L6

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

Small angle x-ray scattering is one of the most flexible and readily available experimental methods for obtaining information on the structure of proteins in solution. In the advent of powerful predictive methods such as the alphaFold and rossettaFold algorithms, this information has become increasingly in demand, owing to the need to characterise the more flexible and varying components of proteins which resist characterisation by these and more standard experimental techniques. To deal with structures about little of which is known a parsimonious method of representing the tertiary fold of a protein backbone as a discrete curve has been developed. It represents the fundamental local Ramachandran constraints through a pair of parameters and is able to generate millions of potentially realistic protein geometries in a short space of time. The data obtained from these methods provides a treasure trove of information on the potential range of topological structures available to proteins, which is much more constrained that that available to self-avoiding walks, but still far more complex than currently understood from existing data. I will introduce this method and its considerations then attempt to pose some questions I think topological data analysis might help answer. Along the way I will ask why roadies might also help give us some insight….

We use Cech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain nine homology and six homotopy theories of closure spaces. We show how metric spaces and more general structures such as weighted directed graphs produce filtered closure spaces. For filtered closure spaces, our homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Cech complexes to closure spaces and prove that their persistent homology is stable.

This is joint work with Nikola Milicevic.

The Persistent Homology Transform, and the Euler Characteristic Transform are topological analogs of the Radon transform that can be used in statsistical shape analysis. In this talk I will consider an interesting variant called the Extended Persistent Homology Transform (XPHT) which replaces the normal persistent homology with extended persistent homology. We are particularly interested in the application of the XPHT to binary images. This paper outlines an algorithm for efficient calculation of the XPHT exploting relationships between the PHT of the boundary curves to the XPHT of the foreground.

Topological data analysis (TDA) is a field with tools to quantify the shape of data in a manner that is concise and robust using concepts from algebraic topology. Persistent homology, one of the most popular tools in TDA, has proven useful in applications to time series data, detecting shape that changes over time and quantifying features like periodicity. In this talk, I will present two applications using tools from TDA to study time series data: the first using zigzag persistence, a generalization of persistent homology, to study bifurcations in dynamical systems and the second, using the shape of weighted, directed networks to distinguish periodic and chaotic behavior in time series data.

Acyclic partial matchings on simplicial complexes play an important role in topological data analysis by facilitating efficient computation of (persistent) homology groups. Here we describe probabilistic properties of critical simplex counts for such matchings on clique complexes of Bernoulli random graphs. In order to accomplish this goal, we generalise the notion of a dissociated sum to a multivariate setting and prove an abstract multivariate central limit theorem using Stein's method. As a consequence of this general result, we are able to extract central limit theorems not only for critical simplex counts, but also for generalised U-statistics (and hence for clique counts in Bernoulli random graphs) as well as simplex counts in the link of a fixed simplex in a random clique complex.

In this talk, I will talk about the category of coherent sheaves on the affine Flag variety of a simply-connected reductive group over $\mathbb{C}$. In particular, I'll explain how the convolution product naturally leads to a construction of the Iwahori-Hecke algebra, and present some combinatorics related to computing duals in this category.

One core aim of (supervised) machine learning is to approximate an unknown function given a dataset containing examples of input-output pairs. Real-world examples of such functions include the mapping from an image to its label or the mapping from a molecule to its energy. For a variety of such functions, while the precise mapping is unknown, we often have knowledge of its properties. For example, the label of an image may be invariant to rotations of the input image. Generally, such properties formally correspond to the function being equivariant to certain actions on its input and output spaces. This has led to much research on building equivariant function classes (aka neural networks). In this talk, we survey this growing field of equivariance in deep learning for a mathematical audience, motivating the need for equivariance, covering concrete examples of equivariant neural networks, and offering a learning theoretic perspective on the benefits of equivariance. 

Ardakov and Wadsley developed a theory of D-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of SL_2 by taking global sections of Drinfeld line bundles. In this talk, we will extend their techniques to SL_3 by studying the Drinfeld upper half-space \Omega^{(3)} of dimension 2.