20th meeting, Oxford

Speakers (confirmed):

Maryam Garba, Northumbria University

Jeff Giansiracusa, Durham University

Miruna Stefana Sorea, Lucian Blaga University of Sibiu

Georgy Scholten, Max Plank Institute of Molecular Cell Biology and Genetics

Sean Dewar, University of Bristol

Abhinav Natarajan, University of Oxford

Date:

November 28-29, 2024

Location:

In person at the Andrew Wiles Building N4.01, Mathematical Institute, University of Oxford

Local organiser:

Gillian Grindstaff, University of Oxford

Registration :

If you wish to attend the meeting, please register by sending an e-mail 📬 to Gillian Grindstaff (grindstaff (at) maths.ox.ac.uk) including any dietary restrictions

Funding:

We have some funding to support the travel costs of PhD students/early career researchers who wish to attend the meeting. Please send an e-mail 📬 to Gillian Grindstaff 📮

Schedule (up to speaker permutation):

Thursday 28th November 2024

10:30-11:00 Welcome and coffee

11:00-12:00 Sean Dewar

12:00-13:00 Georgy Scholten

13:00-14:30 Lunch

14:30-15:30 Jeff Giansiracusa

15:30-16:30 Abhinav Natarajan

18:30 Dinner at Royal Oak

Friday 29th November 2024

9:30-10:00 Morning Coffee

10:00-11:00 Stefana Sorea

11:00-12:00 Maryam Garba

12:00-14:00 Excursion to the Christmas market and Gloucester Green

Title and Abstract collection

Speaker: Sean Dewar

Title: How to count realisations of rigid graphs

Abstract: A graph is d-rigid if, given any generic positioning of its vertices in d-dimensional Euclidean space, there are finitely many other realisations of the graph in d-dimensional Euclidean space (modulo isometries) with the same length edges. Combinatorial characterisations for 1-rigidity (i.e., connectivity) and 2-rigidity (Geiringer-Laman theorem) are known, but it is currently an open problem for d > 2. Interestingly, these combinatorial characterisations remain the same if we switch the geometry we embed into for either spherical or hyperbolic geometries.
My talk will be a survey of results involving a variation of this problem: given a d-rigid graph with a generic realisation, how many other realisations of the graph exist with the same length edges (now allowing for complex realisations also)? For example, given the 2-rigid graph formed by gluing two triangles at an edge, every generic realisation in the plane has exactly one other edge-length equivalent realisation that is formed by flipping one of the triangles. We can similarly consider the realisation number of a graph with respect to spherical and hyperbolic geometries, with the former having close links with the study of cross-ratio degrees in algebraic geometry.
I shall prove that for realisation numbers, the geometry does matter, and in fact the flat Euclidean space realisation number is a lower bound for the realisation number in spherical and hyperbolic geometries. I will also cover recent research of myself and others which uncovered a deep link between the 2-dimensional realisation numbers, tropical geometry and Tutte polynomials. This work is the culmination of multiple projects with Georg Grasegger, Joseph Schicho, Audie Warren and Ayush Kumar Tewari (RICAM), Oliver Clarke (Edinburgh), Daniel Green-Tripp and James Maxwell (Bristol), Anthony Nixon and Ben Smith (Lancaster), and Yue Ren (Durham).

Speaker: Jeffrey Giansiracusa

Title: What TDA can tell us about the nature of matter

Abstract: Physics describes the world in terms of quantum field theories; the inside of protons and neutrons is described by one particular theory called QCD, which says that things are made out of particles called quarks and gluons. It's a very successful theory, except for one annoying detail: nobody has ever seen a quark or gluon! This paradox is explained away by a hypothetical mechanism called 'confinement' that nobody really understands. Experimental physicists see clear evidence of confinement in collider experiments, and computational physicists watch it happen in Monte Carlo simulations on supercomputers, but still a mechanistic understanding eludes us. The Monte Carlo simulations produce vast amounts of data, and so understanding confinement should be a data science problem. I'll explain how persistent homology might just be the tool we need for this problem.

