Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. A special class of flag domains constitute the clas- sifying spaces for variations of Hodge structure, namely period domains or more generally Mumford-Tate domains. In this talk I will consider the prob- lem of classifying all equivariant embeddings of an arbitrary flag domain in a period domain satisfying a certain transversality condition. Satake studied this problem in the weight 1 case in connection to the study of (algebraic) families of abelian varieties where the transversality condition is trivial. In this talk I will describe certain combinatorial structures at the Lie algebra level, called Hodge triples, which are generalisation of sl(2)-triples and show how this structures provide a solution to the classification problem.

In this talk, I will explain how the slope stability condition for the restriction of the tangent bundle of a hermitian symmetric space to a complete intersection leads to a natural question concerning the vanishing of some Dolbeault cohomology groups. Building on Bott's theorem and Snow's interpretation of this theorem in terms of admissible partitions, I will show that the corresponding vanishing theorems hold, proving the stability statement.

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 two 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.

Let G be one of the classical complex groups O(n, C), SO(n, C), and Sp(2k, C) (in the case n = 2k) and let W denote a finite dimensional polynomial GL(n, C)-module. In this talk we consider the symmetric algebra S(W) and describe a method for determining the Hilbert series of the algebra of invariants S(W)^G. This method extends previous results for the case G = SL(n,C). Then we give examples for computing the above Hilbert series for explicit choices of W. As a further application of the described method, we determine also the Hilbert series of the subalgebras of G-invariants of certain exterior algebras. The talk is based on a joint work with Vesselin Drensky.

The study of equivariant embeddings of tori into algebraic varieties, also known as toric varieties, is a well-known topic of algebraic geometry. In a recent work, Geraschenko and Satriano considered the equivariant embeddings of tori into algebraic stacks and proved that they are always quotient stacks of toric varieties. In this talk, I will explain the idea of their proof, give some examples, and also explain how their result might extend to the larger class of equivariant embeddings of horospherical homogenous spaces into algebraic stacks.

I will speak about actions of compact Lie groups on closed manifolds whose cohomogeneity (i.e., codimension of principal orbits) is equal to 1. The equivariant cohomology of such group actions will be discussed. This will be first analysed from the point of view of its canonical structures of module and ring. Concrete descriptions of the ring structure will be presented. The talk is based on joint work with Jeffrey Carlson, Chen He, and Oliver Goertsches.

Given a homogeneous variety X for a complex algebraic group G defined over real numbers, the real Lie group G(R) usually acts non-transitively (but with finitely many orbits) on the real locus X(R). A natural problem, to which many classification problems in algebra and geometry reduce, is to describe the orbits of G(R) on X(R). We address this problem for spherical homogeneous spaces, G being a connected reductive group. We concentrate on two cases: (1) X is a symmetric space; (2) G is split over R. The answer is similar in both cases: the G(R)-orbits are classified by the orbits of a finite reflection group W_X (the "little Weyl group") acting in a fancy way on the set of orbits of T(R) in Z(R), where T is a maximal torus in G and Z is a "Brion-Luna-Vust slice" in X. The latter orbit set can be described combinatorially. We use different tools: Galois cohomology in (1) and action of minimal parabolic subgroups on Borel orbits together with Knop's theory of polarized cotangent bundle in (2). We expect that the second approach can be extended to the non-split case.