Past

Low rank structure is pervasive in real-world datasets.

This talk shows how to accelerate the solution of fundamental computational problems, including eigenvalue decomposition, linear system solves, composite convex optimization, and stochastic optimization (including deep learning), by exploiting this low rank structure.

We present a simple method based on randomized numerical linear algebra for efficiently computing approximate top eigende compositions, which can be used to replace large matrices (such as Hessians and constraint matrices) with low rank surrogates that are faster to apply and invert.

The resulting solvers for linear systems (NystromPCG), composite convex optimization (NysADMM), and stochastic optimization (SketchySGD and PROMISE) demonstrate strong theoretical and numerical support, outperforming state-of-the-art methods in terms of speed and robustness to hyperparameters.

In our first ever joint event with WISOx (Oxford Women in Statistics), we will be having a panel discussion about the hidden labour of minorities, such as extra committee work, editorial work, etc. 

We will be joined by panellists Helen Byrne (Maths) and Gesine Reinert (Stats). 

In this talk I will discuss how phenotypic heterogeneity affects emergent pattern formation in living active matter with chemical communication between cells. In doing so, I will explore how the emergent dynamics of multicellular communities are qualitatively different in comparison to the dynamics of isolated or non-interacting cells. I will focus on two specific projects. First, I will show how genetic regulation of chemical communication affects motility-induced phase separation in cell populations. Second, I will demonstrate how chemotaxis along self-generated signal gradients affects cell populations undergoing 3D morphogenesis.

In this talk I will discuss the representation theory of truncated current Lie algebras in prime characteristic. I will first give an introduction to modular representation theory for general restricted Lie algebras and introduce the Kac-Weisfeiler conjectures. Then I will introduce a family of Lie algebras known as truncated current Lie algebras, and discuss their representation theory and its relationship with the representation theory of reductive Lie algebras in positive characteristic.

A wide variety of solutions have been proposed in order to cope with the deficiencies of Modern Portfolio Theory. The ideal portfolio should optimise the investor’s expected utility. Robustness can be achieved by ensuring that the optimal portfolio does not diverge too much from a predetermined allocation. Information geometry proposes interesting and relatively simple ways to model divergence. These techniques can be applied to the risk budgeting framework in order to extend risk budgeting and to unify various classical approaches in a single, parametric framework. By switching from entropy to divergence functions, the entropy-based techniques that are useful for risk budgeting can be applied to more traditional, constrained portfolio allocation. Using these divergence functions opens new opportunities for portfolio risk managers. This presentation is based on two papers published by the BNP Paribas QIS Lab, `The properties of alpha risk parity’ (2022, Entropy) and `Turning tail risks into tailwinds’ (2020, The Journal of Portfolio Management).

The Mixing Conjecture of Michel-Venkatesh has now taken on additional arithmetic significance via Wiles' new approach to modularity. Inspired by this, we present the best currently available method, pioneered by Khayutin's proof for quaternion algebras over the rationals, which we have successfully applied to totally real fields. The talk will overview the method, which brings a suprising combination of ergodic theory, analysis and geometry to bear on this arithmetic problem.

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.

In this talk, we consider the problem of approximating failure regions. More specifically, given a costly computational model with random parameters and a failure condition, our objective is to determine the parameter region in which the failure condition is likely to not be satisfied. In mathematical terms, this problem can be cast as approximating the level set of a probability density function. We solve this problem by dividing it into two: 1) The design of an efficient Monte Carlo strategy for probability estimation. 2) The construction of an efficient algorithm for level-set approximation. Following this structure, this talk is comprised of two parts:

In the first part, we present a new multi-output multilevel best linear unbiased estimator (MLBLUE) for approximating expectations. The advantage of this estimator is in its convenience and optimality: Given any set of computational models with known covariance structure, MLBLUE automatically constructs a provenly optimal estimator for any (finite) number of quantities of interest. Nevertheless, the optimality of MLBLUE is tied to its optimal set-up, which requires the solution of a nonlinear optimization problem. We show how the latter can be reformulated as a semi-definite program and thus be solved reliably and efficiently.

