• Boris Pasquier, Smooth horospherical projective varieties with small Picard number.

In birational geometry, projective varieties with small Picard number (which is at least 1) appear to be the "smallest" varieties. In particular, one of the principle of the Minimal Model Program is to decrease the Picard number of the variety, and sometimes it terminates with a fibration whose general fiber is of Picard number 1.

I will explain how to classify and study projective varieties with small Picard number in the family of horospherical varieties (second talk). This family is a subfamily of spherical varieties that contains flag varieties and toric varieties and whose geometry properties have nice combinatorial descriptions (first talk).

• Clélia Pech, Mirror symmetry for homogeneous spaces

In these talks I will explain a construction by K. Rietsch of the mirrors of homogeneous spaces. The latter include Grassmannians, quadrics and flag varieties, and their mirrors can be expressed using Lie theory. More precisely the mirrors of homogeneous spaces live on so-called `Richardson varieties', which possess a cluster structure, and the mirror superpotential is defined on these Richardson varieties.

I will start by detailing Rietsch's general construction, then I will present some recent results by Marsh-Rietsch on Grassmannians, as well as joint work with Rietsch (resp. Rietsch and Williams) on Lagrangian Grassmannians (resp. quadrics). In particular I will show how the restriction of the superpotential on each cluster chart is a Laurent polynomial, which changes as we change cluster charts.

• Alexey Petukhov, Coadjoint orbits of Witt Lie algebra

In this talk I would like to discuss my joint work with Susan Sierra about coadjoint orbits of Witt algebra (it is an algebraic version of Lie algebra of vector fields on a circle). I will start from general tools available for coadjoint representation of (infinite-dimensional) Lie algebras and then proceed to the case of the Witt Lie algebra. The final result for the Witt algebra is as follows: a) all coadjoint orbits with non-trivial closure are finite dimensional b) the elements of these orbits can be identified with recurrent sequences of complex numbers c) the orbits themselves can be described through a version of n-jets of circle.

• Michael Pevzner, Symmetry breaking versus holography.

The analysis of branching problems for restriction of representations brings the concept of symmetry breaking transform and holographic transform. Symmetry breaking operators decrease the number of variables in geometric models of representations, whereas holographic operators increase it. From this perspective we investigate two remarkable families of symmetry breaking operators: the Rankin–Cohen operators and the holomorphic Juhl conformally covariant operators, and introduce the corresponding holographic operators.

• Vladimir Popov

1. Algebraic group actions and Zariski's cancellation problem

I call an algebraic variety flattenable if it is isomorphic to an open subset of an affine space. I shall discuss, in the frame of algebraic group actions, several topics which stem from the following question resembling Zariski’s cancellation problem: Are there affine algebraic varieties X and Y such that X and X × Y are flattenable, but Y is not? Among them is the problem of classifying a class of affine algebraic groups naturally singled out in the study of algebraic subgroups of the Cremona groups.

2. Simple algebras and invariants of linear actions

I shall discuss the following topics:

(1) Given an algebraic group G, let V be a finite-dimensional algebraic G-module, which admits a structure of a simple (not necessarily associative) algebra A such that G=Aut (A). Then V admits a close approximation to the analogue of classical invariant theory.

(2) What are the groups G for which such a V exists?

(3) Given G, what are the G-modules V for which (1) holds?

• Dmitry Timashev, Characterizing unipolar flag manifolds by their varieties of minimal rational tangents

A key role in the geometry of Fano manifolds of Picard number one, which are sometimes called unipolar, is played by rational curves of minimal degree. Tangent directions of such curves through a general point form the so-called variety of minimal rational tangents (VMRT) in the projectivized tangent space. In 90-s J.-M. Hwang and N. Mok proposed a program of characterizing unipolar flag manifolds in the class of all unipolar Fano manifolds by their VMRT. Recently the program was successfully completed (J.-M. Hwang, Q. Li, and the speaker). The proof of the main result involves a bunch of ideas and techniques from "pure"algebraic geometry, differential geometry, structure and representation theory of simple Lie groups and algebras, and theory of spherical varieties.