Erlend Grong

Forum mathematicum Doi: https://doi.org/10.1515/forum-2024-0160

In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.

Journal of Lie theory 34(2) p. 339-352

Applied Mathematics and Optimization 89(2) Doi: https://doi.org/10.1007/s00245-024-10110-z

We identify most probable flows for Kunita Brownian motions, i.e. stochastic flows with Eulerian noise and deterministic drifts. Such stochastic processes appear for example in fluid dynamics and shape analysis modelling coarse scale deterministic dynamics together with fine-grained noise. We treat this infinite dimensional problem by equipping the underlying domain with a Riemannian metric originating from the noise. The resulting most probable flows are compared with the non-perturbed deterministic flow, both analytically and experimentally by integrating the equations with various choice of noise structures.

Communications in analysis and mechanics 15(3) p. 515-532 Doi: https://doi.org/10.3934/cam.2023025

Differential geometry and its applications 85:101952 Doi: https://doi.org/10.1016/j.difgeo.2022.101952

With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K-contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound on these structures, we prove a sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian. Then, using a result by Moroianu-Semmelmann [38], we classify the H-type foliations that carry a parallel horizontal Clifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature of these spaces in codimension more than 2.

Archivum Mathematicum 58(5) p. 295-327 Doi: https://doi.org/10.5817/AM2022-5-295

Foundations of Computational Mathematics Doi: https://doi.org/10.1007/s10208-022-09594-4

Annals of Global Analysis and Geometry 61 p. 759-776 Doi: https://doi.org/10.1007/s10455-022-09827-3

Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors in Baudoin and Grong (Ann Glob Anal Geom 56(2):403–428, 2019).

Journal of Differential Equations 340 p. 151-178 Doi: https://doi.org/10.1016/j.jde.2022.08.034

Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For α ∈ (1/3,1/2), we give criteria for when we can approximate Banach space-valued weakly geometric α-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.

Mathematische Zeitschrift 65 p. 1853-1885 Doi: https://doi.org/10.1007/s00209-020-02653-y

We give the complete classification of all sub-Riemannian model spaces with both step and rank three. Model spaces in this context refer to spaces where any infinitesimal isometry between horizontal tangent spaces can be integrated to a full isometry. They will be divided into three families based on their nilpotentization. Each family will depend on a different number of parameters, making the result crucially different from the known case of step two model spaces. In particular, there are no nontrivial sub-Riemannian model spaces of step and rank three with free nilpotentization. We also realize both the compact real form gc2 and the split real form gs2 of the exceptional Lie algebra g2 as isometry algebras of different model spaces.

Communications in analysis and geometry 29(1) p. 77-113 Doi: https://doi.org/10.4310/CAG.2021.V29.N1.A3

Nonlinear Analysis 210 Doi: https://doi.org/10.1016/j.na.2021.112387

We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians of a sub-Riemannian structure with a chosen complement. We introduce a concept of gradient for cylindrical functionals on path space in such a way that the gradient operators are closable in . With this structure in place, we show that a bound on horizontal Ricci curvature is equivalent to several inequalities for functions on path space, such as a gradient inequality, log-Sobolev inequality and Poincaré inequality. As a consequence, we also obtain a bound for the spectral gap of the Ornstein–Uhlenbeck operator.

Electronic Journal of Probability (EJP) 25 Doi: https://doi.org/10.1214/20-EJP501

We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô’s formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng’s type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.

Annales de l'Institut Fourier 69(3) p. 1047-1086 Doi: https://doi.org/10.5802/aif.3265

We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose–Singer’s and Ozeki’s theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle D plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b).

Mathematische Nachrichten 292(9) p. 1956-1971 Doi: https://doi.org/10.1002/mana.201800352

Proceedings of the American Mathematical Society 147(12) p. 5153-5166 Doi: https://doi.org/10.1090/proc/14223

Abstract: For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.

Annals of Global Analysis and Geometry 56(2) p. 403-428 Doi: https://doi.org/10.1007/s10455-019-09672-x

Calculus of Variations and Partial Differential Equations 58(4) Doi: https://doi.org/10.1007/s00526-019-1570-8

Forum mathematicum 30(6) p. 1363-1385 Doi: https://doi.org/10.1515/forum-2017-0029

Potential Analysis 51(2) p. 219-254 Doi: https://doi.org/10.1007/s11118-018-9710-x

Journal of Geometric Analysis 27(2) p. 1260-1273 Doi: https://doi.org/10.1007/s12220-016-9717-8

SIAM Journal of Control and Optimization 54(2) p. 536-566 Doi: https://doi.org/10.1137/15m1008919

Mathematische Zeitschrift 282(1-2) p. 99-130 Doi: https://doi.org/10.1007/s00209-015-1534-4

Mathematische Zeitschrift 282(1-2) p. 131-164 Doi: https://doi.org/10.1007/s00209-015-1535-3

Journal of Geometric Analysis 25(4) p. 2474-2515 Doi: https://doi.org/10.1007/s12220-014-9523-0

Communications on Pure and Applied Analysis 13(1) p. 435-452

SIAM Journal of Control and Optimization 50(4) p. 2462-2485 Doi: https://doi.org/10.1137/110829581

Journal of Dynamical and Control Systems 18(2) p. 181-214 Doi: https://doi.org/10.1007/s10883-012-9139-2

60 years of Analytic Functions in Lublin. In memory of our professors and friends Jan G. Krzyz, Zdzislaw Lewandowski and Wojciech Szapiel p. 97-116

Journal of Geometric Mechanics (JGM) 3(2) p. 225-260 Doi: https://doi.org/10.3934/jgm.2011.3.225

Textos de matemática : mathematical papers in honour of Fátima Silva Leite p. 71-84

Annales Academiae Scientiarum Fennicae Mathematica 34(1) p. 303-314

Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of the Virasoro algebra in the space of univalent functions. 2000 Mathematics Subject Classification: Primary 30C35; Secondary 17B68. Key words: Matching univalent functions, conformal welding, Kirillov's manifold, Virasoro algebra.

Symmetry and differential geometry

The Abel Symposium 2019

Workshop in Analysis and PDE

Young Geometric Analysts’ Forum

Geometric and singular analysis

Dynamical Geometric Analysis in Orsay

Subriemannian structures on Lie groups, differential forms and PDE

Geometric control in control and vision theory

Ninth School of Analysis and Geometry in Metric Spaces

ProbaGeo 2014

Thematic day on Rolling manifold

Harmonic and Complex Analysis and its Applications

Sub-Riemannian geometry and PDEs

Workshop: Geometric analysis and mathematical physics

2011 NCTS Taiwan-Norway Joint Workshop on Analysis and Its Applications

XXX Workshop on Geometric Methods in Physics

Complex Analysis and Mathematical Physics

Controlo 2010: 9th Portuguese conference on automatic control

NCTS Taiwan-Norway Joint Workshop on Geometric Analysis and Mathematical Physics

Norway International workshop in Complex Analysis-dedicated to the 65-th anniversary of professor Arne Stray

Norwegian Fall School in Analysis for PhD Students and Young Researchers

Norway-US international workshop Complex Analysis and Mathematical Physics