Oluwagbenga Joshua Windare (Parma), Gradient Maps associated with actions of Real Reductive Groups

Abstract: This talk is concerned with the Hamiltonian action of a complex reductive group U on a Kahler manifold (Z,ω). The momentum map associated with this action helps to analyze the geometric properties of the action. Our main goal is to investigate a refinement of this setting, namely, the action of a compatible (real) subgroup G of U on a real submanifold X of Z. One can study the geometry of the G-action on X using the associated gradient map and its norm square function, using techniques from convex geometry, complex geometry, Lie group, and Morse theory. In particular, we will discuss the convexity property of the gradient map and the classical Hilbert-Mumford criterion in Geometric Invariant Theory for the G-action on X. The Hilbert-Mumford criterion gives an explicit numerical criterion for testing the (semi, poly)stability of a point in terms of an important G-equivariant function called the maximal weight function.

Tobias Waedt (RUB), A generalisation of LVMB manifolds: a class of compact, complex, non-Kähler manifolds

Abstract: This talk deals with my PhD thesis on a generalization of LVMB-manifolds, a class of non-Kähler compact complex manifolds introduced by López de Medrano and Verjovsky and generalized first by Meersseman and later by Bosio. As proved by Meersseman and Verjovsky, some of these manifolds are closely related to projective simplicial toric varieties. For the generalization I will introduce, it turns out that these new manifolds are naturally related with horospherical varieties, as I shall explain. Some properties and examples of these manifolds will be also discussed.

Christian Karb (RUB), On Batyrev's classification of Fano toric manifolds

Abstract: My talk deals with my master thesis on the classification of Fano compact symplectic toric manifolds in terms of so-called reflexive convex polytopes. This classification was finalized by Batyrev. After a short recollection about the objects appearing in this classification, I will outline the main steps of the proof, namely Atiyah-Guillemin-Sternberg Convexity Theorem and Delzant Classification.

Elizaveta Vishnyakova (Belo Horizonte), About graded coverings

Abstract: In geometry, there is a well-known notion of a covering space. A classical example is the following universal covering: \(p:\mathbb R\to S^1\), \(t \mapsto \exp (it)\). Another example we can find in algebra: a flat covering or torsion-free covering in the theory of modules over rings. All of these covers satisfy some common universal properties.

In the paper ``Super Atiyah classes and obstructions to splitting of supermoduli space'', Donagi and Witten suggested a construction of a first obstruction class for splitting a supermanifold. It appeared that an infinite prolongation of the Donagi-Witten construction satisfies the universal properties of a covering. In other words, this is a covering of a supermanifold in the categorical category of graded manifolds corresponding to the non-trivial homomorphism \(\mathbb Z\to \mathbb Z_2\). In addition, the space of infinite jets can also be regarded as a covering of a (super)manifold in the category of graded manifolds corresponding to the homomorphism \(\mathbb Z\times \mathbb Z_2 \to \mathbb Z_2\), \((m, \bar n)\mapsto \bar n\). (For usual manifolds this homomorphism is trivial \(\mathbb Z \to 0\).) Another example is a multiplicity-free covering of a graded manifold. In fact, similar constructions have appeared before in Poisson geometry. In summary, covers in the category of graded manifolds we call graded. Our talk is devoted to the current state of the theory of graded coverings: general idea, particular examples, notion of a deck transformation group.