21^{st} September 2015
Durham University
09:00 – 10:00 CM211 
Assemble in Mathematical Sciences common room 
10:00 – 11:00 CM103 
Armindo Costa (Queen Mary University of London) Title: The fundamental group of a random simplicial complex (slides) Abstract: Random objects often have desirable properties for which explicit examples are hard to construct. For example often random graphs are good expanders and have strong Ramsey properties. The emerging area of Stochastic Topology is an active research area that studies the topology of random simplicial complexes and the asymptotics of topological phase transitions. Several models of random simplicial complexes have been introduced in the last ten years. In this talk we will have a brief overview of models of
random simplicial complexes and focus on a new model that generalizes all
previously studied models. This is the multiparameter model of random
complexes ie simplicial complexes with randomness
in all dimensions. We will discuss in some detail expected properties of the
fundamental group of a multiparameter random complex. By studying the
fundamental group we also obtain a new model of random groups. 
11:00 – 11:15 
Coffee break 
11:15 – 12:15 CM103 
Ian Jermyn (Durham University) Title: Statistics, computer vision, and shape Abstract: Data
analysis is the process of making inferences from data. This involves, explicitly
or implicitly, constructing a probability distribution describing our
knowledge of the unknown quantity given current information, and then
extracting relevant information from it. In computer vision, the current
information includes image data (optical images or video, IR, radar, MRI,...), and inferences concern both geometric properties of
the world, and the identity and activity of `objects'. 
12:15 – 14:00 
Lunch break 
14:00 – 15:00 CM103 
Problem Session 
15:00 – 16:00 CM103 
Ran Levi
(University of Aberdeen) Title: Neural systems from an algebraic topology point of view (slides) Abstract: The brain is without a doubt the most complicated complex system science ever studied. However, at a basic level, the brain, or any part of it, is a network of neurons which can be described as a directed graph. It is also natural to think about connections among various brain regions in graphical terms. Electrical activity in the brain can similarly be viewed as highlighting certain subgraphs of an ambient graph. This approach has been used by theoretical neuroscientist for a while, employing mostly the tools of classical graph theory. The Blue Brain model is an intricate and biologically accurate computer simulation of the neocortical column  a formation of roughly 31,000 simulated neurons. The data used in our study arises from forty two columns in six clusters of seven columns each, generated by the Blue Brain algorithm. The first five clusters are based on biological data extracted from five individual rats, while the sixth cluster is based on the data averaged across the five individuals. In each case the algorithm is ran seven times to create the columns. Like in a biological brain the resulting models are similar, but not identical, as the algorithm is in part stochastic in nature. The simulation allows scientists to examine questions on the model that are intractable by wet lab techniques. In particular, it is very easy to get the entire connectivity matrices of these columns, as well as activate them and obtain information about emergent chemical and electrical properties. Combinatorial and algebraic topology are naturally suited to associating various
invariants and metrics to directed graphs. In this talk I will report on an
ongoing collaboration with the Blue Brain team. In particular I will show how
rather naive techniques of algebraic topology are used to extract useful
information from the system. These techniques can also be used in the study
of neurological fMRI data and other large networks. This project is the
practical, experimental part of a larger initiative which includes a highly
theoretical component. 

List of participants:
1. Dirk Schuetz
(Durham) 2. David Recio Mitter (Aberdeen) 3. Mark Grant (Aberdeen) 4. Ran Levi (Aberdeen) 5. Vitaliy Kurlin (Durham) 6. Pavel Tumarkin
(Durham) 7. William Mycroft (Sheffield) 8. Andrew Lobb
(Durham) 9. Armindo Costa (Queen Mary) 10. Ian Jermyn (Durham) 11. Vaios Ziogas
(Durham) 12. John Hunton
(Durham) 