Simple examples of distinct Liouville-type symplectic structures

We discuss some examples of open manifolds which admit non-isomorphic symplectic structures of Liouville type.Comment: v2: one remark added, minor change

Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry

We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.Comment: v2: slightly expanded expositio

Exact Lagrangian submanifolds in simply-connected cotangent bundles

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler, using a different approach.Comment: 28 pages, 3 figures. Version 2 -- derivation and discussion of the spectral sequence considerably expanded. Other minor change

THz-range free-electron laser ESR spectroscopy: techniques and applications in high magnetic fields

The successful use of picosecond-pulse free-electron-laser (FEL) radiation for the continuous-wave THz-range electron spin resonance (ESR) spectroscopy has been demonstrated. The combination of two linac-based FELs (covering the wavelength range of 4 - 250 $\mu$m) with pulsed magnetic fields up to 70 T allows for multi-frequency ESR spectroscopy in a frequency range of 1.2 - 75 THz with a spectral resolution better than 1%. The performance of the spectrometer is illustrated with ESR spectra obtained in the 2,2-diphenyl-1-picrylhydrazyl (DPPH) and the low-dimensional organic material (C$_6$H$_9$N$_2$)CuCl$_3$.Comment: 9 pages, 9 figures. Rev. Sci. Instrum., accepte

A beginner's introduction to Fukaya categories

The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology. This text is based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Universit\'e de Nantes in June 2011.Comment: 42 pages, 13 figure

Bimodules in bordered Heegaard Floer homology

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of A(F). We also prove a duality theorem relating the two versions of the 3-manifold invariant. Finally, in the case of a genus one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology.Comment: 153 pages, 29 figures; v4: Address referee comment

Fructose Alters Cell Survival and Gene Expression in Microglia and Neuronal Cells Lines

Purpose: Microglia are macrophages that are found primarily in the CNS and play a crucial role in maintaining a healthy brain by engulfing invading microorganisms, releasing inflammatory mediators, and pruning dead cells. Microglia can become activated in response to certain stimuli which causes them to transition into a pro-inflammatory state, and can sometimes become chronically activated which can result in neuronal damage. Studies have shown a causal relationship between this activation and sugars such as fructose and glucose. We sought to understand the role of sugars in microglial activation and the subsequent effects on neuron health. Methods: Rat microglia (HAPI) and neuronal (B35) cell lines were treated with varying concentrations of fructose (25 mM, 12.5 mM, and 6.25 mM) or glucose (25 mM and 12.5 mM)as a positive control to determine their effects on the cells. Following treatment and incubation for 3 or 24 hours, the cells were analyzed using an MTT assay to measure cell survival or real-time polymerase chain reaction (RT-PCR) to measure gene expression levels. Effects of fructose were measured in HAPI microglia after direct treatment with the sugar. The genes investigated by the RT-PCR in the HAPI cells included: glucose transporter 5 (GLUT5), and the inflammatory markers high mobility group box 1 (HMGB1), and prostaglandin E receptor 2 (Ptger2). To evaluate the effects of microglial activation on neuronal function, the B35 neurons were treated either directly with sugars or with the supernatant collected from fructose-treated HAPI microglia. This allows examination of the effects of soluble neuron-injury factors released by microglia. The genes investigated by RT-PCR in B35 neurons included nuclear factor-κB (NFκB) and enolase 2 (Eno2). Results: Cell survival assays showed that 24-hour direct fructose treatment increased B35 cell survival by up to 13%, while groups treated with microglia supernatant increased cell survival by up to 33%. In HAPI microglia, 3 hours of treatment with fructose caused GLUT5 expression to be suppressed by up to 32% in all treatment groups except for 6.25 mM fructose, while Ptger2 and HMGB1 expression was increased by as much as 65% and 15%, respectively. After 24-hours of treatment with fructose, the HAPI microglia showed a maximum of 80% increased expression of HMGB1, while Ptger2 expression was mostly unchanged. In B35 neurons, 3 hours of treatment with fructose caused a decrease of up to 26% in NFκB and an increase of up to 46% in Eno2 expression. Conclusion: Cell survival results indicate that the microglia may provide a short term protective effect on the B35 neurons. However, data from the gene expression assays show evidence of cellular dysfunction in neurons and pro-inflammatory activity in microglia which may lead to neuronal death on a longer timeline. As seen in the gene expression results, microglia had increased expression of pro-inflammatory genes and B35 neuronal cells had increased expression of markers of cellular damage. Future studies will further explore the effects of fructose on expression of other genes and examine the effects on neuron survival at later time points