Talks
All times are in BST, which is the current time zone for the UK.
Thursday 25 August
Time: 13:00
Speaker: Alexey Pozdnyakov
Title: Murmurations of elliptic curves
Abstract: This talk will review some data scientific experiments involving arithmetic objects, focusing mainly on averaging the Dirichlet coefficients associated to elliptic curves within fixed conductor ranges. In particular, a surprising oscillation that appears in these averages will be discussed. This talk is based on work with He, Lee, and Oliver.
Time: 14:15
Speaker: Margaret Regan
Title: Using data as an input to parameterized polynomial systems
Abstract: Parameterized systems of polynomial equations arise in many applications including computer vision, chemistry, and kinematics. Numerical homotopy continuation methods are a fundamental technique within numerical algebraic geometry for both solving these polynomial systems and determining more refined information about their structure. Imperative to these solving methods is the use of data — either synthetic or from the application itself, such as image pixel data for computer vision and leg length parameters for kinematics. This talk will highlight various uses of data within computer vision and machine learning applications.
Time: 16:00
Speaker: Henry Kvinge
Title: How Deep Learning is Being Made More Robust, More Domain-Aware, and More Capable of Generalization Through the Influence of Algebra and Topology
Abstract: Driven by enormous amounts of data and compute, deep learning-based models continue to surpass yesterday’s benchmarks. In this fast-growing field where machine learning (ML) is applied in more and more domains, there is a constant need for new ways of looking at problems. Recent years have seen the rise of tools derived from topology and algebra, which are not traditionally associated with ML. In this talk I will begin by surveying some of the recent applications of these fields in ML, from hardcoding equivariance into vision models, to using sheaves to better enable learning on graphs. I will argue that pure mathematics will increasingly offer critical tools necessary for a more mature approach to machine learning. I will end by discussing some of my team’s recent work which, inspired by the notion of a fiber bundle, developed a novel deep learning architecture to solve a challenging problem in materials science.
Time: 17:15
Speaker: Jesus De Loera
Title: On the Discrete Geometric Principles of Machine Learning and Statistical Inference
Abstract: In this talk I explain the fertile relationship between inference and learning to combinatorial geometry. My presentation contains several powerful situations where famous theorems in discrete geometry answered natural questions from machine learning and statistical inference: In this tasting tour I will include the problem of deciding the existence of Maximum likelihood estimator in multiclass logistic regression, the variability of behavior of k-means algorithms with distinct random initializations and the shapes of the clusters, and the estimation of the number of samples in chance-constrained optimization models. These obviously only scratch the surface of what one could do with extra free time. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, measures of separability of data, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). All new theorems are joint work with subsets of the following wonderful folks: T. Hogan, D. Oliveros, E. Jaramillo-Rodriguez, and A. Torres-Hernandez.
Friday 26 August
Time: 10:00
Speaker: Geordie Williamson
Title: Equivariant deep learning: a hammer looking for a nail
Abstract: Often one wants to learn quantities which are invariant or equivariant with respect to a group. For example, the decision as to whether there is a tiger nearby should not depend on the precise position of your head and thus this decision should be rotation invariant. Another example: quantities that appear in the analysis of point clouds often do not depend on the labelling of the points, and are therefore invariant under a large symmetric group. I will explain how to build networks which are equivariant with respect to a group action. What ensues is a fascinating interplay between group theory, representation theory and deep learning. Examples based on translations or rotations recover familiar convolutional neural nets, however the theory gives a blueprint for learning in the presence of complicated symmetry. These architectures appear very useful to mathematicians, but I am not aware of any major applications in mathematics as yet. Thus the nail of the title. Most of this talk will be a review of ideas and techniques well-known in to the geometric deep learning community. New material is joint work with Joel Gibson (Sydney) and Sebastien Racaniere (DeepMind).
Time: 11:15
Speaker: Anthea Monod
Title: Approximating Persistent Homology for Large Datasets
Abstract: Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications; it produces a summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its popularity, persistent homology is simply impossible to compute for very large datasets which prohibits its widespread use in many big data settings. What can we do if we would like a representative persistence diagram for a very large dataset whose persistent homology we cannot compute due to size restrictions? We adapt here the classical statistical method of bootstrapping, namely, drawing and studying smaller subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples is a valid approximation of the persistence diagram of the large dataset and derive its convergence rate to the true persistent homology of the large dataset. We demonstrate our approach on synthetic and real data; furthermore, we give an example of the utility of our approach in a shape clustering problem where we are able to obtain accurate results with only 2% subsampled from the original dataset.
Time: 14:00
Speaker: Michael Douglas
Title: Numerical methods for Calabi-Yau and special geometry metrics
Abstract: We discuss recent work in which machine learning techniques and software have been adapted to compute Ricci flat metrics on Calabi-Yau threefolds, and ongoing work to compute G2 metrics and other geometric structures. Based on work with Rodrigo Barbosa, Yidi Qi and Subramananian Lakshinarasimhan.
Time: 15:15
Speaker: Maria Cameron
Title: Quantifying rare events with an aid of diffusion maps
Abstract: Many interesting problems concerned with rare event quantification arise in chemical physics. A typical problem is of finding reaction channels and transition rates for conformal changes in a biomolecule. To reduce the dimensionality and make the description of transition processes more comprehensible, often a set of physically motivated collective variables (dihedral angles, distances between particular pairs of key atoms, etc.) is introduced by means of mapping atomic coordinates to a low-dimensional space and averaging. The dynamics in collective variables remain time-reversible but acquire an anisotropic and position-dependent diffusion tensor. In this talk, I will discuss how one can adapt the diffusion map algorithm with the Mahalanobis kernel to approximate the generator of this diffusion process and use it to compute the committor function, the reactive current, and the transition rate. Applications to alanine-dipeptide and Lennard-Jones-7 in 2D will be presented.