Patterns theory and geodesic automatic structure for a class of groups

We introduce a theory of "patterns" in order to study geodesics in a certain class of group presentations. Using patterns we show that there does not exist a geodesic automatic structure for certain group presentations, and that certain group presentations are almost convex.Comment: Appeared in 2003. I am putting all my past papers on arxi

A non-Hopfian almost convex group

In this article we prove that an "isometric multiple HNN-extension" of a group satisfying the falsification by fellow traveler property is almost convex. As a corollary, Wise's example of a CAT(0) non-Hopfian group is Almost convex.Comment: Appeared in 2004. I am putting all my past papers on arxi

Manitest: Are classifiers really invariant?

Invariance to geometric transformations is a highly desirable property of automatic classifiers in many image recognition tasks. Nevertheless, it is unclear to which extent state-of-the-art classifiers are invariant to basic transformations such as rotations and translations. This is mainly due to the lack of general methods that properly measure such an invariance. In this paper, we propose a rigorous and systematic approach for quantifying the invariance to geometric transformations of any classifier. Our key idea is to cast the problem of assessing a classifier's invariance as the computation of geodesics along the manifold of transformed images. We propose the Manitest method, built on the efficient Fast Marching algorithm to compute the invariance of classifiers. Our new method quantifies in particular the importance of data augmentation for learning invariance from data, and the increased invariance of convolutional neural networks with depth. We foresee that the proposed generic tool for measuring invariance to a large class of geometric transformations and arbitrary classifiers will have many applications for evaluating and comparing classifiers based on their invariance, and help improving the invariance of existing classifiers.Comment: BMVC 201

Integration and conjugacy in knot theory

This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups. In chapter 1 we relate the tree part of the Aarhus integral to the mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum. There is a folklore result in physics saying that the Feynman integration of an exponential is itself an exponential. In chapter 2 we state and prove an exact formulation of this statement in the language which is used in the theory of finite type invariants. The final chapter is concerned with properties of link groups. In particular we study the relationship between known solutions from small cancellation theory and normal surface theory for the word and conjugacy problems of the groups of (prime) alternating links. We show that two of the algorithms in the literature for solving the word problem, each using one of the two approaches, are the same. Then, by considering small cancellation methods, we give a normal surface solution to the conjugacy problem of these link groups and characterize the conjugacy classes. Finally as an application of the small cancellation properties of link groups we give a new proof that alternating links are non-trivial.Comment: University of Warwick Ph.D. thesi

The braided Ptolemy-Thompson group T∗T^* is asynchronously combable

The braided Ptolemy-Thompson group $T^*$ is an extension of the Thompson group $T$ by the full braid group $B_{\infty}$ on infinitely many strands. This group is a simplified version of the acyclic extension considered by Greenberg and Sergiescu, and can be viewed as a mapping class group of a certain infinite planar surface. In a previous paper we showed that $T^*$ is finitely presented. Our main result here is that $T^*$ (and $T$) is asynchronously combable. The method of proof is inspired by Lee Mosher's proof of automaticity of mapping class groups.Comment: 45