What do we use metrics on persistent modules for? Is it only to asure stability of some constructions?
In my talk I will describe why I care about such metrics, show how to construct a rich space of them and illustrate how to use
them for analysis.
We will consider modeling shapes and fields via topological and lifted-topological transforms.
Specifically, we show how the Euler Characteristic Transform and the Lifted Euler Characteristic Transform can be used in practice for statistical analysis of shape and field data. The Lifted Euler Characteristic is an alternative to the. Euler calculus developed by Ghrist and Baryshnikov for real valued functions. We also state a moduli space of shapes for which we can provide a complexity metric for the shapes. We also provide a sheaf theoretic construction of shape space that does not require diffeomorphisms or correspondence. A direct result of this sheaf theoretic construction is that in three dimensions for meshes, 0-dimensional homology is enough to characterize the shape.
Pablo G. Cámara is an Assistant Professor of Genetics at the University of Pennsylvania and a faculty member of the Penn Institute for Biomedical Informatics. He received a Ph.D. in Theoretical Physics in 2006 from Universidad Autónoma de Madrid. He performed research in string theory for several years, with postdoctoral appointments at Ecole Polytechnique, the European Organization for Nuclear Research (CERN), and University of Barcelona. Fascinated by the extremely interesting and fundamental open questions in biology, in 2014 he shifted his research focus into problems in quantitative biology, and joined the groups of Dr. Rabadan, at Columbia University, and Dr. Levine, at the Institute for Advanced Study (Princeton). Building upon techniques from applied topology and statistics, he has devised novel approaches to the inference of ancestral recombination, human recombination mapping, the study of cancer heterogeneity, and the analysis of single-cell RNA-sequencing data from dynamic and heterogeneous cellular populations.
One of the prevailing paradigms in data analysis involves comparing groups of samples to statistically infer features that discriminate them. However, many modern applications do not fit well into this paradigm because samples cannot be naturally arranged into discrete groups. In such instances, graph techniques can be used to rank features according to their degree of consistency with an underlying metric structure without the need to cluster the samples. Here, we extend graph methods for feature selection to abstract simplicial complexes and present a general framework for clustering-independent analysis. Combinatorial Laplacian scores take into account the topology spanned by the data and reduce to the ordinary Laplacian score when restricted to graphs. We show the utility of this framework with several applications to the analysis of gene expression and multi-modal cancer data. Our results provide a unifying perspective on topological data analysis and manifold learning approaches to the analysis of point clouds.
In this talk I will present four case studies of sheaves and cosheaves in topological data analysis. The first two are examples of (co)sheaves in the small:
(1) level set persistence---and its efficacious computation via discrete Morse theory---and,
(2) decorated merge trees and Reeb graphs---enriched topological invariants that have enhanced classification power over traditional TDA methods. The second set of examples are focused on (co)sheaves in the large:
(3) understanding the space of merge trees as a stratified map to the space of barcodes and
(4) the development of a new "sheaf of sheaves" that organizes the persistent homology transform over different shapes.
Oxford Mathematician Álvaro Cartea has been appointed as the new Director of the Oxford-Man Institute of Quantitative Finance (OMI), a world-leading academic research institute at the University of Oxford that specialises in machine learning and data analytics within quantitative finance.
The SI model is the most basic of all compartmental models used to describe the spreading of information through a population. In this talk we will present a mathematical formalism to solve the SI model on generic networks. Our methods rely on a tensor product formulation of the dynamical spreading process, inspired by many-body quantum systems. Here we will focus on time-dependent expectation values for the state of individual nodes, which can be obtained from contributions of subgraphs of the network. We show how to compute these contributions systematically and derive a set of symmetry relations among subgraphs of differing topologies. We conclude by comparing our results for small sample networks to Monte-Carlo simulations and mean-field approximations.
arXiv link: https://arxiv.org/abs/2109.03530
The coach departs the Andrew Wiles Building @ 8am - to University of Cambridge. Returning from Cambridge at 18:30.
The Woolly Owl is a day of short research talks by early career applied mathematics researchers at Oxford and Cambridge, showcasing the outstanding research of the two universities. But there’s a twist: over the course of the day the seven speakers from each side will also be competing as a team to win the coveted - and literal - Woolly Owl trophy.
If you wish to attend please email: @email
Places are limited, so first come, first served.
This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.
Candida Bowtell
Title: Chess puzzles: from recreational maths to fundamental mathematical structures
Abstract:
Back in 1848, in a German chess magazine, Max Bezzel asked how many ways there are to place 8 queens on a chessboard so that no two queens can attack one another. This question caught the attention of many, including Gauss, and was subsequently generalised. What if we want to place n non-attacking queens on an n by n chessboard? What if we embed the chessboard on the surface of a torus? How many ways are there to do this? It turns out these questions are hard, but mathematically interesting, and many different strategies have been used to attack them. We'll survey some results, old and new, including progress from this year.
Joshua Bull
Title: From Cancer to Covid: topological and spatial descriptions of immune cells in disease
Abstract:
Advances in medical imaging techniques mean that we have increasingly detailed knowledge of the specific cells that are present in different diseases. The locations of certain cells, like immune cells, gives clinicians clues about which treatments might be effective against cancer, or about how the immune system reacts to a Covid infection - but the more detailed this spatial data becomes, the harder it is for medics to analyse or interpret. Instead, we can turn to tools from topological data analysis, mathematical modelling, and spatial statistics to describe and quantify the relationships between different cell types in a wide range of medical images. This talk will demonstrate how mathematics can be used as a tool to advance our understanding of medicine, with a focus on immune cells in both cancer and covid-19.
This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.
How can we make maths more accessible, promote its many applications, and encourage more women to enter the field? These are the questions we aim to address with Mathematigals.
Caoimhe Rooney and Jessica Williams met in 2015 at the start of their PhDs in mathematics in Oxford, and in 2020, they co-founded Mathematigals. Mathematigals is an online platform producing content to demonstrate fun mathematical curiosities, showcase ways maths can be used in real life, and promote female mathematicians. Mathematigals primarily produces animated videos that present maths in a way that is engaging to the general public.
In this session, Jess and Caoimhe will talk about their initial motivation to begin Mathematigals, demonstrate the process behind their content creation, and describe their future visions for the platform. The session will end with an opportunity for the audience to provide feedback or ideas to help Mathematigals on their journey to encourage future mathematicians.
This session will take place live in L1 and online. A Teams link will be shared 30 minutes before the session begins.
