Applied Algebra and Geometry Fourth meeting

Date:

31 January 2018

Location:

Room B7, The Hemsley Building (Staff Club), University of Nottingham, University Park, Nottingham, NG7 2RD.

Invited Speakers:

Tom Nye, Newcastle University

Emilie Dufresne, University of Nottingham (last minute replacement)

Jim Smith, University of Warwick

Louis Theran, University of St Andrews

Schedule

10:30-11:00 Welcome and coffee/tea/juice

11:00-12:00 Tom Nye

12:00-13:30 Lunch + discussions

13:30-14:30 Emilie Dufresne

14:30-15:30 Louis Theran

15:30-16:00 coffee/tea/juice

16:00-17:00 Jim Smith

Abstracts

Tom Nye: Geometric statistics in tree space

Phylogenetic trees represent evolutionary relationships between present-day species. Each tree can be considered as a combinatorial object, or when the tree is edge-weighted, as a point in a continuous metric space. Conventional statistical methods cannot be used to analyse data sets of trees since the space of all trees on a common set of species is not a vector space. Instead, tree-space has been shown to be a non-positively curved geodesic metric space and geodesics are computable in polynomial time. In this talk I will give an introduction to the geometry of phylogenetic tree-space, and describe a few ways to translate conventional statistical apparatus into this novel geometrical setting. This brings together ideas from metric geometry and analysis of stochastic processes.

Emilie Dufresne: The geometry of sloppiness

The use of mathematical models in the sciences often require the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. We develop a precise mathematical foundation for sloppiness and define rigorously its key concepts, such as `model manifold' in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric on parameter space. This opens up the possibility of alternative quantification of sloppiness beyond the traditional use of the Fisher Information Matrix, which implicitly assumes infinitesimal measurement error and an Euclidean parameter space. We illustrate the various concepts involved in the proper definition of sloppiness with examples of ordinary differential equation models with time series data arising in mathematical biology. (joint with Heather Harrington and Dhruva Raman)

Louis Theran: Unlabelled rigidity problems

The rigidity theory of frameworks deals with questions of the following type: given n unknown points p_1, …, p_n in R^d and some m pairwise length measurements of the form \ell_ij = ||p_i - p_j||^2, (a) determine the dimension of the set of possible p_i; (b) find the p_i. For example, if all the pairwise distances are measured, problem (b) is solved by classical multi-dimensional scaling. I’ll talk about variants of this problem where: i and j are not provided with the measurements; the number of points in unknown; and the measurements may be lengths of paths and loops instead of pairwise distances. Under the condition that the unknown points are generic, and the measurement sequence is big enough to “allow for trilateration”, we can still provide positive answers to both of the rigidity problems. This is joint work with Steven Gortler, Ioannis Gkioulekas, and Todd Zickler.

Jim Smith: Toric Causation: Using Algebra to search for Causal Explanations

In recent years there has been an explosion in methodologies to search data sets for evidence of causal relationships between different sets of measurement variables, measurable with respect to a particular given sigma algebra. Much of this work first processes data by fitting a graphical model. It then identifies certain invariances associated with the topology of the best fitting graph and uses these to conjecture when one variable might causally influence another. These types of methodology have proved very useful. However their domain of application is severely restricted in the sense that the methods need to assume that the underlying data generating mechanism can be fully described by a graphical model. In practice this is often not the case. I will first review the current graphical methodologies. I will then discuss a possible reframing of these causal conjectures in terms of the existence of various quadratic constraints on the atomic probabilities of a finite state space. I demonstrate that through this generalisation we can redefine causal relationships in a way that does not depend on a graphical model hypothesis. Using examples drawn from public health, criminal internet behaviour and government policy analysis I will illustrate how this algebraic generalisation can free us to discover useful causal relationships that are not expressible graphically. This is joint work with Dr Christiane Goergen currently in the Max Plank Institute in Leipzig.

Local Organizer:

Emilie Dufresne, University of Nottingham