Holomorphic disc, spin structures and Floer cohomology of the Clifford torus

We compute the Bott-Morse Floer cohomology of the Clifford torus in \CP^n with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions there exist two spin structures which give non-vanishing Floer cohomology of the Clifford torus, and in even dimensions, there is only one such spin structure. When the Floer cohomology is non-vanishing, it is isomorphic to the singular cohomology of the torus (with a Novikov ring as its coefficients). As a corollary, we prove that any Hamiltonian deformation of the Clifford torus intersects with it at least at $2^n$ distinct intersection points, when the intersection is transversal. We also compute the Floer cohomology of the Clifford torus with flat line bundles on it and verify the prediction made by Hori using a mirror symmetry calculation.Comment: 31 pages, 2 figure

On the obstructed Lagrangian Floer theory

Lagrangian Floer homology in a general case has been constructed by Fukaya, Oh, Ohta and Ono, where they construct an \AI-algebra or an \AI-bimodule from Lagrangian submanifolds, and studied the obstructions and deformation theories. But for obstructed Lagrangian submanifolds, standard Lagrangian Floer homology can not be defined. We explore several well-known cohomology theories on these \AI-objects and explore their properties, which are well-defined and invariant even in the obstructed cases. These are Hochschild and cyclic homology of an \AI-objects and Chevalley-Eilenberg or cyclic Chevalley-Eilenberg homology of their underlying \LI objects. We explain how the existence of $m_0$ effects the usual homological algebra of these homology theories. We also provide some computations. We show that for an obstructed \AI-algebra with a non-trivial primary obstruction, Chevalley-Eilenberg Floer homology vanishes, whose proof is inspired by the comparison with cluster homology theory of Lagrangian submanifolds by Cornea and Lalonde. In contrast, we also provide an example of an obstructed case whose cyclic Floer homology is non-vanishing.Comment: 43 pages, 1 figur

Counting real pseudo-holomorphic discs and spheres in dimension four and six

First, we provide another proof that the signed count of the real $J$-holomorphic spheres (or $J$-holomorphic discs) passing through a generic real configuration of $k$ points is independent of the choice of the real configuration and the choice of $J$, if the dimension of the Lagrangian submanifold $L$ (fixed points set of the involution) is two or three, and also if we assume $L$ is orientable and relatively spin, and $M$ is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the degree of evaluation maps from the moduli spaces of $J$-holomorphic discs. Then, we define the invariant count of discs intersecting cycles of a symplectic manifold at fixed interior marked points, and intersecting real points at the boundary under certain assumptions. The last result is new and was not proved by Welshinger's method.Comment: 18 pages, 2 figures, typo correcte

On the counting of holomorphic discs in toric Fano manifolds

Open Gromov-Witten invariants in general are not well-defined. We discuss in detail the enumerative numbers of the Clifford torus $T^2$ in \CP^2. For cyclic A-infinity algebras, we show that certain generalized way of counting may be defined up to Hochschild or cyclic boundary elements. In particular we obtain a well-defined function on Hochschild or cyclic homology of a cyclic A-infinity algebra, which has invariance property under cyclic A-infinity homomorphism. We discuss an example of Clifford torus $T^2$ and compute the invariant for a specific cyclic cohomology class.Comment: 17 pages, 2 figures,v2: rewritten using the language of cyclic A-infinity algebra, v3: added an example of a cyclic cohomology class, published versio

Abelian Chern-Simons field theory and anyon equation on a cylinder

We present the anyon equation on a cylinder and in an infinite potential wall from the abelian Chern-Simons theory coupled to non-relativistic matter field by obtaining the effective hamiltonian through the canonical transformation method used for the theory on a plane and on a torus. We also give the periodic property of the theory on the cylinder.Comment: 20pages, REVTE

Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds

We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk-deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential. We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta and Ono.Comment: 75 pages, 4 figures. shortened by reducing repetition of construction from manifold case

Towards Automated Safety Coverage and Testing for Autonomous Vehicles with Reinforcement Learning

The kind of closed-loop verification likely to be required for autonomous vehicle (AV) safety testing is beyond the reach of traditional test methodologies and discrete verification. Validation puts the autonomous vehicle system to the test in scenarios or situations that the system would likely encounter in everyday driving after its release. These scenarios can either be controlled directly in a physical (closed-course proving ground) or virtual (simulation of predefined scenarios) environment, or they can arise spontaneously during operation in the real world (open-road testing or simulation of randomly generated scenarios). In AV testing, simulation serves primarily two purposes: to assist the development of a robust autonomous vehicle and to test and validate the AV before release. A challenge arises from the sheer number of scenario variations that can be constructed from each of the above sources due to the high number of variables involved (most of which are continuous). Even with continuous variables discretized, the possible number of combinations becomes practically infeasible to test. To overcome this challenge we propose using reinforcement learning (RL) to generate failure examples and unexpected traffic situations for the AV software implementation. Although reinforcement learning algorithms have achieved notable results in games and some robotic manipulations, this technique has not been widely scaled up to the more challenging real world applications like autonomous driving

Gradient-like vector fields on a complex analytic variety

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimension $m(p)+n-s$ as was conjectured by Goresky and MacPherson.Comment: 23 pages, 4 figure, v2:major revision, v3:added restriction on Morse function and removed fast conditio

Finite group actions on Lagrangian Floer theory

We construct finite group actions on Lagrangian Floer theory when symplectic manifolds have finite group actions and Lagrangian submanifolds have induced group actions. We first define finite group actions on Novikov-Morse theory. We introduce the notion of a {\em spin profile} as an obstruction class of extending the group action on Lagrangian submanifold to the one on its spin structure, which is a group cohomology class in $H^2(G;\Z/2)$. For a class of Lagrangian submanifolds which have the same spin profiles, we define a finite group action on their Fukaya category. In consequence, we obtain the $s$-equivariant Fukaya category as well as the $s$-orbifolded Fukaya category for each group cohomology class $s$. We also develop a version with $G$-equivariant bundles on Lagrangian submanifolds, and explain how character group of $G$ acts on the theory. As an application, we define an orbifolded Fukaya-Seidel category of a $G$-invariant Lefschetz fibration, and also discuss homological mirror symmetry conjectures with group actions.Comment: 81 pages, 12 figures; comments welcome

Orbifold Morse-Smale-Witten complex

We construct Morse-Smale-Witten complex for an effective orientable orbifold. For a global quotient orbifold, we also construct a Morse-Bott complex. We show that certain type of critical points of a Morse function has to be discarded to construct such a complex, and gradient flows should be counted with suitable weights. The homology of these complexes are shown to be isomorphic to the singular homology of the quotient spaces under the self-indexing assumptions.Comment: 35 pages, 6 figures; The content of the last section is replaced by the explanation on weak group actions. Some other minor change