Fukaya categories of the torus and Dehn surgery

This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded as modules over the Fukaya category of the punctured 2-torus. We extract a short proof of exactness of the Dehn surgery triangle in Heegaard Floer homology. We show that A-infinity structures on the graded algebra A formed by the cohomology of two basic objects in the Fukaya category of the punctured 2-torus are governed by just two parameters (m^6,m^8), extracted from the Hochschild cohomology of A. For the Fukaya category itself, m^6 is nonzero.Comment: 29 pages, 2 figures, a footnote adde

Differential hydrophobicity drives self-assembly in Huntington's disease

Identifying the driving forces and the mechanism of association of huntingtin-exon1, a close marker for the progress of Huntington's disease, is an important prerequisite towards finding potential drug targets, and ultimately a cure. We introduce here a modelling framework based on a key analogy of the physico-chemical properties of the exon1 fragment to block copolymers. We use a systematic mesoscale methodology, based on Dissipative Particle Dynamics, which is capable of overcoming kinetic barriers, thus capturing the dynamics of significantly larger systems over longer times than considered before. Our results reveal that the relative hydrophobicity of the poly-glutamine block as compared to the rest of the (proline-based) exon1 fragment, ignored to date, constitutes a major factor in the initiation of the self-assembly process. We find that the assembly is governed by both the concentration of exon1 and the length of the poly-glutamine stretch, with a low length threshold for association even at the lowest volume fractions we considered. Moreover, this self-association occurs irrespective of whether the glutamine stretch is in random coil or hairpin configuration, leading to spherical or cylindrical assemblies, respectively. We discuss the implications of these results for reinterpretation of existing research within this context, including that the routes towards aggregation of exon1 may be distinct to those of the widely studied homopolymeric poly-glutamine peptides

Lagrangian matching invariants for fibred four-manifolds: I

In a pair of papers, we construct invariants for smooth four-manifolds equipped with `broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the Donaldson-Smith invariants for Lefschetz fibrations. The `Lagrangian matching invariants' are designed to be comparable with the Seiberg-Witten invariants of the underlying four-manifold. They fit into a field theory which assigns Floer homology groups to fibred 3-manifolds. The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres, subject to Lagrangian boundary conditions. Part I is devoted to the symplectic geometry of these Lagrangians.Comment: 72 pages, 4 figures. v.2 - numerous small corrections and clarification

Wrinkled fibrations on near-symplectic manifolds

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow us to pass from any given fibration to any other broken fibration which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory.Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.Comment: 35 pages, 12 figures. Final version. Minor corrections and clarification

Population variation in the life history traits and thermal responses of Atlantic cod, Gadus morhua L.

Studies of the phenotypes of animals at different parts of their geographic range often reveal striking variability. It is of considerable fundamental and applied interest to discover the extent to which such variation depends on genetic as opposed to environmental differences. A first step towards disentangling these effects is to use an empirical approach known as the common environment method in which wild-caught juveniles from different regions are reared under common laboratory conditions. I used this approach to determine the population and thermal responses of Atlantic cod, a species with a wide distribution and geographic variation in life history traits. Life history traits were investigated in cod from three areas around the British Isles of differing thermal regimes, namely St Andrews Bay on the Scottish east coast, the Clyde Sea on the Scottish west coast, and from near Lowestoft in the southern North Sea. Concurrently haemoglobin genotype and behaviour were also studied. Spatially significant differences in life history traits and thermal responses were revealed in juvenile and adult growth rate, gonadal investment and behaviour, suggestive of population differentiation. Behavioural differences between cod of differing haemoglobin genotypes were also demonstrated. Results suggested that juvenile growth rates may be modified by competitive interactions. At a group level, growth rate of cod from the Clyde Sea was suppressed in the presence of cod from St Andrews Bay. Pairwise trials demonstrated that cod from the Clyde Sea consumed a higher prey share than those from St Andrews Bay but that those from St Andrews Bay were more aggressive and thus could potentially restrict feeding of cod from the Clyde Sea, resulting in a reduced growth rate. There were no population differences in the distribution of haemoglobin genotype, but haemoglobin genotype did have a strong influence on behaviour in pairwise contests. Cod of the HbI-2*2 genotype displayed a higher level of aggression than other genotypes and this effect was stronger than the population difference. Juvenile cod from the Clyde Sea exhibited a growth rate 24 % higher than those from St Andrews Bay. Cod from the Clyde Sea and from Lowestoft expressed higher growth rates as adults than those from St Andrews Bay. Body size and thus growth appeared to be the main driver of fecundity in the females and body size and liver were the main influences on gonadosomatic index (GSI) in the males. Females from the Clyde Sea invested more into fecundity than those from St Andrews Bay and males from St Andrews Bay had a higher testis investment than those from the Clyde Sea and Lowestoft. Temperature had a large influence on both the juvenile growth and egg development. Growth rate increased linearly and in parallel over the experimental temperatures, within their normal range. Egg development was strongly affected by temperature, resulting in a decrease in hatch time and an increase in embryonic cardiac rate, and a smaller larval size at hatch for a given temperature at higher temperatures. Temperature did not directly influence fecundity or GSI in males but warmer temperatures resulted in higher growth rates and thus a larger body size, which in turn resulted in a greater fecundity or GSI. These differences in life history traits, demonstrated under controlled environment conditions, raises the possibility that there may be a genetic basis to the variation and that cod may be locally adapted to their thermal environments in areas around the British Isles. However, effects of environmental differences prior to capture, including maternal effects, cannot be ruled out. This greater understanding of life history variation in cod will be important in the conservation of phenotypic diversity, vital for the long-term persistence of the species, while the findings of plasticity in response to temperature will enhance predictions of responses to sea temperature rise

Automatic split-generation for the Fukaya category

We prove a structural result in mirror symmetry for projective Calabi--Yau (CY) manifolds. Let $X$ be a connected symplectic CY manifold, whose Fukaya category $\mathcal{F}(X)$ is defined over some suitable Novikov field $\mathbb{K}$; its mirror is assumed to be some smooth projective scheme $Y$ over $\mathbb{K}$ with `maximally unipotent monodromy'. Suppose that some split-generating subcategory of (a $\mathsf{dg}$ enhancement of) $D^bCoh( Y)$ embeds into $\mathcal{F}(X)$: we call this hypothesis `core homological mirror symmetry'. We prove that the embedding extends to an equivalence of categories, $D^bCoh(Y) \cong D^\pi( \mathcal{F}(X))$, using Abouzaid's split-generation criterion. Our results are not sensitive to the details of how the Fukaya category is set up. In work-in-preparation [PS], we establish the necessary foundational tools in the setting of the `relative Fukaya category', which is defined using classical transversality theory.Comment: 24 pages; v2 updated to include arXiv identifiers of papers posted concurrently in bibliograph