Computing trisections of 4-manifolds

Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a $4$-manifold in terms of the number of pentachora ($4$-simplices) in a triangulation.Comment: 15 pages, 9 figure

Visual perceptual load reduces auditory detection in typically developing individuals but not in individuals with Autism Spectrum Disorders

Objective: Previous studies examining selective attention in individuals with autism spectrum disorder (ASD) have yielded conflicting results, some suggesting superior focused attention (e.g. on visual search tasks), others demonstrating greater distractibility. This pattern could be accounted for by the proposal (derived by applying the Load theory of attention, e.g. Lavie, 2005) that ASD is characterized by an increased perceptual capacity (Remington, Swettenham, Campbell, & Coleman, 2009). Recent studies in the visual domain support this proposal. Here we hypothesize that ASD involves an enhanced perceptual capacity that also operates across sensory modalities, and test this prediction, for the first time using a signal detection paradigm. Method: 17 neurotypical (NT) and 15 ASD adolescents performed a visual search task under varying levels of visual perceptual load while simultaneously detecting presence/absence of an auditory tone embedded in noise. Results: Detection sensitivity (d’) for the auditory stimulus was similarly high for both groups in the low visual perceptual load condition (e.g. 2 items: p = .391, d = 0.31, 95% CI [-.39, 1.00]). However, at a higher level of visual load, auditory d’ reduced for the NT group but not the ASD group leading to a group difference (p = .002, d = 1.2, 95% CI [.44, 1.96]). As predicted, when visual perceptual load was highest, both groups then showed a similarly low auditory d’ (p = .9, d = 0.05, 95% CI [-.65, .74]). Conclusions: These findings demonstrate that increased perceptual capacity in ASD operates across modalities

Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The deformation variety D(T), a subset of which parameterises (incomplete) hyperbolic structures obtained on M using T, is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.Comment: 31 pages, 11 figures; minor changes; to appear in Mathematische Zeitschrif

Imprints of the QCD Phase Transition on the Spectrum of Gravitational Waves

We have investigated effects of the QCD phase transition on the relic GW spectrum applying several equations of state for the strongly interacting matter: Besides the bag model, which describes a first order transition, we use recent data from lattice calculations featuring a crossover. Finally, we include a short period of inflation during the transition which allows for a first order phase transition at finite baryon density. Our results show that the QCD transition imprints a step into the spectrum of GWs. Within the first two scenarios, entropy conservation leads to a step-size determined by the relativistic degrees of freedom before and after the transition. The inflation of the third scenario much stronger attenuates the high-frequency modes: An inflationary model being consistent with observation entails suppression of the spectral energy density by a factor of ~10^(-12).Comment: 11 pages, 13 figure