Abstract: By the Uniformization Theorem a compact Riemann surface other than the Riemann Sphere

or an elliptic curve is uniformized by the unit disk and equivalently by the upper half plane.  The upper

half plane is also the universal covering space of the moduli space of elliptic curves equipped with an

appropriate level structure.  In Several Complex Variables, the Siegel upper half plane is an analogue

of the upper half plane, and it is the universal covering space of moduli spaces of polarized Abelian

varieties with appropriate level structures. The Siegel upper half plane belongs, up to biholomorphic

equivalence, to the set of bounded symmetric domains, on which a great deal of mathematical research

is taking place. Especially, finite-volume quotients of bounded symmetric domains, which are naturally

quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic

Geometry, Arithmetic Geomtry and Number Theory, and an important topic is the study of uniformizations

of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved from methods

of Diophantine Geometry, Model Theory, Hodge Theory and Algebraic Geometry for Shimura varieties,

techniques for the general case of not necessarily arithmetic quotients have just begun to be developed.

We will explain a differential-geometric approach to the study of such algebraic subsets revolving around

the notion of asymptotic curvature behavior and the use of rescaling arguments, and illustrate how this

approach using transcendental techniques leads to various characterization results for totally geodesic

subvarieties of finite-volume quotients without the assumption of arithmeticity. Especially, we will explain

how the study of holomorphic isometric embeddings of the Poincare disk and more generally complex unit

balls into bounded symmetric domains can be further developed to derive uniformization theorems for

bi-algebraic varieties and more generally for the Zariski closure of images of algebraic sets under the

universal covering map.