Starting with the Unit
Disk:
Why are Bounded
Symmetric Domains
and
their quotients interesting ?
Professor Ngaiming Mok
Abstract
The
unit disk, or equivalently the upper half plane, is the simply connected domain
on which there is a homogeneous metric of constant negative Gaussian curvature,
viz., the Poincaré
metric. By the Uniformization Theorem the
universal covering of any Riemann surface other than a few exceptions is conformally equivalent to the unit disk. For the
study of Riemann surfaces it is therefore crucial to make use of the unit disk.
At the same time the upper half plane is a natural parameter space for compact
Riemann surfaces of genus 1, i.e., elliptic curves, and specific quotients can
be interpreted as moduli spaces of elliptic curves
with number-theoretic importance. The lecture is meant to be introductory and
exploratory, moving from the unit disk to bounded symmetric domains together
with their Bergman metrics, and explaining by way of a few examples that these
domains and their quotients serve as the underlying spaces for interesting
problems in Complex Analysis, Differential Geometry, Algebraic Geometry, Lie Theory
and Number Theory. The challenge is to understand the interplay among
methods across a spectrum of areas of expertise, e.g., to use geometric methods
to tackle number-theoretic problems in this context.
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Date: |
November 24, 2005 (Thursday) |
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Time: |
4:00 – 5:00pm |
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Place: |
Room 517, Meng
Wah Complex |
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All are welcome |
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