Ninety Fourth Meeting of the
Transpennine Topology Triangle


School of Mathematics and Statistics

University of Sheffield

Monday 12th January 2015


Supported by the London Mathematical Society 



Everyone who wishes to participate is welcome, particularly postgraduate students. We'll operate the usual criteria for assistance with travel expenses. Beneficiaries will need to complete the standard forms, available from the TTT Homepage.

Programme

All talks will be in room F20 of the Hicks Building. Coffee will be in the common room I15 of the Hicks Building.

11:00 – 11:30
Common Room I15

Tea, Coffee and Biscuits

11:30 – 12:30 

F20

James Cranch
(Sheffield)
Categories in homotopy type theory

14:00 – 15:00 

F20
Mark Bell
(Warwick)
Short proofs of surface homeomorphism properties

15:00 – 16:00
Common Room I15

Tea, Coffee and Biscuits
16:00 –17:00

F20

Vitaliy Kurlin
(Durham and Microsoft Research Cambridge)
Topological Data Analysis. Applications to Computer Vision

There are various places for lunch - follow a group of locals if in doubt.

A group is likely to be going for drinks and an early dinner after the last talk - all are welcome to join.

Abstracts of talks

Mark Bell:  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.

James Cranch:  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.

Vitaliy Kurlin:  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 visualizations 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. The full version of the paper is at
http://kurlin.org/projects/counting-holes-in-noisy-clouds.pdf .

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. The full version of the paper is at http://kurlin.org/projects/auto-completion-closed-contours-full.pdf  .