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.