University of Bath

There is a well-known relationship between finite W-algebras and Yangians. The work of Rogoucy and Sorba on the "rectangular case" in type A eventually led Brundan and Kleshchev to introduce shifted Yangians, which surject onto the finite W-algebras for general linear Lie algebras. Thus, these W-algebras can be realised as truncated shifted Yangians. In parallel, the work of Ragoucy and then Brown showed that truncated twisted Yangians are isomorphic to the finite W-algebra associated to a rectangular nilpotent element in a Lie algebra of type B, C or D. For many years there has been a hope that this relationship can be extended to other nilpotent elements.

I will report on a joint work with Lewis Topley in which we introduced the shifted twisted Yangians, following the work of Lu-Wang-Zhang, and described Poisson isomorphisms between their truncated semiclassical degenerations and the functions Slodowy slices associated with even nilpotent elements in classical simple Lie algebras( which can be viewed as semiclassical W-algebras). I will also mention a work in progress with Lu-Peng-Topley-Wang which deals with the quantum analogue of our theorem.

I will also recall what Poisson algebras and (filtered) quantizations are and give a brief intro to Slodowy slices, finite W-algebras and Yangians so that the talk should be quite accessible.

Namikawa has shown that the functor of flat graded Poisson deformations of a conic symplectic singularity is unobstructed and pro-representable. In a subsequent work, Losev showed that the universal Poisson deformation admits, a quantization which enjoys a rather remarkable universal property. In a recent work, we have repackaged the latter theorem as an expression of the representability of a new functor: the functor of quantizations. I will describe how this theorem leads to an easy proof of the existence of a universal equivariant quantizations, and outline a work in progress in which we describe a presentation of a rather complicated quantum Hamiltonian reduction: the finite W-algebra associated to a nilpotent element in a classical Lie algebra. The latter result hinges on new presentations of twisted Yangians.

A conjecture of Malle predicts an asymptotic formula for the number of number fields with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov-Poisson equation, described a phenomenon he called "violent relaxation," a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov-Poisson equation and, in particular, the critical role of regularity of the steady states in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

Cluster algebras and their quantum counterparts were invented in the early 2000s in an attempt to construct elements of dual canonical bases. This turned out to be a harder goal than first realised. In this talk I will aim to give an introduction and overview of the theory and display the wide range of interesting maths which has gone into making steps in this area. I will try to assume as little possible prior knowledge and instead focus on interesting questions which remain open in this area.

Deep learning has revolutionised many scientific fields and so it is no surprise that state-of-the-art solutions to several inverse problems also include this technology. However, for many inverse problems (e.g. in medical imaging) stability and reliability are particularly important.

Furthermore, unlike other image analysis tasks, usually only a fairly small amount of training data is available to train image reconstruction algorithms.

Thus, we require tailored solutions which maximise the potential of all ingredients: data, domain knowledge and mathematical analysis. In this talk we discuss a range of such hybrid approaches and will encounter along the way connections to various topics like generative models, convex optimization, differential equations and equivariance.

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. We develop a deep transport map that is suitable for sampling concentrated distributions defined by an unnormalised density function. We approximate the target distribution as the push-forward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a tensor train decomposition. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. We propose two bridging strategies suitable for wide use: tempering the target density with a sequence of increasing powers, and smoothing of an indicator function with a sequence of sigmoids of increasing scales. The latter strategy opens the door to efficient computation of rare event probabilities in Bayesian inference problems.

Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.
 

In this talk, I will discuss the notion of quantum limits from different viewpoints: Cordes' work on the Gelfand theory for pseudo-differential operators dating from the 70’s as well as the micro-local defect measures and semi-classical measures of the 90’s. I will also explain my motivation and strategy to obtain similar notions in subRiemannian or subelliptic settings. 

Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions such as the square-root lasso.  Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. The effectiveness of their usage typically depend on proper parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a different route. By over-parameterization and marginalising on certain variables (Variable Projection), we show how many popular non-smooth structured problems can be written as smooth optimization problems. The result is that one can then take advantage of quasi-Newton solvers such as L-BFGS and this, in practice, can lead to substantial performance gains. Another interesting aspect of our proposed solver is its efficiency when handling imaging problems that arise from fine discretizations (unlike proximal methods such as ISTA whose convergence is known to have exponential dependency on dimension). On a theoretical level, one can connect gradient descent on our over-parameterized formulation with mirror descent with a varying Hessian metric. This observation can then be used to derive dimension free convergence bounds and explains the efficiency of our method in the fine-grids regime.