746 research outputs found
A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS
Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S, there exists a countable field F of arbitrary characteristic with the same essential computable-model-theoretic properties as. Along the way, we develop a new computable category theory, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.National Science Foundation (U.S.) (Grant DMS-1069236
Finitely Generated Groups Are Universal
Universality has been an important concept in computable structure theory. A
class of structures is universal if, informally, for any
structure, of any kind, there is a structure in with the same
computability-theoretic properties as the given structure. Many classes such as
graphs, groups, and fields are known to be universal.
This paper is about the class of finitely generated groups. Because finitely
generated structures are relatively simple, the class of finitely generated
groups has no hope of being universal. We show that finitely generated groups
are as universal as possible, given that they are finitely generated: for every
finitely generated structure, there is a finitely generated group which has the
same computability-theoretic properties. The same is not true for finitely
generated fields. We apply the results of this investigation to quasi Scott
sentences
Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories
In this paper we construct invariants of 3-manifolds "\`a la
Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories.
We give concrete examples of such categories which lead to a family of
3-manifold invariants indexed by the integers. We prove this family of
invariants has several notable features, including: they can be computed via a
set of axioms, they distinguish homotopically equivalent manifolds that the
standard Reshetikhin-Turaev-Witten invariants do not, and they allow the
statement of a version of the Volume Conjecture and a proof of this conjecture
for an infinite class of links.Comment: 46 pages, 45 figures, new introduction, some misprints corrected and
an example detailed. An appendix added to correct a proo
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