43 research outputs found
The conjugacy problem for automorphism groups of countable homogeneous structures
We consider the conjugacy problem for the automorphism groups of a number of
countable homogeneous structures. In each case we find the precise complexity
of the conjugacy relation in the sense of Borel reducibility
On the classification of automorphisms of trees
We identify the complexity of the classification problem for automorphisms of
a given countable regularly branching tree up to conjugacy. We consider both
the rooted and unrooted cases. Additionally, we calculate the complexity of the
conjugacy problem in the case of automorphisms of several non-regularly
branching trees
Automorphisms of corona algebras, and group cohomology
In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis,
there exists an outer automorphism of the Calkin algebra. (The Calkin algebra
is the algebra of bounded operators on a separable complex Hilbert space,
modulo the compact operators.) In this paper we establish that the analogous
conclusion holds for a broad family of quotient algebras. Specifically, we will
show that assuming the Continuum Hypothesis, if is a separable algebra
which is either simple or stable, then the corona of has nontrivial
automorphisms. We also discuss a connection with cohomology theory, namely,
that our proof can be viewed as a computation of the cardinality of a
particular derived inverse limit
The complexity of classification problems for models of arithmetic
We observe that the classification problem for countable models of arithmetic
is Borel complete. On the other hand, the classification problems for finitely
generated models of arithmetic and for recursively saturated models of
arithmetic are Borel; we investigate the precise complexity of each of these.
Finally, we show that the classification problem for pairs of recursively
saturated models and for automorphisms of a fixed recursively saturated model
are Borel complete.Comment: 15 page
The Fundamental Theorem on Symmetric Polynomials: History's First Whiff of Galois Theory
We describe the Fundamental Theorem on Symmetric Polynomials (FTSP), exposit
a classical proof, and offer a novel proof that arose out of an informal course
on group theory. The paper develops this proof in tandem with the pedagogical
context that led to it. We also discuss the role of the FTSP both as a lemma in
the original historical development of Galois theory and as an early example of
the connection between symmetry and expressibility that is described by the
theory.Comment: 15 pages, 1 figure. Corrected a misattributio
Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals
We describe the basic theory of infinite time Turing machines and some recent
developments, including the infinite time degree theory, infinite time
complexity theory, and infinite time computable model theory. We focus
particularly on the application of infinite time Turing machines to the
analysis of the hierarchy of equivalence relations on the reals, in analogy
with the theory arising from Borel reducibility. We define a notion of infinite
time reducibility, which lifts much of the Borel theory into the class
in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference,
200
Infinite time decidable equivalence relation theory
We introduce an analog of the theory of Borel equivalence relations in which
we study equivalence relations that are decidable by an infinite time Turing
machine. The Borel reductions are replaced by the more general class of
infinite time computable functions. Many basic aspects of the classical theory
remain intact, with the added bonus that it becomes sensible to study some
special equivalence relations whose complexity is beyond Borel or even
analytic. We also introduce an infinite time generalization of the countable
Borel equivalence relations, a key subclass of the Borel equivalence relations,
and again show that several key properties carry over to the larger class.
Lastly, we collect together several results from the literature regarding Borel
reducibility which apply also to absolutely Delta_1^2 reductions, and hence to
the infinite time computable reductions.Comment: 30 pages, 3 figure