La información de las charlas que dictarán los profesores Stellari y Bernardara se detallan a continuación:

​

​

Expositor: Paolo Stellari

​

Título: Cubic threefolds and fourfolds: geometry and homological algebra

Resumen:
We revisit some classical results concerning the geometry of smooth cubic hypersurfaces of dimension 3 and 4 by means of modern techniques involving derived categories and Bridgeland stability conditions. The examples we want to examine are: the Fano variety of lines, twisted cubics curves on cubic fourfolds and the Torelli theorem for cubic threefolds and fourfolds.

​

​

​

Expositor: Marcello Bernardara

​

Título: Derived categories, cycles, and rationality.

Resumen: 

In the last decades, semiorthogonal decompositions of derived categories of coherent sheaves have been considered as a possible tool to attack birationality questions. Since the seminal work of Bondal and Orlov, the work of many authors, Kuznetsov above all of them, has made clear what we should expect and which are the main technical problems. In particular, one would expect that the derived category of a rational variety must be decomposed in "codimension 2" subcategories.
The aim of these lectures is to set definitions and property that could give a sense to the above considerations, and to show that this conjectural obstruction is stronger than the known classical ones in cases such as surfaces or Fano threefolds, and closely related for some particular fourfold (cubic, Gushel'-Mukai). A particular attention will be put on constructions related to cycles and motives, such as intermediate Jacobians and decomposition of the diagonal.