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

Implementation of Adaptive Neural Networks Controller for NXT SCARA Robot System

Several neural network controllers for robotic manipulators have been developed during the last decades due to their capability to learn the dynamic properties and the improvements in the global stability of the system. In this paper, an adaptive neural controller has been designed with self learning to resolve the problems caused by using a classical controller. A comparison between the improved unsupervised adaptive neural network controller and the P controller for the NXT SCARA robot system is done, and the result shows the improvement of the self learning controller to track the determined trajectory of robotic automated controllers with uncertainties. Implementation and practical results were designed to guarantee online real-time

Growth optimization and structural analysis for ferromagnetic Mn-doped ZnO layers deposited by radio frequency magnetron sputtering

The effect of the deposition temperature on the crystalline quality of (Zn,Mn)O is investigated in thin films prepared by radio frequency magnetronsputtering on c-plane sapphire and GaN substrates. The layers are made of a 0.5μm Mn-doped layer towards the surface on top of a 150nm pure ZnO buffer. Depending on the deposition temperature, the layers can exhibit a columnar structure; the adjacent domains are rotated from one another by 90°, putting [101¯0]and [11¯20] directions face to face. At high Mn concentration the columnar structure is blurred by the formation of Mn rich precipitates. Only one variety of domains is observed at an optimal deposition temperature of 500°C: they are slightly rotated around the [0001] axis (mosaic growth) and bounded by threading dislocations

Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry

We introduce the notion of Wall-Crossing Structure and discuss it in several examples including complex integrable systems, Donaldson-Thomas invariants and Mirror Symmetry. For a big class of non-compact Calabi-Yau 3-folds we construct complex integrable systems of Hitchin type with the base given by the moduli space of deformations of those 3-folds. Then Donaldson-Thomas invariants of the Fukaya category of such a Calabi-Yau 3-fold can be (conjecturally) described in two more ways: in terms of the attractor flow on the base of the corresponding complex integrable system and in terms of the skeleton of the mirror dual to the total space of the integrable system. The paper also contains a discussion of some material related to the main subject, e.g. Betti model of Hitchin systems with irregular singularities, WKB asymptotics of connections depending on a small parameter, attractor points in the moduli space of complex structures of a compact Calabi-Yau 3-fold, relation to cluster varieties, etc.Comment: 111 pages, accepted for Proceedings of the Cetraro Conference "Mirror Symmetry and Tropical Geometry" (Lecture Notes in Mathematics

SYZ mirror symmetry for hypertoric varieties

We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using $T$-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a resolution using the wall and chamber structure of the SYZ base.Comment: v_2: 31 pages, 5 figures, minor revision. To appear in Communications in Mathematical Physic