(Imperial College, London)

We consider the homogenisation problem for the ϕ4/2 equation on the torus T² , i.e. the behaviour as ϵ → 0 of the solutions to the equations suggestively written

∂_tu_ϵ − ∇ · A(x/ϵ, t/ϵ² )∇uϵ = −u³_ϵ + ξ

where ξ denotes space-time white noise and A : T ² × R is uniformly elliptic, periodic and H¨older continuous. Based on joint work with M. Hairer

It is well understood by works of Fukaya and Teleman that the Fukaya category of a symplectic reduction at a regular value of the moment map can be computed before taking the quotient as an equivariant Fukaya category. Informed by mirror calculations,  we will give a new geometric interpretation of the equivariant Fukaya category corresponding to a singular value of the moment map where the equivariance is traded with wrapping.

Joint work in progress with Ed Segal.

 "In this talk, we will discuss invariant measures techniques to establish probabilistic global well-posedness for PDEs. We will go over the limitations that the Gibbs measures and the so-called fluctuation-dissipation measures encounter in the context of energy-supercritical PDEs. Then, we will present a new approach combining the two aforementioned methods and apply it to the energy supercritical Schrödinger equations. We will point out other applications as well."

Modern atmospheric and ocean science require sophisticated geophysical fluid dynamics models. Among them, stochastic partial

differential equations (SPDEs) have become increasingly relevant. The stochasticity in such models can account for the effect

of the unresolved scales (stochastic parametrizations), model uncertainty, unspecified boundary condition, etc. Whilst there is an

extensive SPDE literature, most of it covers models with unrealistic noise terms, making them un-applicable to

geophysical fluid dynamics modelling. There are nevertheless notable exceptions: a number of individual SPDEs with specific forms

and noise structure have been introduced and analysed, each of which with bespoke methodology and painstakingly hard arguments.

In this talk I will present a criterion for the existence of a unique maximal strong solution for nonlinear SPDEs. The work

is inspired by the abstract criterion of Kato and Lai [1984] valid for nonlinear PDEs. The criterion is designed to fit viscous fluid

dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in Holm [2015]. As an immediate application, I show that 

the incompressible SALT 3D Navier-Stokes equation on a bounded domain has a unique maximal solution.

This is joint work with Oana Lang, Daniel Goodair and Romeo Mensah and it is partially supported by European Research Council (ERC)

Synergy project Stochastic Transport in the Upper Ocean Dynamics (https://www.imperial.ac.uk/ocean-dynamics-synergy/

We will report on recent progress regarding the near-critical behavior of certain statistical physics models in dimension 3. Our results deal with the second-order phase transition associated to two percolation problems involving the Gaussian free field in 3D. In one case, they determine a unique ``fixed point'' corresponding to the transition, which is proved to obey one of several scaling relations. Such laws are classically conjectured to hold by physicists on the grounds of a corresponding scaling ansatz. 

Digital data capture is the backbone of all modern day systems and “Digital Revolution” has been aptly termed as the Third Industrial Revolution. Underpinning the digital representation is the Shannon-Nyquist sampling theorem and more recent developments include compressive sensing approaches. The fact that there is a physical limit to which sensors can measure amplitudes poses a fundamental bottleneck when it comes to leveraging the performance guaranteed by recovery algorithms. In practice, whenever a physical signal exceeds the maximum recordable range, the sensor saturates, resulting in permanent information loss. Examples include (a) dosimeter saturation during the Chernobyl reactor accident, reporting radiation levels far lower than the true value and (b) loss of visual cues in self-driving cars coming out of a tunnel (due to sudden exposure to light). 

To reconcile this gap between theory and practice, we introduce the Unlimited Sensing framework or the USF that is based on a co-design of hardware and algorithms. On the hardware front, our work is based on a radically different analog-to-digital converter (ADC) design, which allows for the ADCs to produce modulo or folded samples. On the algorithms front, we develop new, mathematically guaranteed recovery strategies.  

In the first part of this talk, we prove a sampling theorem akin to the Shannon-Nyquist criterion. We show that, remarkably, despite the non-linearity in sensing pipeline, the sampling rate only depends on the signal’s bandwidth. Our theory is complemented with a stable recovery algorithm. Beyond the theoretical results, we will also present a hardware demo that shows our approach in action.

Moving further, we reinterpret the unlimited sensing framework as a generalized linear model that motivates a new class of inverse problems. We conclude this talk by presenting new results in the context of single-shot high-dynamic-range (HDR) imaging, sensor array processing and HDR tomography based on the modulo Radon transform.

Abstract: In this talk we start by introducing a simple model for interbank default contagion in the vein of the  seminal clearing frameworks of Eisenberg & Noe (2001) and Rogers & Veraart (2013). The key feature, and main novelty, consists in combining stochastic dynamics of the external assets with a simple but realistic balance sheet methodology for determining early defaults. After first developing the model for a finite number of banks, we present a natural way of passing to the mean field limit such that the original network structure (of the interbank obligations) is maintained in a meaningful way. Thus, we provide a clear connection between the more classical network-based literature on systemic risk and the recent approaches rooted in stochastic particle systems and mean field theory.

In this talk I will introduce a continuous wetting model consisting of the law of a Brownian meander tilted by its local time at a positive level h, with h small. I will prove that this measure converges, as h tends to 0, to the same weak limit as for discrete critical wetting models. I will also discuss the corresponding gradient dynamics, which is expected to converge to a Bessel SPDE admitting the law of a reflecting Brownian motion as invariant measure. This is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.

Multiplicative preprojective algebras (MPAs) were originally defined by Crawley-Boevey and Shaw to encode solutions of the Deligne-Simpson problem as irreducible representations. 
MPAs have recently appeared in the literature from different perspectives including Fukaya categories of plumbed cotangent bundles (Etgü and Lekili) and, similarly, microlocal sheaves 
on rational curves (Bezrukavnikov and Kapronov.) After some motivation, I'll suggest a purely algebraic approach to study these algebras. Namely, I'll outline a proof that MPAs are 
2-Calabi-Yau if Q contains a cycle and an inductive argument to reduce to the case of the cycle itself.

This work deals with a class of stochastic optimal control problems in the presence of state constraints. It is well known that for such problems the value function is, in general, discontinuous, and its characterisation by a Hamilton-Jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics
of the controlled system. Here, we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions. To this end, the stochastic optimal control problem is first translated into a state-constrained stochastic target problem. Then a level-set approach is used to describe the backward reachable sets of the new target problem. It turns out that these backward reachable sets describe the value function. The main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalisation. However, the target problem involves a new state variable and a new control variable that is unbounded.