Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

Topological Birkhoff

One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtain a representation of B in terms of A, even if B is finite. We show that by considering the natural topology on the functions of A and B in addition to the equations that hold between them, one can do with finite powers even for many interesting infinite algebras A. More precisely, we prove that if A and B are at most countable algebras which are oligomorphic, then the mapping which sends each function from A to the corresponding function in B preserves equations and is continuous if and only if B is a homomorphic image of a subalgebra of a finite power of A. Our result has the following consequences in model theory and in theoretical computer science: two \omega-categorical structures are primitive positive bi-interpretable if and only if their topological polymorphism clones are isomorphic. In particular, the complexity of the constraint satisfaction problem of an \omega-categorical structure only depends on its topological polymorphism clone.Comment: 21 page

Replication and exploratory analysis of 24 candidate risk polymorphisms for neural tube defects.

BackgroundNeural tube defects (NTDs), which are among the most common congenital malformations, are influenced by environmental and genetic factors. Low maternal folate is the strongest known contributing factor, making variants in genes in the folate metabolic pathway attractive candidates for NTD risk. Multiple studies have identified nominally significant allelic associations with NTDs. We tested whether associations detected in a large Irish cohort could be replicated in an independent population.MethodsReplication tests of 24 nominally significant NTD associations were performed in racially/ethnically matched populations. Family-based tests of fifteen nominally significant single nucleotide polymorphisms (SNPs) were repeated in a cohort of NTD trios (530 cases and their parents) from the United Kingdom, and case-control tests of nine nominally significant SNPs were repeated in a cohort (190 cases, 941 controls) from New York State (NYS). Secondary hypotheses involved evaluating the latter set of nine SNPs for NTD association using alternate case-control models and NTD groupings in white, African American and Hispanic cohorts from NYS.ResultsOf the 24 SNPs tested for replication, ADA rs452159 and MTR rs10925260 were significantly associated with isolated NTDs. Of the secondary tests performed, ARID1A rs11247593 was associated with NTDs in whites, and ALDH1A2 rs7169289 was associated with isolated NTDs in African Americans.ConclusionsWe report a number of associations between SNP genotypes and neural tube defects. These associations were nominally significant before correction for multiple hypothesis testing. These corrections are highly conservative for association studies of untested hypotheses, and may be too conservative for replication studies. We therefore believe the true effect of these four nominally significant SNPs on NTD risk will be more definitively determined by further study in other populations, and eventual meta-analysis

The wonderland of reflections

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable \omega-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1). As a consequence, the complexity of the CSP of an at most countable $\omega$-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures. Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.Comment: 24 page

Rate and cost of adaptation in the Drosophila Genome

Recent studies have consistently inferred high rates of adaptive molecular evolution between Drosophila species. At the same time, the Drosophila genome evolves under different rates of recombination, which results in partial genetic linkage between alleles at neighboring genomic loci. Here we analyze how linkage correlations affect adaptive evolution. We develop a new inference method for adaptation that takes into account the effect on an allele at a focal site caused by neighboring deleterious alleles (background selection) and by neighboring adaptive substitutions (hitchhiking). Using complete genome sequence data and fine-scale recombination maps, we infer a highly heterogeneous scenario of adaptation in Drosophila. In high-recombining regions, about 50% of all amino acid substitutions are adaptive, together with about 20% of all substitutions in proximal intergenic regions. In low-recombining regions, only a small fraction of the amino acid substitutions are adaptive, while hitchhiking accounts for the majority of these changes. Hitchhiking of deleterious alleles generates a substantial collateral cost of adaptation, leading to a fitness decline of about 30/2N per gene and per million years in the lowest-recombining regions. Our results show how recombination shapes rate and efficacy of the adaptive dynamics in eukaryotic genomes

A counterexample to the reconstruction of ω\omega-categorical structures from their endomorphism monoids

We present an example of two countable $\omega$-categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids -- in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable $\omega$-categorical structure in a finite relational language which can neither be reconstructed up to first-order bi-interpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone.Comment: 17 page