Applied Algebraic Topology 4

 

21st 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 multi-parameter 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 multi-parameter 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'.

Many of the key problems involve shape, for example `segmentation': the delineation of a specified entity in the image. To make these inferences, we therefore need methods for the representation of shape; probabilistic models of shape that can be adapted to different circumstances; and algorithms capable of making efficient inferences from these models.

In this talk, I will first outline the general inference problem, and describe particular instances that occur in computer vision and image processing. I will then focus on problems involving shape. I will survey briefly various approaches to shape description and modelling, and then focus on two: classical statistical shape modelling, which involves the construction of `shape spaces'; and a complementary approach using random fields capable of describing shape families in which the individual shapes may have very different topologies.

I will describe applications of these approaches to the segmentation of road networks from satellite and aerial images, cells from biomedical images, and to the diagnosis of ADHD from MRI scans.

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)