In the second part, we construct an adaptive level-set approximation algorithm for smooth functions corrupted by noise in $\mathbb{R}^d$. This algorithm only requires point value data and is thus compatible with Monte Carlo estimators. The algorithm is comprised of a criterion for level-set adaptivity combined with an a posteriori error estimator. Under suitable assumptions, we can prove that our algorithm will correctly capture the target level set at the same cost complexity of uniformly approximating a $(d-1)$-dimensional function.

Mathematical modelling can support decontamination processes in a variety of ways.  In this talk, we focus on the contamination step: understanding how much of a chemical spill has seeped into the Earth or a building material, and how far it has travelled, are essential for making good decisions about how to clean it up.  

We consider an infiltration problem in which a chemical is poured on an initially unsaturated porous medium, and seeps into it via capillary action. Capillarity-driven flow through partially-saturated porous media is often modelled using Richards’ equation, which is a simplification of the Buckingham-Darcy equation in the limit where the infiltrating phase is much more viscous than the receding phase.  In this talk, I will explore the limitations of Richards equation, and discuss some scenarios in which predictions for small-but-finite viscosity ratios are very different to the Richards simplification.

I will report on a joint work with Pablo Destic and Nuno Hultberg, about some applications of Globally Valued Fields (GVFs) and I will describe a density result that we needed, which turns out to be connected to Riemann-Zariski and Berkovich spaces.

This talk will serve as an introduction to the outer automorphism group of a free group, its properties and the objects used to study it: especially train track maps (with various adjectives) and Culler--Vogtmann outer space. If time allows I will discuss recent work joint with Hillen, Lyman and Pfaff on stretch factors in rank 3, but the goal of the talk will be to introduce the topic well rather than to speedrun towards the theorem.

Recently, Elliott, Li, and Niu proved a classification theorem for Villadsen-type algebras using the combination of the Elliott invariant and the radius of comparison, an invariant that was introduced by Toms in order to distinguish between certain non-isomorphic AH algebras with the same Elliott invariant. This might have raised the prospect that the Elliott classification program can be extended beyond the Z-stable case by adding the radius of comparison to the invariant. I will discuss a recent preprint in which we show that this is not the case: we construct an uncountable family of nonisomorphic AH algebras with the same Elliott and same radius of comparison. We can distinguish between them using a finer invariant, which we call the local radius of comparison. This is joint work with N. Christopher Phillips.

A cluster graph is a disjoint union of complete graphs. We consider the random $G(n,p)$ graph on $n$ vertices with connection probability $p$, conditioned on the rare event of being a cluster graph. There are three main motivations for our study.

1. For $p = 1/2$, each random cluster graph occurs with the same probability, resulting in the uniform distribution over set partitions. Interpreting such a partition as a graph adds additional structural information.
2. To study how the law of a well-studied object like $G(n,p)$ changes when conditioned on a rare event; an evidence of this fact is that the conditioned random graph overcomes a phase transition at $p=1/2$ (not present in the dense $G(n,p)$ model).
3. The original motivation was an application to community detection. Taking a random cluster graph as a model for a prior distribution of a partition into communities leads to significantly better community-detection performance.

This is joint work with Martijn Gösgens, Lukas Lüchtrath, Elena Magnanini and Élie de Panafieu.

We consider quadratic deformations of the q-symmetric algebras A_q given by x_i x_j = q_{ij} x_j x_i, for q_{ij} in C*.   We explicitly describe the Hochschild cohomology and compute the weights of the torus action (dilating the x_i variables). We describe new families of filtered deformations of A_q, which are Koszul and Calabi—Yau algebras. This also applies to abelian category deformations of coh(P^n), and for n=3 we give examples having no homogeneous coordinate ring.  We then focus on the case where n is even and the deformations are obtainable from deformation quantisation of toric log symplectic structures on P^n.  In this case we construct formally universal families of quadratic algebras deforming A_q, obtained by tensoring filtered deformations and Feigin—Odesskii elliptic algebras. The universality is a consequence of a beautiful combinatorial classification of deformations via "smoothing diagrams", a collection of disjoint cycles and segments in the complete graph on n vertices, viewed as the dual complex for the coordinate hyperplanes in P^{n-1}.  Already for n=5 there are 40 of these, mostly entirely new. Our proof also applies to deformations of Poisson structures, recovering the P^n case of our previous results on general log symplectic varieties with normal crossings divisors, which motivated this project.  This is joint work with Mykola Matviichuk and Brent Pym.

