Past

Electron cryomicroscopy (cryo-EM) is an imaging technique widely used in structural biology to determine the three-dimensional structure of biological molecules from noisy two-dimensional projections with unknown orientations. As the typical pipeline involves processing large amounts of data, efficient algorithms are crucial for fast and reliable results. The stochastic gradient descent (SGD) algorithm has been used to improve the speed of ab initio reconstruction, which results in a first, low-resolution estimation of the volume representing the molecule of interest, but has yet to be applied successfully in the high-resolution regime, where expectation-maximization algorithms achieve state-of-the-art results, at a high computational cost. 
In this work, we investigate the conditioning of the optimisation problem and show that the large condition number prevents the successful application of gradient descent-based methods at high resolution. 
Our results include a theoretical analysis of the condition number of the optimisation problem in a simplified setting where the individual projection directions are known, an algorithm based on computing a diagonal preconditioner using Hutchinson's diagonal estimator, and numerical experiments showing the improvement in the convergence speed when using the estimated preconditioner with SGD. The preconditioned SGD approach can potentially enable a simple and unified approach to ab initio reconstruction and high-resolution refinement with faster convergence speed and higher flexibility, and our results are a promising step in this direction.

Abstract: I will introduce and explain a new symmetry structure for type IIA string theory, called string^h. Using string^h I will explain  how some objects of stable homotopy theory relating to elliptic cohomology enter into type IIA string theory.

Persistent homology is infeasible to compute when a dataset is very large. Inspired by the bootstrapping method, Chazal et al. (2014) proposed a multiple subsampling approach to approximate the persistence landscape of a massive dataset. In this talk, I will present an extension of the multiple subsampling method to a broader class of vectorizations of persistence diagrams and to persistence diagrams directly. First, I will review the statistical foundation of the multiple subsampling approach as applied to persistence landscapes in Chazal et al. (2014). Next, I will talk about how this analysis extends to a class of vectorized persistence diagrams called Hölder continuous vectorizations. Finally, I will address the challenges in applying this method to raw persistence diagrams for two measures of centrality: the mean persistence measure and the Fréchet mean of persistence diagrams. I will demonstrate these methods through simulation results and applications in estimating data shapes. 

This week's Fridays@2 will feature a panel discussion on how to manage your time during your degree. The panel will share their thoughts and experiences in a Q&A session, discussing some of the practicalities of juggling lectures, the many ways to study independently and non-maths activities. 

The Bruhat-Tits building is a crucial combinatorial tool in the study of reductive p-adic groups and their representation theory. Given a p-adic group, its Bruhat-Tits building is a simplicial complex upon which it acts with remarkable properties. In this talk I will give an introduction to the Bruhat-Tits building by sketching its definition and going over some of its basic properties. I will then show the usefulness of the Bruhat-Tits by determining the maximal compact subgroups of a p-adic group up to conjugacy by using the Bruhat-Tits building.

At the heart of both cross-section calculations at the Large Hadron Collider and gravitational wave physics lie the evaluation of Feynman integrals. These integrals are meromorphic functions (or distributions) of the parameters on which they depend and understanding their analytic structure has been an ongoing quest for over 60 years. In this talk, I will demonstrate how these integrals fits within the framework of generalized hypergeometry by Gelfand, Kapranov, and Zelevinsky (GKZ). In this framework the singularities are simply calculated by the principal A-determinant and I will show that some Feynman integrals can be used to generate Cohen-Macaulay rings which greatly simplify their analysis. However, not every integral fits within the GKZ framework and I will show how the singularities of every Feynman integral can be calculated using Whitney stratifications.

Cells must reliably coordinate responses to noisy external stimuli for proper functionality whether deciding where to move or initiate a response to threats. In this talk I will present a perspective on such cellular decision making problems with extreme statistics. The central premise is that when a single stochastic process exhibits large variability (unreliable), the extrema of multiple processes has a remarkably tight distribution (reliable). In this talk I will present some background on extreme statistics followed by two applications. The first regards antigen discrimination - the recognition by the T cell receptor of foreign antigen. The second concerns directional sensing - the process in which cells acquire a direction to move towards a target. In both cases, we find that extreme statistics explains how cells can make accurate and rapid decisions, and importantly, before any steady state is reached.

This study introduces a novel suite of historical large language models (LLMs) pre-trained specifically for accounting and finance, utilising a diverse set of major textual resources. The models are unique in that they are year-specific, spanning from 2007 to 2023, effectively eliminating look-ahead bias, a limitation present in other LLMs. Empirical analysis reveals that, in trading, these specialised models outperform much larger models, including the state-of-the-art LLaMA 1, 2, and 3, which are approximately 50 times their size. The findings are further validated through a range of robustness checks, confirming the superior performance of these LLMs.

Quantum computers are becoming a reality and current generations of machines are already well beyond the 50-qubit frontier. However, hardware imperfections still overwhelm these devices and it is generally believed the fault-tolerant, error-corrected systems will not be within reach in the near term: a single logical qubit needs to be encoded into potentially thousands of physical qubits which is prohibitive.

Due to limited resources, in the near term, hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage and these need to resort to quantum error mitigation techniques. I will explain the basic concepts and introduce hybrid quantum-classical protocols are the most promising candidates for achieving early quantum advantage. These have the potential to solve real-world problems---including optimisation or ground-state search---but they suffer from a large number of circuit repetitions required to extract information from the quantum state. I will finally identify the most likely areas where quantum computers may deliver a true advantage in the near term.

Bálint Koczor

Associate Professor in Quantum Information Theory

Mathematical Institute, University of Oxford

webpage

In recent years, some progress has been made in the development of finite element complexes, particularly in the discretization of BGG complexes in two and three dimensions, including Hessian complexes, elasticity complexes, and divdiv complexes. In this talk, I will discuss distributional complexes in two and three dimensions. These complexes are simply constructed using geometric concepts such as vertices, edges, and faces, and they share the same cohomology as the complexes at the continuous level, which reflects that the discretization is structure preserving. The results can be regarded as a tensor generalization of the Whitney forms of the finite element exterior calculus. This talk is based on joint work with Snorre Christiansen (Oslo), Kaibo Hu (Edinburgh), and Qian Zhang (Michigan).

The fundamental group of a hyperbolic surface has an infinite number of rank k subgroups. What does it mean, therefore, to pick a 'random' subgroup of this type? In this talk, I will introduce a method for counting subgroups and discuss how counting allows us to study the properties of a random subgroup and its associated cover.

We will present some of the remarkable properties of eigenvalue trajectories for rank-one perturbations of random matrices, with an emphasis on two models of particular interest, namely weakly non-Hermitian and weakly non-unitary matrices. In both cases, precise estimates can be obtained for the critical timescale at which an outlier can be observed with high probability. We will outline the proofs of these results and highlight their significance in connection with quantum chaotic scattering. (Based on joint works with L. Erdös and J. Reker)

The notion of semi-uniform stability of a strongly continuous semi-group refers to the stability of classical solutions of a linear evolution equation, and this has analogues with the classical Katznelson-Tzafriri theorem. The co-generator of a strongly continuous semigroup is a bounded linear operator that comes from a particular discrete approximation to the semigroup. After reviewing some background on (quantified) stability theory for semigroups and the Katznelson-Tzafriri theorem, I will present some results relating the stability of a strongly continuous semigroup with that of its cogenerator. This talk is based on joint work with David Seifert.

The conjugacy growth function of a finitely generated group is a variation of the standard growth function, counting the number of conjugacy classes intersecting the n-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invariant in general, but the conjugacy growth of finite extensions can be understood via the twisted conjugacy growth function, counting automorphism-twisted conjugacy classes. I will discuss what is known about the asymptotic and formal power series behaviour of (twisted) conjugacy growth, in particular some relatively recent results for certain groups of polynomial growth (i.e. virtually nilpotent groups).

Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as the maximal minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees form a regular matroid. In this talk, I will give a short historical overview of the tree-counting problem and a related quantity from electrical circuit theory: the effective resistance. I will describe a characterization of effective resistances in terms of a certain polytope and discuss some recent applications to discrete notions of curvature on graphs. More details can be found in the recent preprint: https://arxiv.org/abs/2410.07756

The classical lower-tail problem for triangles in random graphs asks the following: given $\eta\in[0,1)$, what is the probability that $G(n,p)$ contains at most $\eta$ times the expected number of triangles?  When $p=o(n^{-1/2})$ or $p = \omega(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and regularity or container methods in the latter case.

We prove for the first time asymptotic formulas for the logarithm of the lower tail probability when $p=c n^{-1/2}$ for $c$ constant.  Our results apply for all $c$ when $\eta \ge 1/2$ and for $c$  small enough when $\eta < 1/2$.  For the special case $\eta=0$ of triangle-freeness, our results prove that a phase transition occurs as $c$ varies (in the sense of a non-analyticity of the rate function), while for $\eta \ge 1/2$ we prove that no phase transition occurs.

Our method involves ingredients from algorithms and statistical physics including rapid mixing of Markov chains and the cluster expansion.  We complement our asymptotic formulas with efficient algorithms to approximately sample from $G(n,p)$ conditioned on the lower tail event.

Joint work with Will Perkins, Aditya Potukuchi and Michael Simkin.

Let G be a reductive group over a finite field F of characteristic p. I will present work with Tzu-Jan Li in which we determine the endomorphism algebra of the Gelfand-Graev representation of the finite group G(F) where the coefficients are taken to be l-adic integers, for l a good prime of G distinct from p. Our result can be viewed as a finite-field analogue of the local Langlands correspondence in families. 

We will be joined by Dr Brigitte Stenhouse from the Open University to discuss harassment, particularly in academic settings. 

We prove the local Lipschitz continuity of energy minimizing harmonic maps between singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.  Joint work with Yaoting Gui and Jürgen Jost.