Geometric Flows on Manifolds with G_2 Structures

Speaker: 

Spiro Karigiannis

Institution: 

MSRI

Time: 

Tuesday, February 27, 2007 - 3:00pm

Location: 

MSTB 254

I will discuss geometric flows of G_2 structures on manifolds. These are flows of a 3-form on a 7-manifold with a certain non-degeneracy condition. The form determines a Riemmannian metric in a non-linear way. There is an associated tensor, called the torsion of the G_2 structure, which vanishes if and only if the manifold has G_2 holonomy.

Polar actions on Hilbert spaces

Speaker: 

Professor Ernst Heintze

Institution: 

University of Augsburg, Germany

Time: 

Tuesday, February 20, 2007 - 4:00pm

Location: 

MSTB 254

An isometric action of a Lie group is called polar if it admits
sections, i.e. submanifolds which meet all orbits and always
perpendicularly. Polarity is a very restrictive condition. For example,
in case of linear actions on *R^*n polarity characterizes the isotropy
representations of symmetric spaces (Dadok).

The aim of this talk is to report on work in progress to prove an
infinite dimensional analogue of Dadok's theorem. C.-L. Terng has
constructed interesting examples of polar actions on Hilbert spaces by
affine isometries, the so called P(G,H) actions. Here G is a compact Lie
group, H a closed subgroup of G \times G, and P(G,H) consists of all
paths in G with end points in H. The action of P(G,H) on the Hilbert
space of L^2-curves in the Lie algebra of G is by gauge transformations.
Surprisingly the actions correspond also to isotropy actions of
symmetric spaces which are now infinite dimension and quotients of a
Kac-Moody group by the fixed point set of an involution. We conjecture
that the P(G,H) actions exhaust all polar actions on a Hilbert space.

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