Topic:
A differential complex is a sequence of linear maps between vector spaces such that the composition of two subsequent maps always vanishes. Important examples come from differential operators defined on sections (of some regularity) of vector bundles over smooth manifolds, for example on tensor fields of different types. Many important representatives fall into the class of Hilbert complexes.
These abstract concepts play a fundamental role in many fields of (pure and applied) mathematics and (theoretical) physics like differential and discrete geometry, algebra and topology, continuum modeling, relativity and gravity, functional and numerical analysis. For instance, they encode key structures in PDE-based models like crucial conservation principles and constraints. The language of differential complexes can also be used to express fundamental rigidity and compatibility results in differential geometry. The perspective of differential complexes reveals deep connections between these areas, connections that are only emerging, and whose discovery and further elaboration will require scientists with diverse backgrounds to talk to each other and start collaborating.
The topic of the programme is at the intersection of differential geometry, topology, continuum modeling, and numerical analysis. This translates into its central and natural goal of encouraging exchange and fostering collaboration among researchers with these diverse backgrounds. The field of the proposed thematic program has seen a surge of research activity in recent years and numerous exciting new results have been found already, with, in our opinion, many more awaiting discovery. This momentum offers great opportunities for a thematic program and it also vindicates the huge potential of approaching (numerical) modeling tasks from the direction of differential/Hilbert complexes.
This programme is dedicated to Doug Arnold, eminent numerical analyst and pioneer of Finite Element Exterior Calculus (FEEC)
Activities:
Workshops:
May 4-8: Workshop on "Theory of Differential Complexes and Related Models"
May 18-22: Workshop on "Discretization of Differential Complexes"
Short Courses and Labs:
April 21-24: Short course on "Finite Element Exterior Calculus (FEEC)" (R. Hiptmair), Link to tablet notes (PDF).
Contents:
Perspectives on Differential Forms
The L2 de Rham Hilbert complex
Chains, Cochains on meshes
Finite Element Spaces of Discrete Differential Forms
(Stable) Commuting Projections
Discrete Compactness
See details in the "Schedule" section of the ThP website.
April 27-30: Short course on "The Bernstein-Gelfand-Gelfand (BGG) Construction: Algebra, Geometry, and Analysis" (A. Cap and K. Hu)
May 11-15: Short course on "Continuum and Geometric Mechanics" (J. Schöberl, M. Neunteufel)
May 25-27: Finite Element Tensor Calculus (FETC) Implementation Lab (P. Brubeck Martinez et al.)