Jarod Alper, Evolution of Luna's etale slice theorem

Luna's etale slice theorem is a beautiful result in equivariant geometry with important applications to moduli theory. Luna's result has inspired recent investigations into the local structure of algebraic stacks. The goal of this talk is to explain various stacky generalizations of Luna's slice theorem and its applications both to equivariant geometry and moduli theory. This talk is mainly based on joint work with J. Hall and D. Rydh.

Roman Avdeev, Combinatorial invariants of spherical subgroups: computational aspects

The existing combinatorial description of spherical homogeneous spaces and their equivariant embeddings is given (essentially) in terms of the following three combinatorial invariants: weight lattice, spherical roots, and colors. In view of the importance of these invariants in the whole theory, a natural problem is to find explicit formulas or algorithms for computing them for a given spherical homogeneous space G/H starting from an explicit form of the subgroup H. It is known that the full information about the weight lattice and colors of a spherical homogeneous space is encoded in one more invariant, called the extended weight semigroup. The goal of the talk is to present recent results on computing the spherical roots and extended weight semigroup for an arbitrary spherical subgroup H specified by a regular embedding in a parabolic subgroup of G.

Stephanie Cupit-Foutou, Gromov-width of wonderful manifolds

We will discuss results on the Gromov-width of the underlying symplectic manifolds of Luna's wonderful varieties.

Jacopo Gandini, Nipotent orbits of height 2 and involutions in the affine Weyl group

Let G be a semisimple algebraic group, with a fixed Borel subgroup B and maximal torus T in B. To any set of pairwise strongly orthogonal roots I will attach a nilpotent B-orbit in the Lie algebra of G, and will explain how the combinatorics of the involutions in the affine Weyl group of G relates to the geometry of such B-orbits. The talk is based on a joint work with P. Moseneder Frajria and P. Papi.

Daniel Greb, Moduli of sheaves that are semistable with respect to a Kähler polarisation

In recent work with M. Toma (Nancy) we constructed a proper moduli space of sheaves that are Gieseker-Maruyama-semistable with respect to a not necessarily rational Kähler class on a given complex projective manifold. I will explain why algebraic geometers should/could be interested in these moduli spaces and how techniques related to Luna's slice theorem crucially enter the proof via an existence criterion for good moduli spaces of Artin stacks by Alper-Fedorchuk-Smyth.

Jochen Heinloth, Existence of good moduli spaces for algebraic stacks and applications

Recently Alper, Hall and Rydh gave general criteria when a moduli problem can locally be described as a quotient and thereby clarified the local structure of algebraic stacks. We report on a joint project with Jarod Alper and Daniel Halpern-Leistner in which we use these results to show general existence and completeness results for good coarse moduli spaces.

In the talk we will focus on aspects that illustrate how the geometry of algebraic stacks gives a new point of view on classical methods for the construction of moduli spaces. Namely we explain how one-parameter subgroups in automorphism groups allow to formulate a version of Hilbert-Mumford stability in stacks that are not global quotients and sketch how one can reformulate Langton's proof of semistable reduction for coherent sheaves in geometric terms. This allows to apply the method to an interesting class of moduli problems.

Peter Heinzner, Equivariant embeddings of Cauchy-Riemann manifolds

A Cauchy-Riemann manifold is a real manifold with a partial complex structure. In general Cauchy-Riemann manifolds can not be realized as real submanifolds of complex manifolds. In the case where we have a Lie group G acting properly by Cauchy-Riemann maps and the action is transversal to the complex direction one can show that a realization as a sub-manifold of a complex manifold exists. Moreover if X is strongly pseudo-convex, then one can embed the Cauchy-Riemann manifold equivariantly into some C^m in the case of an action of a reductive Lie group.

Valentina Kiritchenko, Newton-Okounkov polytopes of flag varieties for classical groups

Theory of Newton-Okounkov convex bodies extends methods of toric geometry to non-toric varieties. The Newton-Okounkov convex body of a line bundle on a variety depends on a valuation on the field of rational functions. For flag varieties, different valuations yield several well-known families of polytopes from representation theory such as string polytopes and Nakashima-Zelevinsky polyhedral realizations of crystal bases. We survey results in this direction, and define a new valuation in the case of flag varieties for classical groups SL(n), SO(n) and Sp(2n). The valuation in every type comes from a natural coordinate system on the open Schubert cell, and is combinatorially related to the Gelfand-Zetlin pattern in the same type. For SL(n) and Sp(2n), we identify the corresponding Newton-Okounkov polytopes with the Feigin-Fourier-Littelmann-Vinberg polytopes. For SO(n), we compute low-dimensional examples and formulate open problems. All necessary definitions will be given in the talk.

Friedrich Knop, (Quasi-)Hamiltonian manifolds of cohomogeneity one

We present the classification of (quasi-)Hamiltonian manifolds M for (simply) connected compact Lie groups K where dim M/K = 1. This is joint work with Kay Paulus.

Hanspeter Kraft, Invariant Subsets, First Integrals, and Endomorphisms

Abstract

Lucy Moser-Jauslin, Real structures of horospherical and symmetric varieties

In this talk, I will describe some results on real structures of horospherical and of symmetric varieties. In these two special cases of spherical varieties, I will discuss the classification of all equivariant real structures for a given complex quasi-homogeneous variety. This is done first by considering the homogeneous case, and then by determining which real structures on the open orbit extend, using the combinatorics of the Luna-Vust theory.

Finally, I will show how the study of quasi-homogeneous varieties of complexity one differs from the spherical case.

This is joint work with R. Terpereau.

Dmitri Panyushev, Casimir elements associated with Z-gradings of simple Lie algebras

For any reductive subalgebra H of a simple Lie algebra L, one can define a Casimir element C using the restriction of the Killing form to H. In my talk, I consider the case, in which H=L(0) is the zero part of a Z-grading of L. I compute the eigenvalues of C in L and provide some applications to maximal abelian subspaces of L(1) and to the strange formula of Freudenthal--de Vries.

Claudio Procesi, Trace identities and a Luna stratification

I will discuss a generalization of a classical theorem of Amitsur on polynomial identities to trace identities in non commutative algebras. This topic is connected with the geometry of closed orbits, under the action of the projective linear group conjugating m-tuples of matrices.