Applied Algebraic Topology 8

Monday 21^st November 2016

University of Southampton

Ketley Room, Mathematics (Building 54)

10:00 – 11:00	Coffee
11:00-11:45	Paweł Dłotko (INRIA Orsay) Title: Topological data analysis - from diagrams to information, from information to knowledge. (slides) Abstract: In this talk I will discuss various techniques in topological data analysis which are based on persistent homology. After a mild introduction to persistence, by using simple examples we will show various ways to turn persistence diagrams into a useful tool in statistics and machine learning. During this part of the talk, I will give a demonstration of a recently developed software package for topological statistics. At the end, I will show a few examples where the presented techniques are used in practice: we will discuss biological analysis of various types of embedded trees, and a couple of problems from material sciences.
12:00 – 12:30	Francisco “Kiko” Belchí Guillamón (University of Southampton) Title: Understanding respiratory diseases through the use of persistent homology. (slides) Abstract: In a work in collaboration with the Southampton Respiratory Biomedical Research Unit we aim at better diagnosing and treating pulmonary diseases. In this talk I will explain how, as a preliminary step, we are using topological tools to figure out the relation between some respiratory diseases and how the bronchi bend and occupy space in our lungs.
12:30 – 14:15	Break for Lunch and Discussions
14:15 – 15:00	Matteo Rucco (CNR/IMATI, Genova) Title: Topological data analysis and formal methods in computer science for modeling complex systems. Abstract: In this talk I report on a new methodology for modelling complex systems. The methodology can be considered the first characterization of the S[B] paradigm and it is based on the following pillars: formal methods in computer science, computational topology, information theory, and automata theory. The S[B] paradigm represents a complex system by two entangled levels: the structural S level and the behavioural B level. In terms of data volume a complex system can be associated to a big heterogeneous dataset, and in order to extract from it useful information new techniques have been recently introduced, most of them in the area of Topological Data Analysis (TDA). TDA is a sub-area of computational topology that develops and applies topological based techniques for achieving robust analysis of scientific data. TDA basically performs the construction and the analysis of a topological space from data. TDA geometrically represents a dataset D with a family of simplicial complexes C, that are obtained from D by completion, namely constructs the simplicial complex C which has D as 1-skeleton (scaffold). The features of this new topological space are extracted by computing persistent homology. We used jHoles, a java high performance tool, that implements the Clique Weight Rank Persistent Homology algorithm for computing persistent homology. The dynamics of a complex system can be modelled by using measures that are formally defined in information theory, e.g. entropy, etc… .We defined a new entropy, the so-called persistent entropy, that is based on the persistent bar-code. The newly defined Persistent Entropy Automaton is the formal model used for representing the dynamics of a complex systems within the S level of the S[B] paradigm. Complex systems have the attitude to execute simultaneously two or more actions against same resources. This feature is known in computer science as concurrency. A new runtime verification is defined for dealing with the formal verification of S[B] concurrent models. We successfully applied the methodology to a real case study: a network-based model of the Immune System, the so-called Idiotypic Network. The methodology is able to identify when the system performs the immunization against antigens and which are the optimal executions for reaching the immune memory.
15:15 – 16:00	Conrad D’Souza (University of Southampton) Title: Applying HodgeRank to Predict the Outcome of Competitive Events - A Case Study of Horse Racing (slides) Abstract: In this talk, we will explore how topology can be applied to past performance data of horse races to make profitable predictions. We employ HodgeRank, a topologically-inspired ranking algorithm for pairwise comparison data between alternatives which is particularly useful for incomplete and inconsistent data, to measure the underlying quality of horses. We can construct a simplicial complex which represents the relative performances of pairs of horses. By applying a discrete version of the Hodge Decomposition Theorem to our complex, we can separate the dataset into consistent and inconsistent data (in other words cycles). From the consistent data, we can measure the underlying quality of horses with the induced pairwise comparisons minimising the weighted square error with respect to the observed comparisons. These measures of quality are fed into a conditional logit model to calculate the probability of each horse winning a race. We show that these measurements improve the accuracy and profitability of the predictions made by the betting market.

List of participants: Will appear here.