Applied Algebraic Topology 3

 

8th June 2015

University of Aberdeen

09:00 10:00

Fraser Noble Annex

Coffee in Mathematical Sciences common room

10:00 11:00

Fraser Noble 156

Nina Otter (University of Oxford)

Title: A roadmap for the computation of persistent homology (slides)

Abstract: Persistent homology is a method from algebraic topology used in data analysis to study qualitative features of data, which is robust to noise, dimension independent and provides statistical summaries of the outputs. Despite recent progress, the computation of persistent homology for large datasets remains an open problem. We investigate the challenges of computing persistent homology by evaluating the different algorithms and data structures, using the open source implementations currently available and a wide range of synthetic and real world datasets.

11:00 11:30

Fraser Noble 156

Short break for discussions

11:30 12:30

Fraser Noble 156

Subramanian Ramamoorthy (University of Edinburgh)

Title: Topological Trajectory Classification with Persistent Homology (slides)

Abstract: A long-standing and important problem for autonomous robotics is to devise encodings of tasks that span the hierarchy from the quantitative sensorimotor signals to more qualitative task specifications. Despite significant recent activity around methods for learning hierarchical representations, e.g., in the area of computer vision, the problem of defining and learning action-oriented symbols remains open.

Motivated in this way, we address the problem of trajectory classification. We present a sampling-based approach to trajectory classification which enables automated high-level reasoning about topological classes of trajectories. Our approach is applicable to general configuration spaces and relies only on the availability of collision free samples. Unlike previous sampling-based approaches in robotics which use graphs to capture information about the path-connectedness of a configuration space, we construct a multiscale approximation of neighborhoods of the collision free configurations based on filtrations of simplicial complexes. Our approach thereby extracts additional homological information which is essential for a topological trajectory classification. We propose a multiscale classification algorithm for trajectories in configuration spaces of arbitrary dimension and for sets of trajectories starting and ending in two fixed points. Using a cone construction, we then generalize this approach to classify sets of trajectories even when trajectory start and end points are allowed to vary in path-connected subsets. We furthermore show how an augmented filtration of simplicial complexes based on an arbitrary function on the configuration space, such as a costmap, can be defined to incorporate additional constraints. We present an evaluation of our approach in 2-, 3-, 4- and 6-dimensional configuration spaces in simulation and in real-world experiments using a Baxter robot and motion capture data.

We view this work as a step towards a broader family of algorithms that maintain beliefs and plan actions in a multiscale fashion. I will conclude my talk with a discussion on this, in the process giving a high level outline of some related algorithms from our recent work addressing interactive decision making.

12:30 14:00

Lunch break

14:00 15:00

Fraser Noble 156

Problem Session

15:00 16:00

Fraser Noble 156

David Quinn (University of Aberdeen)

Title: Finite domination and combinatorial Novikov completions

Abstract: Joint work with Thomas Huttemann. We say a cochain complex C of modules over a Laurent polynomial ring in several indeterminates is finitely dominated over the ground ring R if it is a homotopy retract of a bounded cochain complex of finitely generated free R-modules. 

In the talk we discuss `Novikov' completions of the Laurent ring, where, given a polytope P, we associate a completion of the Laurent ring to each flag of faces of P. We can then show that if C, upon tensoring with each of these completions, becomes an acyclic complex then C is finitely dominated. The reverse implication also holds but uses different techniques.

 

List of participants:

 

1.       Mark Grant, Aberdeen

2.       Jarek Kędra, Aberdeen

3.       Rachael Boyd, Aberdeen

4.       Richard Hepworth, Aberdeen

5.       David Quinn, Aberdeen

6.       Subramanian Ramamoorthy, Edinburgh

7.       Zur Izhakian, Aberdeen

8.       John Hubbuck, Aberdeen

9.       Diarmuid Crowley, Aberdeen

10.   Brendan Owens, Glasgow

11.   Assaf Libman, Aberdeen

12.   Hassan Hamdoun, Aberdeen

13.   Nina Otter, Oxford

14.   Adriana Marciuk, Aberdeen

15.   Mark Paulin, Aberdeen