Workshop on Transformation Groups

In honor of Peter Heinzner

 

 

July 18^th, 2025 at Ruhr-Universität Bochum

 

 

Speakers

 

Leonardo Biliotti (Parma)

Jürgen Hausen (Tübingen)

Gerald Schwarz (Brandeis)

 

 

Venue

All talks will take place in IA 01/473. A campus map can be found here

 

 

Titles and abstracts

 

Leonardo Biliotti	Stability, analytic stability for real reductive Lie groups
	In this talk, we study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We assume that the action of G extends holomorphically to an action of a complex reductive group and is Hamiltonian with respect to a compatible maximal compact subgroup of the complex group. There is a corresponding gradient map arising from a Cartan decomposition of G. Under mild assumptions on the G-action on X, we characterize the G-orbits in X that intersect the zero level set of the gradient map in terms of the maximal weight functions. If time permits, we also study the natural G-action on the space of probability measures on X. We define an analogue of the gradient map and the usual concepts of stability, proving numerical criteria for stability, semi-stability, a polystability. The result are proved in collaboration with Oluwagbenga Joshua Windare. In the classical case of a group action on a Kahler manifold these characterizations are due to Mundet i Riera, Teleman, Kapovich, Leeb and Millsos and probably many others. In fact many of these ideas go back as far as Mumford.
Jürgen Hausen	Title
	Abstract
Gerald Schwarz	Equivariant Oka principles
	The equivariant Oka principle of Heinzner and Kutzschebauch implies  the following. Let $K$ be a compact Lie group acting holomorphically on a Stein manifold $X$. Let $E$ and $E'$ be topological $K$-vector bundles over $X$. Then $E$ and $E'$ have structures as holomorphic $K$-vector bundles over $X$, and these are $K$-equivariantly biholomorphic if and only if they are $K$-equivariantly homeomorphic. We report on work with F. Kutzschebauch and F. Lárusson where we prove analogues of these results for various classes of $G$-equivariant mappings between Stein $G$-manifolds (where $G$ is the complexification of $K$). This includes criteria for a positive solution to the holomorphic linearization problem: when is a holomorphic action of $G$ on complex $n$-space  linear after a holomorphic change of coordinates? In the last 40 years,  work of M. Gromov, F. Forstnerič and others has shown that there are various Oka principles for mappings from  Stein manifolds into Oka manifolds $Y$, generalizing results of Grauert where $Y$ is a complex homogeneous space. We report on  $G$-equivariant versions of these results.

 

 

 

Schedule

 

11:00	Welcome by the Dean
11:30 - 12:20	Jürgen Hausen:
Lunch break
14:00 - 14:50	Leonardo Biliotti: Stability, analytic stability for real reductive Lie groups
Coffee break
15:30 - 16:20	Gerald Schwarz: Equivariant Oka principles

 

Talks will take place in IA 01/473.

 

 

 

Organisers

 

Stéphanie Cupit-Foutou (Bochum)

Daniel Greb (Essen)

Christian Miebach (Calais)

Stefan Nemirovski (Wuppertal)

 

 

Financial support

The workshop is supported by CRC/TRR 191 Symplectic Structures in Geometry, Algebra and Dynamics.