TTT94: Sheffield
School of Mathematics and Statistics, the University of Sheffield.
Supported by the London Mathematical Society.
Date
12 January 2015
Speakers
James Cranch (Sheffield)
Categories in homotopy type theory
Categories in homotopy type theory
Abstract: Homotopy type theory is a new set of ideas, currently under vigorous development. From the point of view of pure mathematics, it offers an novel foundation for homotopy theory, with the intriguing feature that spaces are more fundamental than sets. I'll introduce the ideas involved in a manner geared up for topologists, and speculate on the problems of defining categories in this new theory.
Mark Bell (Warwick)
Short proofs of surface homeomorphism properties
Short proofs of surface homeomorphism properties
Abstract: We will look at how difficult it is to compute various properties of surface homeomorphisms. We will mainly focus on deciding whether a given map is 'reducible', that is, fixes a simple closed loop. If there is such a loop then we can reduce studying the map to studying how it acts on the the complement of the loop, a simpler surface. In this case, such a loop acts as an easy to check proof that the map is reducible.
We will discuss ideas using train tracks, initially suggested by Agol and Thurston, for the irreducible case to again show that an easy to check proof exists.
Vitaliy Kurlin (Durham and Microsoft Research Cambridge)
Topological data analysis: Applications to computer vision
Topological data analysis: Applications to computer vision
Abstract: Topological data analysis is a new research area on the interface between algebraic topology, computational geometry, machine learning and statistics. The key aims are to efficiently represent real-life shapes and to measure shapes by using topological invariants such as homology groups.
The usual input is a big unstructured point cloud, which is a finite metric space. The desired outputs are persistent topological structures hidden in the cloud. The flagship method is persistent homology describing the evolution of homology classes in the filtration on data points over all possible scales. After reviewing basic concepts and results, we consider two applications.
The first result was presented at CVPR 2014: Computer Vision and Pattern Recognition. We study the problem of counting holes in noisy 2D clouds. Such clouds emerge as visualisations of high-dimensional data through so-called delay embeddings.
We design a fast algorithm to find the probability distribution for the number of holes in the given cloud. We prove theoretical guarantees when the algorithm outputs the true number of holes of an unknown shape approximated by a noisy sample.
Full version of the paper (PDF, 1.6MB)
The second result was presented at CTIC 2014: Computational Topology in Image Context. We extend the previous approach to auto-complete closed contours in unstructured 2D clouds. The new algorithm requires more complicated data structures to maintain adjacency relations of persistent regions, but has the same time O(n log n) and space O(n) for any n points in the plane.
For a noisy sample C of a 'good' graph G, the algorithm correctly finds all contours of the graph within a small neighborhood of G using only the cloud C without any extra input parameters.