Moduli of map germs, Thom polynomials and the Green-Griffiths conjecture

This survey paper is based on my IMPANGA lectures given in the Banach Center, Warsaw in January 2011. We study the moduli of holomorphic map germs from the complex line into complex compact manifolds with applications in global singularity theory and the theory of hyperbolic algebraic varieties

M\"obius Invariants of Shapes and Images

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images

Correlation functions, null polygonal Wilson loops, and local operators

We consider the ratio of the correlation function of n+1 local operators over the correlator of the first n of these operators in planar N=4 super-Yang-Mills theory, and consider the limit where the first n operators become pairwise null separated. By studying the problem in twistor space, we prove that this is equivalent to the correlator of a n-cusp null polygonal Wilson loop with the remaining operator in general position, normalized by the expectation value of the Wilson loop itself, as recently conjectured by Alday, Buchbinder and Tseytlin. Twistor methods also provide a BCFW-like recursion relation for such correlators. Finally, we study the natural extension where n operators become pairwise null separated with k operators in general position. As an example, we perform an analysis of the resulting correlator for k=2 and discuss some of the difficulties associated to fixing the correlator completely in the strong coupling regime.Comment: 34 pages, 6 figures. v2: typos corrected and references added; v3: published versio