The aim of the proposed research is to study the homotopy theory of algebraic varieties
and other algebraically defined geometric objects, especially over fields other than the complex numbers.
The main areas they cover are algebraic varieties
containing An-cylinders, algebraic varieties
with fibrations, algebraic group actions and orbit stratifications on algebraic varieties
, and automorphism groups and birational automorphism groups of algebraic varieties
Profound notions in mathematics and quantum mechanics -- dealing with algebraic varieties
and space-time curvatures in relativity -- are captured by elegant equations and tight notations, wonderfully packaged in a handful of pages.
On classification of algebraic varieties
. Shigefumi MORI, M.J.A.
We assume all algebraic varieties
are defined over C.
Bott-Samelson varieties are smooth, irreducible and [absolute value of Q]-dimensional algebraic varieties
. These varieties come equipped with a natural map
One thing is to use category theory in order to construct a unified homology theory like Eilenberg and Steenrod did in their book, or introduce schemes and the etale cohomology as Grothendieck did for the purpose of finding suitable invariants for algebraic varieties
over finite fields and in order to prove the Weil conjectures.
Topology of algebraic varieties
and singularities; proceedings.
The main focus is rational points on algebraic varieties
over non-algebraically closed fields.
Zak, Tangents and Secants of Algebraic Varieties
, Translations of Mathematical Monographs 127 (1993).
This sits among a bigger project that aims to explicitly classify algebraic varieties
in dimension 3.
The summer 2015 graduate program of the Park City Mathematics Institute concentrated on a combination of interrelated topics in algebraic geometry, the topology of algebraic varieties
, and representation theory.