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.

We assume all

algebraic varieties are defined over C.

Objective: In relation with the study of both moduli and enumerative problems in complex algebraic geometry, we propose the geometric study of various families of subvarieties of certain complex

algebraic varieties of small dimension, and mainly of families of (possibly singular) curves.

Bott-Samelson varieties are smooth, irreducible and [absolute value of Q]-dimensional

algebraic varieties.

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).

These are so-called

algebraic varieties, which are the zero-sets of polynomials, and certain geometric objects called manifolds," he added.

This edition has new chapters on observable actions of affine algebraic groups, quotient varieties, and

algebraic varieties.

One of the most important open problems in the minimal model theory for higher-dimensional

algebraic varieties is the abundance conjecture.

The FACE project will focus on particular types of non-linear geometric objects: arcs, (n,r)-arcs, caps, saturating sets, and

algebraic varieties.