One of the major insights gained from holographic duality is the relation between the physics of black holes and quantum chaotic systems. This relation is made precise in the duality between two dimensional JT gravity and random matrix theory.  In this work, we generalize this to a duality between AdS3 gravity and a random ensemble of approximate CFT's.  The latter is described by a combined tensor and matrix model, describing the OPE coefficients and spectrum of a theory that approximately satisfies the bootstrap constraints.   We show that the Feynman diagrams of the random ensemble produce a sum over 3 manifolds that agrees with the partition function of 3d gravity.  A crucial element of this dictionary is the Virasoro TQFT, which defines the bulk gravitational path integral via the cutting and sewing relations of topological field theory.  Time permitting, we will explain why this TQFT has gravitational edge modes degrees of freedom whose entanglement gives rise to gravitational entropy.

In this talk, we present a graph discretized approximation scheme for diffusions with drift and killing on a complete Riemannian manifold M. More precisely, for a given Schrödinger operator with drift on M having the form A = — Δ — b + V , we introduce a family of discrete time random walks in the  ow generated by the drift b with killing on a sequence of proximity graphs, which are constructed by partitions cutting M into small pieces. As a main result, we prove that the drifted Schrodinger semigroup {e^—tA}_t≥0 is approximated by discrete semigroups generated by the family of random walks with a suitable scale change. This result gives a  nite dimensional summation approximation of a Feynman-Kac type functional integral over M. Furthermore, when M is compact, we also obtain a quantitative error estimate of the convergence.
This talk is based on a joint work with Satoshi Ishiwata (Yamagata University), and the full paper can be found on https://doi.org/10.1007/s00208-024-02809-9.

Course Overview
The course gives an introduction to a topic of current interest in Lorentzian geometic analysis and mathematical General Relativity: an approach to nonregular spacetimes based on a “metric” point of view.
 

Learning Outcomes
Becoming acquainted with Lorentzian length spaces, sectional and Ricci curvature bounds for non-regular Lorentzian spaces and the appropriate techniques.
 

Course Synopsis
Lecture 1a: Review of Lorentzian geometry, spaces of constant curvature, causality theory, singularity theorems.
Lecture 1b: Introduction to Lorentzian length spaces, timelike sectional curvature bounds.

Lecture 2a: Optimal transport, timelike Ricci curvature bounds
Lecture 2b: Sobolev calculus for time functions. Literature: [O’N83, KS18, CM20].
 

Reading group: Depending on student’s interest one could discuss the papers [GKS19, AGKS21, ABS22].

References
[ABS22] L. Aké Hau, S. Burgos, and D. A. Solis. Causal completions as Lorentzian pre-length spaces. General Relativity and Gravitation, 54(9), 2022. doi:10.1007/s10714-022-02980-x.
[AGKS21] S. B. Alexander, M. Graf, M. Kunzinger, and C. Sämann. Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems. Comm. Anal. Geom., to appear, 2021. doi:10.48550/arXiv.1909.09575. arXiv:1909.09575 [math.MG].
[CM20] F. Cavalletti and A. Mondino. Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Cambridge Journal of Mathematics, to appear, arXiv:2004.08934 [math.MG], 2020. doi:10.48550/arXiv.2004.08934.
[GKS19] J. D. E. Grant, M. Kunzinger, and C. Sämann. Inextendibility of spacetimes and Lorentzian length spaces. Ann. Global Anal. Geom., 55(1):133–147, 2019. doi:10.1007/s10455-018-9637-x.
[KS18] M. Kunzinger and C. Sämann. Lorentzian length spaces. Ann. Glob. Anal. Geom., 54(3):399–447, 2018. doi:10.1007/s10455-018-9633-1.
[O’N83] B. O’Neill. Semi-Riemannian geometry with applications to relativity, volume 103 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.

Should you be interested in taking part in the course, please send an email to @email  by 10 May 2024. 

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighborhood” given the automorphy of something in that neighborhood. The “neighborhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

We consider the general framework of distributionally robust optimization under a martingale restriction. We provide explicit expressions for model risk sensitivities in this context by considering deviations in the Wasserstein distance and the corresponding adapted one. We also extend the dual formulation to this context.