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

Weak Sequential Composition in Process Algebras

n this paper we study a special operator for sequential composition, which is defined relative to a dependency relation over the actions of a given system. The idea is that actions which are not dependent (intuitively because they share no common resources) do not have to wait for one another to proceed, even if they are composed sequentially. Such a notion has been studied before in a linear-time setting, but until recently there has been no systematic investigation in the context of process algebras. We give a structural operational semantics for a process algebraic language containing such a sequential composition operator, which shows some interesting interplay with choice. We give a complete axiomatisation of strong bisimilarity and we show consistency of the operational semantics with an event-based denotational semantics developed recently by the second author. The axiom system allows to derive the communication closed layers law, which in the linear time setting has been shown to be a very useful instrument in correctness preserving transformations. We conclude with a couple of examples

Introduction: new research in monetary history - A map

This handbook aims to provide a comprehensive (though obviously not exhaustive) picture of state-of-the-art international scholarship on the history of money and currency. The chapters of this handbook cover a wide selection of research topics. They span chronologically from antiquity to nowadays and are geographically stretched from Latin America to Asia, although most of them focus on Western Europe and the USA, as a large part of the existing research does. The authors of these chapters constitute, we hope, a balanced sample of various generations of scholars who contributed to what Barry Eichengreen defined as "the new monetary and financial history" – an approach that combines the analysis of monetary aggregates and policies with the structure and dynamics of the banking sector and financial markets. We have structured this handbook in ten broad thematic parts: the historical origins of money; money, coinage, and the state; trade, money markets, and international currencies; money and metals; monetary experiments; Asian monetary systems; exchange rate regimes; monetary integration; central banking and monetary policy; and aggregate price shocks. In this introduction, we offer for each part some historical context, a few key insights from the literature, and a brief analytical summary of each chapter. Our aim is to draw a map that hopefully will help readers to organize their journey through this very wide and diverse research area

Slicing Object-Z Specifications for Verification

Abstract. Slicing is the activity of reducing a program or a specification with respect to a given condition (the slicing criterion) such that the condition holds on the full program if and only if it holds on the reduced program. Originating from program analysis the entity to be sliced is usually a program and the slicing criterion a value of a variable at a certain program point. In this paper we present an approach to slicing Object-Z specifications with temporal logic formulae as slicing criteria and show the correctness of our approach. The underlying motivation is the goal to substantially reduce the size of the specification and subsequently facilitate verification of temporal logic properties.

The late Cretaceous Klepa basalts in Macedonia (FYROM) - constraints on the final stage of tethys closure in the balkans

The waning stage(s) of the Tethyan ocean(s) in the Balkans are not well understood. Controversy centres on the origin and life-span of the Cretaceous Sava Zone, which is allegedly a remnant of the last oceanic domain in the Balkan Peninsula, defining the youngest suture between Eurasia- and Adria-derived plates. In order to investigate to what extent late-Cretaceous volcanism within the Sava zone is consistent with this model, we present new age data together with trace-element and Sr–Nd–Pb isotope data for the Klepa basaltic lavas from the central Balkan Peninsula. Our new geochemical data show marked differences between the Cretaceous Klepa basalts (Sava Zone) and the rocks of other volcanic sequences from the Jurassic ophiolites of the Balkans. The Klepa basalts mostly have Sr–Nd–Pb isotopic and trace-element signatures that resemble enriched within-plate basalts, substantially different from Jurassic ophiolite basalts with MORB, BAB and IAV affinities. Trace-element modelling of the Klepa rocks indicates 2–20% polybaric melting of a relatively homogeneously metasomatised mantle source that ranges in composition from garnet lherzolite to ilmenite+apatite bearing spinel–amphibole lherzolite. Thus, the residual mineralogy is characteristic of a continental rather than oceanic lithospheric mantle source, suggesting an intracontinental within-plate origin for the Klepa basalts. Two alternative geodynamic models are internally consistent with our new findings: i) if the Sava Zone represents remnants of the youngest Neotethyan Ocean, magmatism along this zone would be situated within the forearc region and triggered by ridge subduction; ii) if the Sava Zone delimits a diffuse tectonic boundary between Adria and Europe, which had already collided in the Late Jurassic, the Klepa basalts together with a number of other magmatic centres represent volcanism related to transtensional tectonics