Speaker: Maryam Garba

Title: The origin of wald space

Abstract: Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. We propose a new space of phylogenetic trees which we call wald space. The motivation is to develop a space suitable for statistical analysis of phylogenies, but with a geometry based on more biologically principled assumptions than existing spaces: in wald space, trees are close if they induce similar distributions on genetic sequence data. As a point set, wald space contains the previously developed Billera–Holmes–Vogtmann (BHV) tree space; it also contains disconnected forests, like the edge-product (EP) space but without certain singularities of the EP space. We investigate two related geometries on wald space. The first is the geometry of the Fisher information metric of character distributions induced by the two-state symmetric Markov substitution process on each tree. Infinitesimally, the metric is proportional to the Kullback–Leibler divergence. The second geometry is obtained analogously but using a related continuous-valued Gaussian process on each tree, and it can be viewed as the trace metric of the affine-invariant metric for covariance matrices. For both geometries we derive computational methods to compute geodesics in polynomial time and show numerically that the two information geometries (discrete and continuous) are very similar. In particular, geodesics are approximated extrinsically. Comparison with the BHV geometry shows that our canonical and biologically motivated space is substantially different.

Speaker: Miruna Stefana Sorea

Title: The Disguised Toric Locus and Affine Equivalence of Reaction Networks

Abstract: In this talk, we consider families of polynomial dynamical systems inspired by (biochemical) reaction networks. We start by focusing on toric dynamical systems, which are also called complex balanced mass-action systems. From the applications point of view, these systems exhibit important dynamical properties, such as local and global stability, existence and uniqueness of positive steady states. In the context of mass-action kinetics a dynamical system may be realized by different reaction networks and/or parameters, thus it might happen that a mass-action system enjoys the strong dynamical behavior mentioned above, without being toric; we call such systems disguised toric dynamical systems. We show that the parameters that give rise to disguised toric dynamical systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations and show that there is a canonical bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially. This is based on joint work with Sabina J. Haque (Harvard University), Matthew Satriano (University of Waterloo) and Polly Y. Yu (Harvard University).

Speaker: Georgy Scholten

Title: Sparse moments of univariate step functions and the coalescence manifold

Abstract: We study the univariate moment problem of piecewise-constant density functions on the interval [0,1] and its consequences for an inference problem in population genetics. We show that, up to closure, any collection of n moments is achieved by a step function with at most n − 1 breakpoints and that this bound is tight. We use this to show that any point in the n th coalescence manifold in population genetics can be attained by a piecewise constant population history with at most n − 2 changes. Both the moment cones and the coalescence manifold are projected spectrahedra and we describe the problem of finding a nearest point on them as a semidefinite program

Speaker: Abhinav Natarajan

Title: Morse Theory for Chromatic Delaunay Triangulations

Abstract: The chromatic alpha filtration is a generalization of the alpha filtration that can encode spatial relationships among classes of labelled point cloud data, and has applications in topological data analysis of multi-class data. In recent work with Thomas Chaplin, Adam Brown, and Maria-Jose Jimenez, we introduced the chromatic Delaunay–Cech and chromatic Delaunay–Rips filtrations, which are built on the same underlying simplicial complex but have filtration values that are easier to compute. Our main result is an application of generalised discrete Morse theory to show that the Cech, chromatic Delaunay-Cech, and chromatic alpha filtrations are related by simplicial collapses. This result generalizes a result of Bauer and Edelsbrunner from the non-chromatic to the chromatic setting. We also show that the chromatic Delaunay–Rips filtration is locally stable to perturbations of the underlying point cloud. This local stability, in conjunction with the Morse-theoretic result, means that the chromatic Delaunay-Rips filtration is a viable approximation to the chromatic alpha filtration for persistent homology calculations in low dimensional data, with the advantage of being much faster to compute. In this talk I will give a sketch of the proofs of the main results, and elaborate on how these results provide theoretical justification for the use of chromatic Delaunay–Cech and chromatic Delaunay–Rips filtrations in practical applications. I will also show the data from numerical experiments to compare the computational efficiency of the various constructions.

Sponsors:

We are grateful for the financial support from the Isaac Newton Institute (we acknowledge support from the INI and the EPSRC grant with reference EP/V521929/1), the Glasgow Mathematical Journal Learning and Research Support Fund, from the Edinburgh Mathematical Society and the London Mathematical Society.

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