A01=Shmuel Weinberger
Algebraic topology
Algorithm
Algorithmic
Arithmetic
Arithmetic group
Arithmetical hierarchy
Author_Shmuel Weinberger
Automorphism
Betti number
Calculation
Category=PBM
Characterization (mathematics)
Cohomology
Commutator
Commutator subgroup
Complex dimension
Computability
Computable function
Computation
Configuration space
CW complex
Diffeomorphism
Dimension (vector space)
Disk (mathematics)
Dot product
Duality (mathematics)
Embedding
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Euclidean space
Function (mathematics)
Function space
Fundamental group
Geometry
H-cobordism
H-space
Homology (mathematics)
Homology sphere
Homotopy
Hypersurface
Information theory
JSJ decomposition
Linear topology
Lipschitz continuity
Manifold
Mathematics
Maxima and minima
Minkowski space
Moduli space
Multivariable calculus
Polynomial
Projection (mathematics)
Riemann surface
Riemannian manifold
Semialgebraic set
Simplicial complex
Simply connected space
Special case
String (computer science)
Subgroup
Superadditivity
Superperfect group
Surgery theory
Symplectic geometry
Tangent space
Theorem
Theory of computation
Transversality (mathematics)
Trichotomy (mathematics)
Triviality (mathematics)
Turing degree
Turing machine
Variable (mathematics)
Word problem (mathematics)
Product details
- ISBN 9780691118895
- Weight: 425g
- Dimensions: 152 x 235mm
- Publication Date: 19 Dec 2004
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Hardback
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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.
Shmuel Weinberger is Professor of Mathematics at the University of Chicago. He is the author of "The Topological Classification of Stratified Spaces".
