Departmental Seminars & Events

I will present a recent work with G. Kocharyan, where we show the undecidability of the following two problems: given a finitely generated subgroup G of GL(n,Q), a) determine whether G has a non-identity element whose (i,j) entry is equal to zero, and b) determine whether the stabilizer of a given vector in G is non-trivial. Undecidability of problem b) answers a question of Dixon from 1985. The proofs reduce to the undecidability of the word problem for finitely presented groups.

What do fiber polymers and ice sheets have in common? They both flow with a directionally dependent - anisotropic - viscosity. This behaviour occurs in other geophysical flows, such as the Earth's mantle, where a material's microstructure affects its large-scale flow. In ice, the alignment of crystal orientations can cause the viscosity to vary by an order of magnitude, consequently having a strong impact on the flow of ice sheets and glaciers. However, the effect of anisotropy on large-scale flow is not well understood, due to a lack of understanding of a) the best physical approximations to model crystal orientations, and b) how crystal orientations affect rheology. In this work, we aim to address both these questions by linking rheology to crystal orientation predictions, and testing a range of models against observations from the Greenland ice sheet. The results show assuming all grains experience approximately the same stress provides realistic predictions, and we suggest a set of equations and parameters which can be used in large-scale models of ice sheets. 

In this talk, we discuss continuous in time dynamics for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V . Such problems are motivated by minimax convexconcave and, in particular, by convex optimization problems with linear constraints. The central role is played by a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. We show that these methods exhibit fast convergence rates for kV (z(t))k as t ! +1, where z( ) denotes the generated trajectory, and for the restricted gap function, and that z( ) converges to a zero of the operator V . For the corresponding implicit and explicit discrete time models with Nesterov’s momentum, we prove that they share the asymptotic features of the continuous dynamics.

Extensions to variational inequalities and fixed-point problems are also addressed. The theoretical results are illustrated by numerical experiments on bilinear games and the training of generative adversarial networks.

In this seminar we introduce a portfolio optimisation framework, in which the use of rough path signatures (Lyons, 1998) provides a novel method of incorporating path-dependencies in the joint signal-asset dynamics, naturally extending traditional factor models, while keeping the resulting formulas lightweight, tractable and easily interpretable. Specifically, we achieve this by representing a trading strategy as a linear functional applied to the signature of a path (which we refer to as “Signature Trading” or “Sig-Trading”). This allows the modeller to efficiently encode the evolution of past time-series observations into the optimisation problem. In particular, we derive a concise formulation of the dynamic mean-variance criterion alongside an explicit solution in our setting, which naturally incorporates a drawdown control in the optimal strategy over a finite time horizon. Secondly, we draw parallels between classical portfolio stategies and Sig-Trading strategies and explain how the latter leads to a pathwise extension of the classical setting via the “Signature Efficient Frontier”. Finally, we give explicit examples when trading under an exogenous signal as well as examples for momentum and pair-trading strategies, demonstrated both on synthetic and market data. Our framework combines the best of both worlds between classical theory (whose appeal lies in clear and concise formulae) and between modern, flexible data-driven methods (usually represented by ML approaches) that can handle more realistic datasets. The advantage of the added flexibility of the latter is that one can bypass common issues such as the accumulation of heteroskedastic and asymmetric residuals during the optimisation phase. Overall, Sig-Trading combines the flexibility of data-driven methods without compromising on the clarity of the classical theory and our presented results provide a compelling toolbox that yields superior results for a large class of trading strategies.

This is based on works with Blanka Horvath and Magnus Wiese.

There have been a lot of interests in understanding the behaviour of random multiplicative functions, which are probabilistic models for deterministic arithmetic functions such as the Möbius function and Dirichlet characters. Despite recent advances, the limiting distributions of partial sums of random multiplicative functions remain mysterious even at the conjectural level. In this talk, I shall discuss the so-called $L^2$ regime of twisted sums and provide a precise answer to the distributional problem. This is based on ongoing work with Ofir Gorodetsky.

