Algebraic and/or Arithmetic and/or Complex Geometry, Number Theory
1) Between 1989-98, we developed a theory of
relative Bott-Chern secondary characteristic classes,
based on which we established an arithmetic Grothendieck-Riemann-Roch
theorem for l.c.i. morphisms.
2) We also develop an Arakelov theory for surfaces
with respect to singular metrics by establishing an arithmetic
Deligne-Riemann-Roch 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 Weil-Petersson metrics, Takhtajan-Zograf
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 non-abelain L functions for global fields,
based on a new cohomology, stability and Langlands' theory of
Eisenstein series, and expose the relation
between these non-abelian 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 non-abelian
zetas for number fields satisfy the Riemann Hypothesis.
4) We develop a Program on what we call Geometric Arithmetic,
in which an approach to non-abelian Class Field Theory using stability
and an approach to the Riemann Hypothesis using intersection,
together with a study on non-abelian L functions, are included.
5) We initiated an Arakelov approach to the study of what we call
Kobayashi-Hitchin 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.
7) We are studying zeta functions and general class field theory
using stability and Galois representations. In particular, together
with Zagier, we establish the Riemann hypothesis for non-abelian zeta
functions of elliptic curves on finite fields.
8) We have just published a book on "Zeta Functions for Reductive Groups and Their Zeros"
with World Scientific in February 2018. In this book, we develop a basic theory for these functions,
establish the spacial uniformity of zeta functions on the equivalence of rank n non-abelian
zeta functions and SL(n)-zeta functions, based on Siegel-Langland' theory of Eisenstein series.
In particular, we confirm a central conjecture on "Parabolic Reduction, Stability and the Volumes".
The key to this is an analytic version of the Mumford'S GIT correspondence between un-stable principle
bundles and the parabolic subgroups of the associated reductive groups. This itself is based on
an equivalence between Arthur's analytic truncation on the adelic spaces and the geo-arithmetic
truncation of stability on principal bundles. Finally, we prove the Riemann hypothesis
for our zeta functions. The book consists of 7 parts:
Part 1 Non-Abelian Zeta Function
Part 2 Rank 2 Zeta Functions
Part 3 Eisensetin Periods and Multiple Zeta Functions
Part 4 Zeta Functions for Reductive Groups
Part 5 Algebraic and Analytic Structures and Riemann Hypothesis
Part 6 Geometric Structures and Riemann Hypothesis
Appendices (with K. Sugahara) Five Essays On Arithmetic Cohomology
Recently, I develop a theory of high rank algebraic codes and construct arithmetic characteristic curves.
Most of the works listed above can be found either at xxx.lanl.gov
or at MathSciNet.
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Educational Activities
For the undergraduate study,
we currently follow the tradition of this institute in choosing textbooks and
in evaluating studends. Particularly, we assign many tutorial problems for our
students in helping them to understand what has been taught in the class.
Normally, we spend lots of time to explain many of these problems as well.
The results are quite encouraging. The first half of 2014 was a tough one.
Simply too heavy.
For graduate study, we bring the students to the up-most frontier of the
current research while explaining the basic materials, and encourage the students
to have their independent thinking towards mathematics.
