L3

Let M be a closed spin manifold.

Gromov and Lawson have shown that the presence of certain "enlargeable"

submanifolds of codimension 2 is an obstruction to the existence of a Riemannian metric with positive scalar curvature on M.

In joint work with Hanke, we refine the geoemtric condition of

"enlargeability": it suffices that a K-theoretic index obstruction of the submanifold doesn't vanish.

A "folk conjecture" asserts that all index type obstructions to positive scalar curvature should be read off from the corresponding index for the ambient manifold M (this this is equivalent to a small part of the strong Novikov conjecture). We address this question for the obstruction above and discuss partial results.

We show that the leading terms of the number of absolutely indecomposable representations of a quiver over a finite field are related to counting graphs. This is joint work with Geir Helleloid.

A major issue in general relativity, from its earliest days to the

present, is how to extract physical information from any solution or

class of solutions to the Einstein equations. Though certain

information can be obtained for arbitrary solutions, e.g., via geodesic

deviation, in general, because of the coordinate freedom, it is often

hard or impossible to do. Most of the time information is found from

special conditions, e.g., degenerate principle null vectors, weak

fields close to Minkowski space (using coordinates close to Minkowski

coordinates) or from solutions that have symmetries or approximate

symmetries. In the present work we will be concerned with

asymptotically flat space times where the approximate symmetry is the

Bondi-Metzner-Sachs (BMS) group. For these spaces the Bondi

four-momentum vector and its evolution, found from the Weyl tensor at

infinity, describes the total energy-momentum of the interior source

and the energy-momentum radiated. By generalizing certain structures

from algebraically special metrics, by generalizing the Kerr and the

charged-Kerr metric and finally by defining (at null infinity) the

complex center of mass (the real center of mass plus 'i' times the

angular momentum) with its transformation properties, a large variety

of physical identifications can be made. These include an auxiliary

Minkowski space viewed from infinity, kinematic meaning to the Bondi

momentum, dynamical equations of motion for the center of mass, a

geometrically defined spin angular momentum and a conservation law with

flux for total angular momentum.

This is joint work with Diaz, Glesser and Park.

In Proc. Instructional Conf, Oxford 1969, G. Glauberman shows that

several global properties of a finite group are determined by the properties

of its p-local subgroups for some prime p. With Diaz, Glesser and Park, we

reviewed these results by replacing the group by a saturated fusion system

and proved that the ad hoc statements hold. In this talk, we will present

the adapted versions of some of Glauberman and Thompson theorems.