## Noncommutative algebraic varieties, their properties and geometric realizations

Dmitry Orlov

Steklov Mathematical Institute, Russian Academy of Sciences; Member, School of Mathematics

February 1, 2017

We are will discuss a notion of noncommutative and derived algebraic variety. This approach comes from a generalization of derived categories of (quasi)coherent sheaves on usual algebraic varieties and their enhancements. We are going to talk about different properties of noncommutative varieties such as regularity, smoothness and properness. A new operation of gluing of noncommutative varieties will be introduced.