The goal of topological data analysis is to apply tools form algebraic topology to reveal geometric structures hidden within high dimensional data. Mapper is among its most widely and successfully applied tools providing, a framework for the geometric analysis of point cloud data. Given a number of input parameters, the Mapper algorithm constructs a graph, giving rise to a visual representation of the structure of the data. The Mapper graph is a topological representation, where the placement of individual vertices and edges is not important, while geometric features such as loops and flares are revealed.
However, Mappers method is rather ad hoc, and would therefore benefit from a formal approach governing how to make the necessary choices. In this talk I will present joint work with Francisco Belchì, Jacek Brodzki, and Mahesan Niranjan. We study how sensitive to perturbations of the data the graph returned by the Mapper algorithm is given a particular tuning of parameters and how this depend on the choice of those parameters. Treating Mapper as a clustering generalisation, we develop a notion of instability of Mapper and study how it is affected by the choices. In particular, we obtain concrete reasons for high values of Mapper instability and experimentally demonstrate how Mapper instability can be used to determine good Mapper outputs.
Our approach tackles directly the inherent instability of the choice of clustering procedure and requires very few assumption on the specifics of the data or chosen Mapper construction, making it applicable to any Mapper-type algorithm.
- Topological Data Analysis Seminar
Would you like to meet some of your fellow students, and some graduate students and postdocs, in an informal and relaxed atmosphere, while building your communication skills? In this Friday@2 session, you'll be able to play a selection of board games, meet new people, and practise working together. What better way to spend the final Friday afternoon of term?! We'll play the games in the south Mezzanine area of the Andrew Wiles Building, outside L3.
Arctic sea ice is one of the most sensitive components of the Earth’s climate system. The underlying ocean plays an important role in the evolution of the ice cover through its heat flux at the ice-ocean interface. Despite its importance, the spatio-temporal variations of this heat flux are not well understood. In this talk, I will take the following approach to study the variations in the heat flux. First, I will consider the problem of classical Rayleigh-Bénard convection and systematically explore the effects of fractal boundaries on heat transport using direct numerical simulations. And second, I will analyze time-series data from the Surface Heat Budget of the Arctic Ocean (SHEBA) program using Multifractal Detrended Fluctuation Analysis (MFDFA) to understand the nature of fluctuations in the heat flux. I will also discuss developing simple stochastic ODEs using results from these studies.
- Mathematical Geoscience Seminar
This challenge relates to problems (of a mathematical nature) in generating optimal solutions for natural flood management. Natural flood management involves large numbers of small scale interventions in a much larger context through exploiting natural features in place of, for example, large civil engineering construction works. There is an optimisation problem related to the catchment hydrology and present methods use several unsatisfactory simplifications and assumptions that we would like to improve on.
- Industrial and Interdisciplinary Workshops
- Mathematical Biology and Ecology Seminar
Stevenhagen conjectured that the density of d such that the negative Pell equation x^2-dy^2=-1 is solvable over the integers is 58.1% (to the nearest tenth of a percent), in the set of positive squarefree integers having no prime factors congruent to 3 modulo 4. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we use a recent breakthrough of Smith to prove that the infimum of this density is at least 53.8%, improving previous results of Fouvry and Klüners, by studying the distribution of the 8-rank of narrow class groups of quadratic number fields.
- Number Theory Seminar
The problem of a bubble moving steadily in a Hele-Shaw cell goes back to Taylor and Saffman in 1959. It is analogous to the well-known selection problem for Saffman-Taylor fingers in a Hele-Shaw channel. We apply techniques in exponential asymptotics to study the bubble problem in the limit of vanishing surface tension, confirming previous numerical results, including a previously predicted surface tension scaling law. Our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane.
- Industrial and Applied Mathematics Seminar
- String Theory Journal Club
I will present an analysis of a continuous version of the compressed sensing problem, where the l^1 norm is replaced by the total variation of measures, and one aims to recover the positions and amplitudes of Dirac masses. We show that provided that the Diracs are sufficiently separated under a Fisher metric (which accounts for the geometry of the problem), stable recovery can be achieved when the number of random samples scales linearly with sparsity (up to log factors). This is joint work with Nicolas Keriven and Gabriel Peyre.
- Computational Mathematics and Applications Seminar