Mathematical Institute University of Oxford

Understanding how living systems dynamically self-organise across spatial and temporal scales is a fundamental problem in biology; from the study of embryo development to regulation of cellular physiology. In this talk, I will discuss how we can use mathematical modelling to uncover the role of microscale physical interactions in cellular self-organisation. I will illustrate this by presenting two seemingly unrelated problems: environmental-driven compartmentalisation of the intracellular space; and self-organisation during collective migration of multicellular communities. Our results reveal hidden connections between these two processes hinting at the general role that chemical regulation of physical interactions plays in controlling self-organisation across scales in living matter

Underlying many biological models are chemical reaction networks (CRNs), and identifying allowed and forbidden dynamics in reaction networks may 
give insight into biological mechanisms. Algebraic approaches have been important in several recent developments. For example, computational 
algebra has helped us characterise all small mass action CRNs admitting certain bifurcations; allowed us to find interesting and surprising 
examples and counterexamples; and suggested a number of conjectures.   Progress generally involves an interaction between analysis and 
computation: on the one hand, theorems which recast apparently difficult questions about dynamics as (relatively tractable) algebraic problems; 
and on the other, computations which provide insight into deeper theoretical questions. I'll present some results, examples, and open 
questions, focussing on two domains of CRN theory: the study of local bifurcations, and the study of multistationarity.

We consider a system of interacting particles as a model for a foraging ant colony, where each ant is represented as an active Brownian particle. The interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of a pheromone field. Unlike conventional models, our study introduces a parameter that enables the reproduction of two distinctive behaviours: the conventional Keller-Segel aggregation and the formation of travelling clusters without relying on external constraints such as food sources or nests. We consider the associated mean-field limit of this system and establish the analytical and numerical foundations for understanding these particle behaviours.

Predicting real-world phenomena often requires an understanding of their causal relations, not just their statistical associations. I will begin this talk with a brief introduction to the field of causal inference in the classical case of structural causal models over directed acyclic graphs, and causal discovery for static variables. Introducing the temporal dimension results in several interesting complications which are not well handled by the classical framework. The main component of a constraint-based causal discovery procedure is a statistical hypothesis test of conditional independence (CI). We develop such a test for stochastic processes, by leveraging recent advances in signature kernels. Then, we develop constraint-based causal discovery algorithms for acyclic stochastic dynamical systems (allowing for loops) that leverage temporal information to recover the entire directed graph. Assuming faithfulness and a CI oracle, our algorithm is sound and complete. We demonstrate strictly superior performance of our proposed CI test compared to existing approaches on path-space when tested on synthetic data generated from SDEs, and discuss preliminary applications to finance. This talk is based on joint work with Georg Manten, Cecilia Casolo, Søren Wengel Mogensen, Cristopher Salvi and Niki Kilbertus: https://arxiv.org/abs/2402.18477 .

Mathematical methods are developed to characterize the asymptotics of recurrent neural networks (RNN) as the number of hidden units, data samples in the sequence, hidden state updates, and training steps simultaneously grow to infinity. In the case of an RNN with a simplified weight matrix, we prove the convergence of the RNN to the solution of an infinite-dimensional ODE coupled with the fixed point of a random algebraic equation. 
The analysis requires addressing several challenges which are unique to RNNs. In typical mean-field applications (e.g., feedforward neural networks), discrete updates are of magnitude O(1/N ) and the number of updates is O(N). Therefore, the system can be represented as an Euler approximation of an appropriate ODE/PDE, which it will converge to as N → ∞. However, the RNN hidden layer updates are O(1). Therefore, RNNs cannot be represented as a discretization of an ODE/PDE and standard mean-field techniques cannot be applied. Instead, we develop a fixed point analysis for the evolution of the RNN memory state, with convergence estimates in terms of the number of update steps and the number of hidden units. The RNN hidden layer is studied as a function in a Sobolev space, whose evolution is governed by the data sequence (a Markov chain), the parameter updates, and its dependence on the RNN hidden layer at the previous time step. Due to the strong correlation between updates, a Poisson equation must be used to bound the fluctuations of the RNN around its limit equation. These mathematical methods allow us to prove a neural tangent kernel (NTK) limit for RNNs trained on data sequences as the number of data samples and size of the neural network grow to infinity.

Optimization problems constrained on manifolds are prevalent across science and engineering. For example, they arise in (generalized) eigenvalue problems, principal component analysis, and low-rank matrix completion, to name a few problems. Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a (Riemannian) manifold.  Algorithms designed in this framework usually require some geometrical description of the manifold, i.e., tangent spaces, retractions, Riemannian gradients, and Riemannian Hessians of the cost function. However, in some cases, some of the aforementioned geometric components cannot be accessed due to intractability or lack of information.

In this talk, we present methods that allow for overcoming cases of intractability and lack of information. We demonstrate the case of intractability on canonical correlation analysis (CCA) and on Fisher linear discriminant analysis (FDA). Using Riemannian optimization to solve CCA or FDA with the standard geometric components is as expensive as solving them via a direct solver. We address this shortcoming using a technique called Riemannian preconditioning, which amounts to changing the Riemannian metric on the constraining manifold. We use randomized numerical linear algebra to form efficient preconditioners that balance the computational costs of the geometric components and the asymptotic convergence of the iterative methods. If time permits, we also show the case of lack of information, e.g., the constraining manifold can be accessed only via samples of it. We propose a novel approach that allows approximate Riemannian optimization using a manifold learning technique.

We present a method for obtaining approximate solutions to the problem of optimal execution, based on a signature method. The framework is general, only requiring that the price process is a geometric rough path and the price impact function is a continuous function of the trading speed. Following an approximation of the optimisation problem, we are able to calculate an optimal solution for the trading speed in the space of linear functions on a truncation of the signature of the price process. We provide strong numerical evidence illustrating the accuracy and flexibility of the approach. Our numerical investigation both examines cases where exact solutions are known, demonstrating that the method accurately approximates these solutions, and models where exact solutions are not known. In the latter case, we obtain favourable comparisons with standard execution strategies.