My research interests lie primarily in the geometry and analysis of non-linear partial differential equations arising in fluid dynamics and other areas of mathematical physics. The bulk of this work takes place in the setting of infinite-dimensional Riemannian geometry and incorporates tools from finite-dimensional geometry, Lie theory, functional and harmonic analysis, as well as partial differential equations. Much of my work is concerned with the geometry of diffeomorphism groups equipped with weak Sobolev metrics, with a particular focus on the kinetic energy metric associated with the Euler equations of ideal hydrodynamics.
Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves
Martin Bauer, Patrick Heslin and Cy Maor. Journal of Geometric Analysis 34 (2024), Article 214.
Journal article
Geometric analysis of the generalized surface quasi-geostrophic equations
Martin Bauer, Patrick Heslin, Gerard Misiołek and Stephen C. Preston. Mathematische Annalen 390 (2024), 4639–4655.
Journal article
Two-point boundary value problems on diffeomorphism groups
Patrick Heslin. International Mathematics Research Notices 2023, no. 23, 19902–19931.
Journal article
Distance Geometry and a Hydrodynamical Hopf-Rinow Criterion
Theodore D. Drivas, Patrick Heslin and Gerard Misiołek (2026).
arXiv:2610.03926
Exact non-stationary solutions of the Euler equations in two and three dimensions
Patrick Heslin and Stephen C. Preston (2026).
arXiv:2602.13929
Singular sets of Riemannian exponential maps in hydrodynamics
James Benn, Patrick Heslin, Leandro Lichtenfelz and Gerard Misiołek (2025).
arXiv:2509.03454