# Foundations of Numerical Differential Geometry

### TMS Workshop on

# Foundations of Numerical Differential Geometry

The central aim of the workshop is to bring together researchers from different backgrounds in geometry and applied mathematics, to discuss and advance both fields. Particular emphasis will be put on aspects of the following topics: Numerical mathematics, applied differential geometry, geometric methods for partial differential equations, and numerical methods for analysis on Lie groups and manifolds.

## Invited Speakers

- Nicolas Boumal (EPFL)
- David Martin de Diego (ICMAT)
- Ana Djurdjevac (FU Berlin)
- Hanne Hardering (TU Dresden)
- Annika Lang (Chalmers University of Technology)
- Klas Modin (Chalmers University of Technology)

## Venue

Thw workshop will take place at NTNU in Trondheim, Norway.

# Talks

**Abstract.** We study a higher-order surface finite-element (SFEM) penalty-based discretization of the tangential surface Stokes problem. We analyze the inf-sup stability of the discrete scheme in a generic approach by lifting stable finite-element pairs known from the literature. The discretization errors depend on the best-approximation error of the finite element pairs, the choice of the penalty parameters, and the geometric errors of the surface approximation.

We discuss how this influences several different discrete formulations, and show optimal order convergence in tangential norms of an isogeometric setting that requires only geometric knowledge of the discrete surface.

**Abstract.** Looking around us, many surfaces including the Earth are no plain Euclidean domains but special cases of Riemannian manifolds. One way of describing uncertain physical phenomena on these surfaces is via stochastic partial differential equations. In this talk, I will introduce how to compute approximations of solutions to such equations and give convergence results to characterize the quality of the approximations. Furthermore, I will show how these solutions on surfaces are a first step towards the computation of time-evolving stochastic manifolds.

**Abstract.** The “reversibility paradox” refers to the seeming contradiction between two major discoveries made during the end of the 19th century: the law of entropy, given by Boltzmann, and the recurrence theorem, given by Poincaré. The discourse is captured in the following question: how can reversible dynamics give rise to irreversible behavior?

In this talk, I shall discuss how the same enigma shows up in a finite-dimensional model of the 2-D incompressible Euler equations. The model, suggested by V. Zeitlin in 1991, uses quantization theory to replace the vorticity formulation of the 2-D Euler equations by an isospectral flow of matrices. This way, a spatial discretization preserving all the geometric structure is obtained. Recent advances on Zeitlin’s model propose a connection between the long-time behavior of (generic) 2-D Euler flows and integrability conditions for “blob dynamics””. A major point at issue, however, is whether simulations based on Zeitlin’s model truly reflect the dynamics of 2-D Euler. It is within this context the reversibility paradox enters.