This book is the first to present a new area of mathematical
research that combines topology, geometry, and logic. Shmuel
Weinberger seeks to explain and illustrate the implications of the
general principle, first emphasized by Alex Nabutovsky, that
logical complexity engenders geometric complexity. He provides
applications to the problem of closed geodesics, the theory of
submanifolds, and the structure of the moduli space of isometry
classes of Riemannian metrics with curvature bounds on a given
manifold. Ultimately, geometric complexity of a moduli space forces
functions defined on that space to have many critical points, and
new results about the existence of extrema or equilibria
follow.
The main sort of algorithmic problem that arises is recognition:
is the presented object equivalent to some standard one? If it is
difficult to determine whether the problem is solvable, then the
original object has doppelgangers--that is, other objects that are
extremely difficult to distinguish from it.
Many new questions emerge about the algorithmic nature of known
geometric theorems, about "dichotomy problems," and about the
metric entropy of moduli space. Weinberger studies them using tools
from group theory, computability, differential geometry, and
topology, all of which he explains before use. Since several
examples are worked out, the overarching principles are set in a
clear relief that goes beyond the details of any one problem."
General
Imprint: |
Princeton University Press
|
Country of origin: |
United States |
Series: |
Porter Lectures |
Release date: |
December 2004 |
First published: |
December 2004 |
Authors: |
Shmuel Weinberger
|
Dimensions: |
229 x 152 x 20mm (L x W x T) |
Format: |
Hardcover
|
Pages: |
174 |
Edition: |
New |
ISBN-13: |
978-0-691-11889-5 |
Categories: |
Books >
Science & Mathematics >
Mathematics >
Geometry >
General
|
LSN: |
0-691-11889-2 |
Barcode: |
9780691118895 |
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