11:00 – 11:30 Common Room I15 | Tea, Coffee and Biscuits |
11:30 – 12:30 F20 | (Sheffield) Categories in homotopy type theory |
14:00 – 15:00 F20 | Mark Bell (Warwick) Short proofs of surface homeomorphism properties |
15:00 – 16:00 | Tea, Coffee and Biscuits |
| 16:00 –17:00 F20 | Vitaliy Kurlin (Durham and Microsoft Research Cambridge) Topological Data Analysis. Applications to Computer Vision |
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/
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/