In May, Kaibo Hu, Jia Jia Qian, Puchun Zhou and I will be hosting a workshop, here at the University of Oxford, exploring common structures arising in finite element exterior and tensor calculus, discrete differential geometry, differential complexes, splines, topology and geometric PDEs. A central aim is to bring together researchers from different backgrounds and career stages, identify connections between different approaches, and chart new research directions.
The workshop will be held from the 24th to 27th of May 2027. We are currently finalising the venue and practical arrangements, and will send further details in due course.
Finite element exterior calculus (FEEC) has shown the value of preserving geometric and topological structures in the numerical approximation of PDEs. In particular, differential forms have provided a generic stepping stone towards, for instance, elasticity problems. More generally, finite element tensor calculus (FETC) extends this perspective to arbitrary tensor fields with general algebraic symmetries and differential structures.
This development points towards a broader problem: identifying universal discrete structures across mathematics and computation. Closely related questions arise in differential and finite element complexes; forms and fields valued in vector bundles or Lie algebras; discrete metrics, connections and curvature; the algebraic and combinatorial structure and dimension theory of splines; cohomology, spectral sequences and persistence; representation theory; and tensor computation. Across these subjects, common questions concern which structures should survive discretization, how they should be represented in finite-dimensional models, and how they influence analysis and computation.
For geometric PDEs, including Yang–Mills and Einstein equations, geometry itself becomes part of the numerical discretization, leading to a form of discrete geometric analysis. In evolutionary problems, preserving geometric and topological structures may also be essential for stability and faithful long-time dynamics. The workshop will bring together researchers from different backgrounds and career stages to exchange methods, identify common principles, and chart new research directions in FETC, discrete geometric structures and compatible discretizations.
Note, the European Finite Element Fair will take place in Oxford immediately afterwards, on 28th and 29th of May 2027.