Computers, Rigidity, and Moduli

Computers, Rigidity, and Moduli

Number of page: 174
Author: Shmuel Bochner, Shmuel Weinberger
Publisher: Princeton University Press
Rating:
Category: Mathematics

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 doppelgängers–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