This talk addresses the efficient Monte Carlo estimation of rare-event probabilities associated with a broad class of McKean--Vlasov stochastic differential equations (MV-SDEs), which arise in the analysis of mean-field systems in statistical physics, mathematical finance, and collective behaviour models. Standard Monte Carlo methods become computationally infeasible in this setting due to the rapid growth of the estimator's relative variance (coefficient of variation) in the rare-event regime. Using stochastic optimal control, an optimal importance sampling measure change is constructed to minimise the variance of the resulting estimator. The resulting double-loop Monte Carlo (DLMC) estimator with importance sampling significantly mitigates this growth in the coefficient of variation. The framework is further extended to the multilevel Monte Carlo setting to reduce computational complexity, leveraging propagation-of-chaos and strong antithetic coupling to ensure that the level differences vanish in the mean-field limit. To address the discontinuity of the probability observable, a numerical smoothing technique is introduced to recover optimal variance convergence rates. Numerical experiments on linear mean-field, Kuramoto, and Cucker--Smale models demonstrate computational savings of several orders of magnitude compared with standard Monte Carlo.
Abstract - A fundamental notion in quantum topology is that of topological quantum field theory (TQFT) formulated by Witten and Atiyah. This notion originates in ideas from quantum physics and constitutes a framework that organizes certain topological invariants of manifolds, called quantum invariants, which are defined by means of quantum groups.
Homotopy quantum field theories (HQFTs) are a generalization of TQFTs. The idea is to use TQFT techniques to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a (fixed) topological space called the target. The resulting invariants are called quantum homotopy invariants.
Turaev and Virelizier have constructed quantum homotopy invariants of 3-manifolds (by state sum) when the target space is a 1-type, and Sözer and Virelizier have recently constructed quantum homotopy invariants of 3-manifolds when the target space is a 2-type. Using state sum techniques, Douglas and Reutter have constructed quantum invariants of 4-manifolds using fusion 2-categories. In this talk, we combine both of these approaches: we construct quantum homotopy invariants of 4-manifolds with a 3-type target from 3-group extensions of fusion 2-categories.
Abstract - A fundamental notion in quantum topology is that of topological quantum field theory (TQFT) formulated by Witten and Atiyah. This notion originates in ideas from quantum physics and constitutes a framework that organizes certain topological invariants of manifolds, called quantum invariants, which are defined by means of quantum groups.
Homotopy quantum field theories (HQFTs) are a generalization of TQFTs. The idea is to use TQFT techniques to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a (fixed) topological space called the target. The resulting invariants are called quantum homotopy invariants.
Turaev and Virelizier have constructed quantum homotopy invariants of 3-manifolds (by state sum) when the target space is a 1-type, and Sözer and Virelizier have recently constructed quantum homotopy invariants of 3-manifolds when the target space is a 2-type. Using state sum techniques, Douglas and Reutter have constructed quantum invariants of 4-manifolds using fusion 2-categories. In this talk, we combine both of these approaches: we construct quantum homotopy invariants of 4-manifolds with a 3-type target from 3-group extensions of fusion 2-categories.
NOTES: Note the unusual room.
In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold is compact and spacelike-convex (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace $\mathbb{R}^{n+1,0}\subset \mathbb{R}^{n+1,k}$.
Becoming and succeeding as an elite mathematics student involves navigating a demanding academic and cultural environment. Women and disadvantaged students are underrepresented on the mathematics courses with the highest entry requirements compared to mathematics courses more broadly, raising questions about how admissions systems operate and about the social and learning consequences for those who are admitted.
This presentation draws on my qualitative doctoral research examining access to elite mathematics courses across four universities. Through in-depth interviews with 45 participants, including students, teachers and lecturers, I develop a multi-dimensional account of how elite mathematics ability is understood, assessed and experienced within course cultures.
Using ‘ability’ as an organising concept, I show how its construction varies across the four universities, with different emphases on competition, independence and resilience. I present findings from the perspective of women participants that challenge common constructions of ‘confidence’ and highlight the importance of relational forms of resilience and the social aspects of learning.
NOTES: online.
In a recent paper written jointly with Boaz Klartag, we prove that the variance of the Euclidean norm of any isotropic log-concave random vector is bounded above by a universal constant, not depending on the dimension. Thus, most of the mass of the random vector is concentrated in a thin spherical shell, whose width is order 1, while its radius is order root of the dimension. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of a given log-concave measure related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture. In this talk, I'll first review the context and the history of the problem, before laying out the main steps of our proof.
Thermal therapies such as hyperthermia, laser ablation, and high-intensity focused ultrasound rely on delivering controlled heat to biological tissue. The temperature propagation in the tissue cannot be modelled accurately with the Fourier's law, but should be modelled with the Maxwell–Cattaneo law instead, which gives rise to the hyperbolic bio-heat equation, that incorporates the assumption of the finite speed propagation. In this talk I will discuss inverse source problems for such a hyperbolic model, where the goal is to recover the unknown space-dependent component of the heat source.
Abstract: Although Évariste Galois died at the age of 20 after a duel, the ideas he introduced continue to influence modern mathematics nearly two centuries later. In this talk, we will begin with a short refresher on classical Galois theory before moving to differential Galois theory, introducing Picard–Vessiot theory to study linear differential equations. We will then discuss the difference-differential setting, where differential equations are considered together with additional endomorphism(s) acting on the base field, and briefly outline my current research on extending this theory to several commuting endomorphisms..
Advances in sensing technology have made it possible to collect large volumes of high-dimensional time-series data. In fields like genetics and neuroscience, key questions concern whether directed relationships between variables can be learned from these data. To this end, graphical vector autoregressions are a popular tool because zeros among the autoregressive coefficients and error precision matrix have natural interpretations in terms of Granger non-causality and contemporaneous conditional independence. In many applications where system dynamics are subject to functional or structural constraints, assuming the process is stable can be advantageous. However, enforcing stability demands restricting the autoregressive coefficients to lie in a constrained space with a complex geometry called the stationary region. The resulting inferential challenges are compounded when sparsity is also a requirement. Working in the Bayesian paradigm, we tackle the problem through a parameter expansion approach, constructing a spike-and-slab prior with support constrained to the stationary region. A mixture of G-Wishart distributions provides a sparse prior for the error precision matrix. Computational inference is carried out via a Metropolis-within-Gibbs scheme which exploits the No-U-Turn Sampler and reversible-jump steps. We demonstrate the benefits, both inferential and predictive, of our approach through simulation experiments and an application in neuroscience.
