India Marsden will talk about: 'Expanding the definition of a finite element: groups, complexes and software'

The finite element method is a flexible framework to discretise and solve partial differential equations which has been applied to many problems across science and engineering, for example weather modelling and battery design. A core feature of the success of the finite element method, the Ciarlet definition of the components of a finite element has been used for many years. The experience of these decades (and the subsequent implementations) has exposed several key deficiencies. In particular, Ciarlet’s definition is missing information about the global continuity of the mesh and how the degrees of freedom map to each other under the relative orientation of the mesh entities. This information is necessary to implement the finite element method, leaving scope for a new definition.

We propose a new definition to handle these issues and incorporate the constantly growing landscape of new elements. This new definition also aims to encapsulate more information about the elements, such as the symmetries, incorporating ideas from Group Theory. Through this work, we hope to produce a robust, thorough definition that allows processes such as implementation-independent serialisation of finite element data.

Alongside this new definition, we will discuss the new software FUSE, which provides a domain specific language for the definition and enables elements defined in this way to be used in high performance simulation using the finite element package Firedrake. 

A difficult question in the local Langlands framework is to understand the interplay between the characters of irreducible smooth representations of a reductive group over a local field and the geometry of the dual space of Langlands parameters. An important invariant of the character (viewed as a distribution, i.e, a continuous linear functional on the space of smooth compactly supported functions) is the wavefront set, a measure of its singularities along with their directions. Motivated by the work of Adams, Barbasch, and Vogan for real reductive groups, it is natural to expect that the wavefront set is dual (in a certain sense) to the geometric singular support of the Langlands parameter. Dan Ciubotaru will give an overview of these ideas and describe recent progress in establishing a precise connection for representations of reductive p-adic groups. 

Modern ML methods for derivatives sit at a delicate interface between market prices, implied-volatility (IV) surfaces, and the simulated environments produced by market generators. To date, these models have largely operated in one of two coordinate systems: price space, where markets quote and no-arbitrage constraints are most naturally enforced, and IV space, where surfaces are smoothed, regularized, and evaluated. This talk presents a technique that unifies learning across both coordinates — using gradients from each via a differentiable Jäckel operator and a low-vega gating mechanism — enabling end-to-end batch training without the error-prone, expensive, hand-engineered filtering usually needed to discard incompatible IV values. I will present PIVOT (Price-Implied Volatility Operator Transform), a differentiable Jäckel IV operator that preserves the accuracy of the standard "Let's Be Rational" (LBR) solver in the forward pass while supplying implicit gradients through the Black–Scholes/Black-76 price map. This gives neural volatility-surface models a principled bridge between price-space and IV-space objectives, with explicit handling of the low-vega singular regime. Second, I will  present Fast-Vollib ( https://pypi.org/project/fast-vollib/), a CUDA-accelerated option-pricing library with NumPy, PyTorch, and JAX interfaces, built for high-throughput pricing-label generation in AI/ML batch training.