Applied Algebraic Topology 1


23rd January 2015

Queen Mary University of London



09:00 – 10:00


Coffee in the Mathematics Common Room

10:00 – 11:00

ArtsTwo, Room 3.20

Dr Bernd Schulze (University of Lancaster)

Title: Combinatorial characterisations of rigid frameworks (slides)

Abstract: Rigidity Theory is concerned with the study of bar-joint frameworks and related constraint

 systems of geometric objects. This area has a rich history which can be traced back to classical work of Euler, Cauchy and Maxwell on the rigidity of polyhedra and skeletal frames.

The combinatorial side of the theory for generic structures began with Maxwell's counting conditions from 1864, and in 1970 Laman obtained the fundamental result that these necessary conditions are also sufficient for generic rigidity in the Euclidean plane. Combinatorial characterisations of generically rigid graphs in dimension 3 or higher have not yet been found. However, there exist significant partial results for the special classes of body-bar, body-hinge and molecular frameworks. In the first part of this talk I will give a survey of these key results in Combinatorial Rigidity Theory.

In the second part of the talk we will consider the impact of symmetry on the rigidity properties of frameworks. In particular, we present combinatorial characterisations of rigid symmetric frameworks (of various types) which are as generic as possible subject to the given symmetry constraints. This relatively new research area draws on the combinatorics of group-labeled quotient graphs and their associated matroids, symmetry-adapted Maxwell counting, and methods from group representation theory.

11:00 – 11:30

Coffee break

11:30 – 12:30

ArtsTwo, Room 3.20

Dr Vitaliy Kurlin (Durham University and Microsoft Research, Cambridge)

Title: Topological Data Analysis. Applications to Computer Vision (slides)

Abstract: We briefly review Topological Data Analysis defining basic concepts and

explaining the stability of persistence under noise. Then we describe 3 applications to

Computer Vision.

(1) Given only a noisy n-point sample C of a 2D shape, we count topological persistent holes

of C in time O(n log n). The output coincides with the true number of holes in the original

shape for any dense enough sample.

(2) For a noisy sample C of unknown closed contours in the plane, we provably reconstruct

the unknown contours with the same running time O(n log n) without using any extra input

parameters except the sample C.

(3) We introduce a homologically persistent skeleton that depends only on a point cloud C,

optimally extends a minimum spanning tree of C and provably approximates a good enough

graph represented by the cloud C.

12:30 – 14:00

Lunch break

14:00 – 15:00

Laws, Room 1.12

Problem Session

15:30 – 16:30

Laws, Room 1.12

Dr Piotr Beben (University of Southampton)

Title: Configuration Spaces and Polyhedral Products (slides)

Abstract: We use configuration space models for spaces of maps into certain subcomplexes of

product spaces (including polyhedral products) to obtain a single suspension splitting for the

loop space of certain polyhedral products, and show that the summands in these splittings

have a very direct bearing on the topology of polyhedral products, and moment-angle

complexes in particular.


List of participants


1.       Bernd Schulze, Lancaster

2.       Jacek Brodzki, Southampton

3.       Jon Selig, London South Bank

4.       Mark Jerrum, QMUL

5.       Vitaliy Kurlin, Durham and Microsoft

6.       Dirk Schόtz, Durham

7.       Armindo Costa, QMUL

8.       Michael Farber, QMUL

9.       Matthew Burfitt, Southampton

10.   Larry So, Southampton

11.   Mark Grant, Aberdeen

12.   Conor Smyth, Southampton

13.   Hakan Guler, QMUL

14.   Katie Clinch, QMUL

15.   Bill Jackson, QMUL

16.   Alex Fink, QMUL

17.   Hugo Maruri, QMUL

18.   Jelena Grbic, Southampton

19.   Piotr Beben, Southampton

20.   Ingrid Membrillo, Southampton