Applied Algebraic Topology 6

27^th May 2016

University of Aberdeen

09:30 – 10:15 Fraser Noble Annex	Coffee in Mathematical Sciences common room
10:15 – 11:15 Fraser Noble 156	Ben Martin (University of Aberdeen) Title: Manipulation and comparison of voxel images Abstract: Medical scans often provide information in terms of a voxel image (voxels are the 3-dimensional analogue of pixels). If the image is of a body organ - say, a brain - then it is reasonable to assume that the voxel image is homeomorphic to a solid ball. I will discuss recent work with Raazesh Sainudiin and Josh Voorkamp on how to store and manipulate such images. We can represent the images using edge-labelled graphs and use graph-theoretic algorithms to tell when two images are the same (up to translation and exchange of co-ordinate axes). The eventual aim is to develop an automated system to detect whether the scanned organ is healthy or diseased by comparing it to a standard database of images. To do this, one needs a way to measure whether two images are similar. This seems to be a difficult problem.
11:30 – 12:30 Fraser Noble 156	Silke Henkes (University of Aberdeen) Title: Applications of Topology in Soft Matter Physics (slides) Abstract: Liquid crystals are classified by their orientational symmetry into nematic, smectic and chiral, equivalent to a line field, parallel sheets, and twisted helical phases respectively. Recently, two new areas have emerged where those symmetries mix with the symmetries of the environment: Liquid crystals on curved surfaces as well as in spherical shells, and active matter, where components move themselves in either polar (vector) or nematic fashion. The main tool in this area are topological considerations like the Poincare-Hopf theorem and the resulting classification of topological defects (aka zeroes of the field). In thermodynamics equilibrium, the Frank free energy can additionally be used to predict the lowest energy topological structure. I will give an introduction to and an overview of these applications of topology.
12:30 – 14:00	Lunch break
14:00 – 15:00 Fraser Noble 156	Piotr Beben (University of Southampton) Title: Cohomology and LS-category of Moment-Angle Complexes (slides) Abstract: Given a simplicial complex $K$ with $n$ vertices, the moment-angle complex $\mathcal{Z}_K$ is a certain subcomplex of $(D^2)^{\times n}$ defined purely in terms of $K$. Despite their simple description, they have connections to several areas of mathematics. At the same time their topology is very intricate. I will discuss their cohomology, Lusternik-Schnirelmann category, and homotopy type, and various mathematical and real-world applications of these.
15:15 – 16:15 Fraser Noble 156	Gregory Lupton (Cleveland State University) Title: Topological Complexity: From Robots to Social Choice Abstract: Suppose the possible states of some system are represented as points in an appropriate topological space---the configuration space of the system.  Then a path in the configuration space corresponds to a transition, through intermediate states of the system, from an initial to a final state.  A continuous assignment of paths to each pair of states is called a motion planner for the system.  Topological Complexity is a numerical homotopy invariant (of the configuration space) that may be viewed as an index of the necessary discontinuity in a motion planner for the system.  In this talk, I will begin with an overview of the basic ideas of the topic.  I will describe some work---joint with Grant and Oprea----that provides useful lower bounds for the Topological Complexity.  I will indicate a possible direction for future work, namely the topic of social choice from mathematical economics.

List of participants: 1. Dave Benson (Aberdeen} 2. Piotr Beben (Southampton) 3. Ben Martin (Aberdeen) 4. Dirk Schuetz (Durham) 5. Mel Chen (Glasgow) 6. John Hubbuck (Aberdeen) 7. Greg Lupton (Cleveland State) 8. Silke Henkes (Aberdeen) 9. Diarmuid Crowley (Aberdeen) 10. David Quinn (Edinburgh) 11. Mark Grant (Aberdeen) 12. Rachael Boyd (Aberdeen) 13. Aaron Tikuisis (Aberdeen) 14. David Recio Mitter (Aberdeen) 15. Ran Levi (Aberdeen) 16. Csaba Nagy (Aberdeen)