Joining The Dots - Applied and Computational Topology, sponsored by the LMS and the Joining The Dots programme Seminar

Speaker information

Rachel Jeitziner, EPFL Lausanne. Abstract: In biology and medicine, since sequencing technology is booming, there is a growing need for unbiased clustering methods that work for a wide range of dataset sizes, in an automated fashion and without user-induced bias. To identify distinct subgroups in global gene expression datasets and determine the features differentiating them, we developed a topology-based analytical framework called Two-Tiers Mapper (TTMap). TTMap is unbiased since all parameters are data-driven. The output is given as a colored graph describing distinct clusters, allowing quick detection of outliers and subgroups. We will go through the mathematics underlying this new tool, TTMap, illustrated them with a toy example, and mention some aspects of its stability (work done in collaboration with Steve Oudot and Mathieu Carrière, INRIA). Then, we will provide full comprehension of its utility through real data applications of it. The method is developed as an open source package in R deposited at the Bioconductor.

Mariam Pirashvilli, University of Southampton. Abstract: Topological data analysis is a family of recent mathematical techniques seeking to understand the `shape' of data, and has been used to understand the structure of the descriptor space produced from a standard chemical informatics software from the point of view of solubility. We have used the mapper algorithm, a TDA method that creates low-dimensional representations of data, to create a network visualization of the solubility space. While descriptors with clear chemical implications are prominent features in this space, reflecting their importance to the chemical properties, an unexpected and interesting correlation between chlorine content and rings and their implication for solubility prediction is revealed. A parallel representation of the chemical space was generated using persistent homology applied to molecular graphs. Links between this chemical space and the descriptor space were shown to be in agreement with chemical heuristics.

Ginestra Bianconi, QMUL. Abstract: Simplicial complexes naturally describe discrete topological spaces and when their links are assigned a length they describe discrete geometries. As such simplicial complexes have been widely used in quantum gravity approaches that involve a discretization of spacetime. Recently they are becoming increasingly popular to describe complex interacting systems such a brain networks or social networks. In this talk we present equilibrium and non-equilibrium statistical mechanics approaches to model large simplicial complexes. By extending our knowledge of growing complex networks we propose the simplicial complex model of Network Geometry with Flavor and we explore the hyperbolic nature of its emergent geometry. Finally we reveal the rich interplay between Network Geometry with Flavor and synchronization of coupled oscillators. We show that the skeleton of these simplicial complexes display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network. This result can shed light on the recent experimental finding that neuronal cultures have a dynamics that is strongly dependent on the network geometry and in particular on their dimensionality of the neuronal networks.

Grzegorz Muszynski, Liverpool. Abstract: Massive climate simulation products are produced due to development of high-performance computational technology. Thus, there are many efforts to provide automated methods for post-processing of generated data. We focus on one particular type of such methods, i.e. methods for local detection of weather patterns. Detecting such patterns that very often lead to extreme weather events is a first step in understanding how they may vary under different climate change scenarios. We will present an automated method for the identification of weather events/patterns called Atmospheric Rivers (ARs) in large climate model output. ARs are narrow filaments of concentrated water vapor in the lower part of the troposphere where wind speed can excess 12.5 meters per second. A single AR can carry as much water as the Amazon river. They bring much of the precipitation in many mid-latitude regions. Their presence has been associated with major flooding events in California, United States and Cumbria, United Kingdom. This method adapts an Union-Find algorithm of topological data analysis to extract numerical features of topological descriptors called connected components in 2D scalar field snapshots. The features then feed a supervised machine learning classifier (Support Vector Machine) which performs a binary classification task to identify AR and non-AR patterns. We utilize the parallel toolkit for extreme climate events analysis (TECA: the Toolkit for Extreme Climate Analysis) for comparison (it is assumed that events identified by TECA are ground truth). The obtained results indicate that our approach brings new insight into the study of ARs and provides quantitative information about the relevance of topological feature descriptors in analyses of large climate datasets. We will also show future directions of this work in the context of analysis of atmospheric blocks.