Induction of Topological Environment Maps from Sequences of Visited Places

In this paper we address the problem of topologically mapping environments which contain inherent perceptual aliasing caused by repeated environment structures. We propose an approach that does not use motion or odometric information but only a sequence of deterministic measurements observed by traversing an environment. Our algorithm implements a stochastic local search to build a small map which is consistent with local adjacency information extracted from a sequence of observations. Moreover, local adjacency information is incorporated to disambiguate places which are physically different but appear identical to the robots senses. Experiments show that the proposed method is capable of mapping environments with a high degree of perceptual aliasing, and that it infers a small map quickly

James bundles

We study cubical sets without degeneracies, which we call {square}-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a {square}-set C has an infinite family of associated {square}-sets Ji(C), for i = 1, 2, ..., which we call James complexes. There are mock bundle projections pi: |Ji(C)| -> |C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of {Omega}(S2). The algebra of these classes mimics the algebra of the cohomotopy of {Omega}(S2) and the reduction to cohomology defines a sequence of natural characteristic classes for a {square}-set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation

Model Checking Spatial Logics for Closure Spaces

Spatial aspects of computation are becoming increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification techniques are well suited to analyse the temporal evolution of programs; however, properties of space are typically not taken into account explicitly. We present a topology-based approach to formal verification of spatial properties depending upon physical space. We define an appropriate logic, stemming from the tradition of topological interpretations of modal logics, dating back to earlier logicians such as Tarski, where modalities describe neighbourhood. We lift the topological definitions to the more general setting of closure spaces, also encompassing discrete, graph-based structures. We extend the framework with a spatial surrounded operator, a propagation operator and with some collective operators. The latter are interpreted over arbitrary sets of points instead of individual points in space. We define efficient model checking procedures, both for the individual and the collective spatial fragments of the logic and provide a proof-of-concept tool

Universal Polynomials for Tautological Integrals on Hilbert Schemes

We show that tautological integrals on Hilbert schemes of points can be written in terms of universal polynomials in Chern numbers. The results hold in all dimensions, though they strengthen known results even for surfaces by allowing integrals over arbitrary "geometric" subsets (and their Chern-Schwartz-MacPherson classes). We apply this to enumerative questions, proving a generalised G\"ottsche Conjecture for all singularity types and in all dimensions. So if L is a sufficiently ample line bundle on a smooth variety X, in a general subsystem P^d of |L| of appropriate dimension the number of hypersurfaces with given singularity types is a polynomial in the Chern numbers of (X,L). When X is a surface, we get similar results for the locus of curves with fixed "BPS spectrum" in the sense of stable pairs theory.Comment: 44 pages, minor changes and correction