariant. The intrinsic spectral interpretation demonstrate further similarity and paves the way for deeper understanding of the invariant and future applications. The general result is also an even dimensional analog of the famous Witten's holonomy theorem. In addition we studied the Bergman kernel on manifolds with orbifold singularity, which is a special case of conical singularity. Bergman kernel and conical singularity have played essential roles in the recent spectacular solution of Yau-Tian-Donaldson conjecture. We also investigated adiabatic limit of analytic torsion and introduced a new analytic invariant related to the Bismut-Freed connection whose adiabatic limit will produce the Bismut-Lott analytic torsion form. Besides teaching a number of graduate topics courses related to my research, I also supervised several PhD students and REU projects. Three of my PhD students have graduated during the period (Xiaoling Huang, Jeffrey Case, NSF postdoc, Princeton, Peng Wu (co-chair), postdoc at Cornell). In addition a joint training PhD (Jianqing Yu) successively completed one year research here at UCSB. I supervised two REU projects (Lynette Cortes, McNair scholar, and Eric Ling, CCS student). The PI organized a special lecture series on the Millenium Prize Problems which attracted undergraduate students, graduate students, scientists and engineers, as well as community members. During the period, the PI gave nineteen research talks, including Chern Centennial Conference, invited address, Chern Institute, 10/11, Workshop on Stratefied Spaces, Oberwolfach, 12/11, Workshop on Analysis and Geometric Singularities, Oberwolfach, 05/12, Analytic Torsion and its Applications, Orsay, 06/12, Spectral Invariants for Singular and Noncompact Manifolds, CRM, 07/12, Conformal and CR Geometry, Banff, 07/12. On several occasions, the PI has given math club talks and graduate colloquia introducing research areas and frontiers to undergraduate and graduate students here at UCSB. He organizes the Colloquium, the RTG seminar and helps run the differential geometry seminar and his NSF funding has enabled him to invite speakers from all over the country and abroad which benefited greatly the postdocs and graduate students by exposing them to new ideas and new research directions and facilitating interactions with leaders of the field. In 2012, I organized the fourth ``International Conference on Geometry and Topology on Manifolds" in Santa Barbara. The conference is particularly beneficial to the postdocs and graduate students here at UCSB. The PI has co-edited three volumes, one of them the proceedings for the ``International Conference on Metric and Differential Geometry". These made the current development available to the whole mathematical community (e.g. the important work of M""uller on using the analytic torsion to detect the hyperbolic volume appears in that volume). The PI has been actively pursuing and promoting international collaboration, especially among UCSB and the Chern Institute of Mathematics in China. The activities include exchange of postdocs and PhD students, workshops, and research collaborations. The PI served in the Semifinalist Selection Committee for the oversea regions competition of the Yau High School Mathematics Award.