The chiralization in the title denotes a certain procedure which turns cluster X-varieties into q-W algebras. Many important notions from cluster and q-W worlds, such as mutations, global functions, screening operators, R-matrices, etc emerge naturally in this context. In particular, we discover new bosonizations of q-W algebras and establish connections between previously known bosonizations. If time permits, I will discuss potential applications of our approach to the study of 3d topological theories and local systems with affine gauge groups. This talk is based on a joint project with J. Shiraishi, J.E. Bourgine, B. Feigin, A. Shapiro, and G. Schrader.

The orbit method is a fundamental tool to study a finite dimensional solvable Lie algebra g. It relates the annihilators of simple U(g)-module to the coadjoint orbits of the adjoint group on g^* . In my talk, I will extend this story to the Witt algebra – a simple (non-solvable) infinite dimensional Lie algebra which is important in physics and representation theory. I will construct an induced module from an element of W^* and show that its annihilator is a primitive ideal. I will also construct an algebra homomorphism that allows one to relate the orbit method for W to that of a finite dimensional solvable algebra.

The role of tissue stiffness in controlling cell behaviours ranging from proliferation to signalling and activation is by now well accepted. A key focus of experimental studies into mechanotransduction are focal adhesions, localised patches of strong adhesion, where cell signalling has been established to occur. However, these adhesion sites themselves alter the mechanical equilibrium of the system determining the force balance and work done. To explore this I have developed an active matter continuum description of cellular contractility and will discuss recent results on the specific role of spatial positioning of adhesions in mechanotransduction. I show using energy arguments why the experimentally observed arrangements of focal adhesions develop and the implications this has for stiffness sensing and cellular contractility control. I will also show how adhesions play distinct roles in single cells and tissue layers respectively drawing on recent experimental work with Dr JR Davis (Manchester University) and Dr Nic Tapon (Crick Institute) with applications to epithelial layers and organoids.

What makes a good talk? This year, graduate students and postdocs will give a series talks on how to give talks! There may even be a small prize for the audience’s favourite.

If you’d like to have a go at informing, entertaining, or just have an axe to grind about a particularly bad talk you had to sit through, we’d love to hear from you (you can email Ric Wade or ask any of the organizers).
 

Data that have an intrinsic network structure can be found in various contexts, including social networks, biological systems (e.g., protein-protein interactions, neuronal networks), information networks (computer networks, wireless sensor networks),  economic networks, etc. As the amount of graphical data that is generated is increasingly large, compressing such data for storage, transmission, or efficient processing has become a topic of interest. In this talk, I will give an information theoretic perspective on graph compression. 

The focus will be on compression limits and their scaling with the size of the graph. For lossless compression, the Shannon entropy gives the fundamental lower limit on the expected length of any compressed representation. I will discuss the entropy of some common random graph models, with a particular emphasis on our results on the random geometric graph model. 

Then, I will talk about the problem of compressing a graph with side information, i.e., when an additional correlated graph is available at the decoder. Turning to lossy compression, where one accepts a certain amount of distortion between the original and reconstructed graphs, I will present theoretical limits to lossy compression that we obtained for the Erdős–Rényi and stochastic block models by using rate-distortion theory.

The quadratic Euler characteristic of an algebraic variety is a (virtual) symmetric bilinear form which refines the topological Euler characteristic and contains interesting arithmetic information when the base field is not algebraically closed. For smooth projective varieties, it has a quite concrete expression in terms of the cup product and Serre duality for Hodge cohomology. However, for singular varieties, it is defined abstractly (using either cut and paste relations or motivic homotopy theory) and is still rather mysterious. I will first introduce this invariant and place it in the broader context of quadratic enumerative geometry. I will then explain some progress on concrete computations, first for symmetric powers (joint with Lenny Taelman) and second for conductor formulas for hypersurface singularities (older results with Marc Levine and Vasudevan Srinivas on the one hand, and joint work in progress with Ran Azouri, Niels Feld, Yonathan Harpaz and Tasos Moulinos on the other).

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.

TBC

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. 

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.

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.

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.

TBA

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.

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.

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.

TBA

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).

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.

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.

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.