 Message from the Dean
 History
 Education and Research

Staff Introduction
 Seminars & Events
 Distinctive Programs
 Access
 Job Openings
 Publications
 Related Links
 Contacts
Staff Introduction
1) Between 198998, we developed a theory of relative BottChern secondary characteristic classes, based on which we established an arithmetic GrothendieckRiemannRoch theorem for l.c.i. morphisms.
2) We also develop an Arakelov theory for surfaces with respect to singular metrics by establishing an arithmetic DeligneRiemannRoch isometry for them. Consequently, we study arithmetic aspect of the moduli spaces of punctured Riemann surfaces by introducing certain natural metrized line bundles related with WeilPetersson metrics, TakhtajanZograf metrics. Intrinsic relations among them, some of which are open problems, are exposed as well. The difficulty here is that classical approach on determinant metric does not work.3) We introduce genuine nonabelain L functions for global fields, based on a new cohomology, stability and Langlands' theory of Eisenstein series, and expose the relation between these nonabelian Ls and what we call the Arthur periods. Basic properties such as meromorphic continuation and functional equation(s) are established as well. In particular we show that the rank two nonabelian zetas for number fields satisfy the Riemann Hypothesis.
4) We develop a Program on what we call Geometric Arithmetic, in which an approach to nonabelian Class Field Theory using stability and an approach to the Riemann Hypothesis using intersection, together with a study on nonabelian L functions, are included.
5) We initiated an Arakelov approach to the study of what we call KobayashiHitchin correspondence for manifolds aiming at establishing the equivalence between intersection stability and existence of KE metrics. I spent several years in discussion with Mabuchi. These almost weekly discussions prove to be quite crucial to problems involved. I have no formal publication in it. But one can trace them from some papers of Mabuchi.
6) Other works such as metrized version of projective flatness of certain bundles and degenerations of Riemann surfaces are of some importance to the related fields.
Most of the works listed above can be found either at xxx.lanl.gov or at MathSciNet.
Keywords  Algebraic and/or Arithmetic and/or Complex Geometry, Number Theory 

Faculty , Department  Faculty of Mathematics , Department of Mathematics 