Some speculations on pairs-of-pants decompositions and Fukaya categories

This is partly a survey and partly a speculative article, concerning a particular question about Fukaya categories of symplectic manifolds. Namely, can we decompose a symplectic manifold into standard pieces, and then reconstruct its Fukaya category by gluing together categories depending only on the geometry of each piece, in a (loosely understood) sheaf-theoretic way? For this to work, some degree of control over pseudo-holomorphic curves is required, an issue which depends on the geometry of the decomposition under consideration. Sheaf-theoretic ideas have been successfully applied to the symplectic geometry of cotangent bundles, starting with the work of Fukaya-Oh [17], and followed by Kasturirangan-Oh [26, 39] and Nadler-Zaslow [35,36] (for a survey of the last-mentioned work and related ideas of Fukaya-Smith, see [19]). Recently, Kontsevich [28] has proposed a generalization to Stein manifolds whose Lagrangian skeleta have certain singularities. However, that is not quite the direction we wish to take here

Fukaya A∞-structures associated to Lefschetz fibrations. II 1/2

We consider a version of the relative Fukaya category for anticanonical Lefschetz pencils. There are direct connections between the behaviour of this category and enumerative geometry: some of these are results announced here, others remain conjectural. Among the conjectural goals is a formula for the dependence of the Fukaya category of Calabi-Yau hypersurfaces on the Novikov (area) parameter

The equivariant pair-of-pants product in fixed point Floer cohomology

We use equivariant methods and product structures to derive a relation between the fixed point Floer cohomology of an exact symplectic automorphism and that of its square.National Science Foundation (U.S.) (Grant DMS-1005288)Simons Foundation (Simons Investigator grant)Radcliffe Institute for Advanced Stud

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements