162 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Recent Advances in ÎŁ-definability over Continuous Data Types
The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas
Finite and infinite quotients of discrete and indiscrete groups
These notes are devoted to lattices in products of trees and related topics.
They provide an introduction to the construction, by M. Burger and S. Mozes, of
examples of such lattices that are simple as abstract groups. Two features of
that construction are emphasized: the relevance of non-discrete locally compact
groups, and the two-step strategy in the proof of simplicity, addressing
separately, and with completely different methods, the existence of finite and
infinite quotients. A brief history of the quest for finitely generated and
finitely presented infinite simple groups is also sketched. A comparison with
Margulis' proof of Kneser's simplicity conjecture is discussed, and the
relevance of the Classification of the Finite Simple Groups is pointed out. A
final chapter is devoted to finite and infinite quotients of hyperbolic groups
and their relation to the asymptotic properties of the finite simple groups.
Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups
updated; more examples added in Section 4; new Proposition 5.1
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