Mathematical Institute (University of Oxford)

Optimization problems constrained to a smooth manifold can be solved via the framework of Riemannian optimization. To that end, a geometrical description of the constraining manifold, e.g., tangent spaces, retractions, and cost function gradients, is required. In this talk, we present a novel approach that allows performing approximate Riemannian optimization based on a manifold learning technique, in cases where only a noiseless sample set of the cost function and the manifold’s intrinsic dimension are available.

Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this talk, I will introduce a novel class of scoring rules on path space based on signature kernels and use them as an objective for training Neural SDEs non-adversarially. The strict properness of such kernel scores and the consistency of the corresponding estimators, provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite-dimensional spaces of functions, this framework can be easily extended to generate spatiotemporal data. This procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics.

Join us for our first Fridays@4 session of Trinity about different academic routes people take post-PhD, with a particular focus on fellowships and grants. We’ll hear from Jason Lotay about his experiences on both sides of the application process, as well as hear about the experiences of ECRs in the South Wing, North Wing, and Statistics. Towards the end of the hour we’ll have a Q+A session with the whole panel, where you can ask any questions you have around this topic!

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

Identifying the hidden organizational principles and relevant structures of complex networks is fundamental to understand their properties. To this end, uncovering the structures involving the prominent nodes in a network is an effective approach. In temporal networks, the simultaneity of connections is crucial for temporally stable structures to arise. In this work, we propose a measure to quantitatively investigate the tendency of well-connected nodes to form simultaneous and stable structures in a temporal network. We refer to this tendency as the temporal rich club phenomenon, characterized by a coefficient defined as the maximal value of the density of links between nodes with a minimal required degree, which remain stable for a certain duration. We illustrate the use of this concept by analysing diverse data sets and their temporal properties, from the role of cohesive structures in relation to processes unfolding on top of the network to the study of specific moments of interest in the evolution of the network.

Article link: https://www.nature.com/articles/s41567-022-01634-8

The structure of complex networks can be characterized by counting and analyzing network motifs, which are small graph structures that occur repeatedly in a network, such as triangles or chains. Recent work has generalized motifs to temporal and dynamic network data. However, existing techniques do not generalize to sequential or trajectory data, which represents entities walking through the nodes of a network, such as passengers moving through transportation networks. The unit of observation in these data is fundamentally different, since we analyze observations of walks (e.g., a trip from airport A to airport C through airport B), rather than independent observations of edges or snapshots of graphs over time. In this work, we define sequential motifs in trajectory data, which are small, directed, and sequenced-ordered graphs corresponding to patterns in observed sequences. We draw a connection between counting and analysis of sequential motifs and Higher-Order Network (HON) models. We show that by mapping edges of a HON, specifically a kth-order DeBruijn graph, to sequential motifs, we can count and evaluate their importance in observed data, and we test our proposed methodology with two datasets: (1) passengers navigating an airport network and (2) people navigating the Wikipedia article network. We find that the most prevalent and important sequential motifs correspond to intuitive patterns of traversal in the real systems, and show empirically that the heterogeneity of edge weights in an observed higher-order DeBruijn graph has implications for the distributions of sequential motifs we expect to see across our null models.

ArXiv link: https://arxiv.org/abs/2112.05642

When you ask someone what maths is, their answer will massively depend on how they’ve been exposed to maths up until that point. From a 10-year-old who will tell you it’s adding up numbers, to a Fields medalist who may say to you about the idea of abstraction of logical ideas, there is no clear consensus as to the “right” answer to this question. Our individual journeys as mathematicians give us a clear idea about what it means to us, and this affects how we then communicate our ideas to an audience of other mathematicians and the general public. However, a pitfall that we easily fall into as a result is forgetting that others can understand maths in a different way to ourselves, and by only offering our preferred perspective, we are missing out on the chance to effectively communicate our ideas.

In this talk, I will explore how our individual understanding of what mathematics is can shape our methods of communication. I will review which methods of communication mathematicians utilise, and show examples where each method does well, and not so well.  Examples of communication methods include writing equations, plotting graphs, creating diagrams and storytelling. Given this, I will cover how by using a collection of these different methods, you can increase the impact of your research by engaging with the various different mindsets your audience may have on what mathematics is.

The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons.