Mathematical Sciences: Manifolds and Homotopy Theory

Investigator(s):

Thomas Goodwillie (PI)

Sponsor:

Brown University, RI 02912 4018632777

Start Date/Expiration Date

1995-07-15 to 1998-06-30 (amended 1997-06-20)

Awarded Amount to Date:

$89,900

Abstract: 9509744 Goodwillie This project uses several kinds of 'functorial calculus' to investigate the homotopy types of spaces of diffeomorphisms and spaces of smooth embeddings. These techniques exploit multirelative connectivity estimates to describe continuous functors of various types in terms of special values; for example, a functor of topological spaces might be recovered from its values at highly connected spaces, or a functor of subspaces of a manifold, from its values at zero-dimensional subspaces, or a functor of real inner product spaces, from its values at high-dimensional spaces. The project is to refine and combine these techniques and to apply them to various questions in differential topology. Each 'functorial calculus' mentioned above is so called because of a not-entirely-fanciful resemblance to the ordinary 'functional calculus' of Newton and Leibniz. Sometimes a fact about numbers is best proved by placing it in a context where a number is part of a huge family of numbers -- a numerical function. Properties of the function then lead, by general theorems of calculus that may seem a bit magical on first encountering them, to a computation of the number. So it is here: sometimes a fact about some geometrically defined object is best proved by placing it in a context where the object is part of a huge family of such objects -- a functor -- and using some magic of a more modern kind. This analogy may convey something of the flavor of the research; the content is harder to convey, because most of the 'geometric' objects in question are connected to everyday reality only by rather long chains of abstract ideas. (Despite the intervention of similar long chains of abstract ideas, however, one finds that the 'objects' physicists study predict real physical phenomena with uncanny precision.) ***

NSF Org:

DMS - Division of Mathematical Sciences

Award Number:

9509744

Award Instrument:

Continuing grant

Program Manager:

Ralph M. Krause
DMS Division of Mathematical Sciences
MPS Directorate for Mathematical & Physical Sciences

NSF Program(s):

TOPOLOGY, TOPOLOGY

Field Application(s):

Mathematics, Other nsf.applications NEC

Program Reference Code(s):

UNASSIGNED, 0000

Program Element Code(s):

1267
, 1267