# Research

I obtained my PhD in 2012 at Utrecht University under supervision of Hans Duistermaat, Erik van den Ban, and Heinz Hanßmann. I worked on persistence of noncompact normally hyperbolic invariant manifolds; I have extended this theory to a setting of general Riemannian manifolds under the assumption of bounded geometry. My PhD thesis is available below, see my CRmath article for a brief statement of the results.

My recent work includes applications of normal hyperbolicity to network dynamics and biomechanics. At Imperial College I furthermore worked on geometric mechanics approaches to image registration and fluid dynamics. In a current project with Heinz Hanßmann, we look at persistence of normally hyperbolic KAM tori, where we pay special attention to the boundary of the invariant manifold.

Another long-standing research project originating from my PhD work is to investigate nonholonomically constrained systems as reduced systems and the question of which friction forces realize the Lagrange-d'Alembert description of such systems in the limit of infinite friction. See my talk given at the AIMS 2016 conference and my recent paper on realizing nonholonomic dynamics. I also work on this in a Poisson geometry setting with Paula Balseiro and Larry Bates.

## Publications

Research monograph:

**Normally hyperbolic invariant manifolds — the noncompact case**,

Springer Atlantis Series in Dynamical Systems,**vol 2**, xii+189 pages, August 2013. (pre-proof PDF, MathRev)

Articles:

**Weak dual pairs and jetlet methods for ideal incompressible fluid models in n ≥ 2 dimensions**,

with Colin Cotter, Darryl Holm, Henry Jacobs & David Meier,

Journal of Nonlinear Science**26.6**(2016) pp. 1723-1765. (article, arXiv, MathRev)**Realizing nonholonomic dynamics as limit of friction forces**,

Regular and Chaotic Dynamics**21.4**(2016) pp. 390-409. (article, arXiv, MathRev)**The role of symmetry and dissipation in biolocomotion**,

with Henry Jacobs,

SIAM J. Applied Dynamical Systems**15.1**(2016), pp. 24-59. (article, arXiv, MathRev)**A distance on curves modulo rigid transformations**,

with Joris Vankerschaver,

Differential Geometry and its Applications**36**(2014) pp. 149-164. (article, arXiv, MathRev)**Towards a general theory for coupling functions allowing persistent synchronization**,

with Tiago Pereira, Martin Rasmussen & Alexei Veneziani,

Nonlinearity**27.3**(2014) pp. 501-525. (article, arXiv, MathRev)**Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry**,

Comptes Rendus Mathematique**350.11-12**(2012) pp. 617-620. (article, arXiv, MathRev)

See also my ORCID and ResearcherID, and my articles on arXiv and MathSciNet.

A list of errata to my published works is available. Additional corrections are gratefully welcomed!

PhD thesis: **Persistence of noncompact normally hyperbolic
invariant manifolds in bounded geometry**, 229 pages, August 2012.

Written under supervision of Prof.dr. Hans Duistermaat, Dr. Heinz
Hanßmann, and Prof.dr. Erik van den Ban at Utrecht University.
(UU archive,
PDF,
arXiv)

MSc thesis: **The polygon model for 2+1D gravity: the constraint
algebra and problems of quantization**, 67 pages.

Written under supervision of Prof.dr. Renate Loll at Utrecht University.
(PDF,
arXiv)

## Presentations

I have given or will give the following talks/presentations:

- 26 April 2017, a talk on a chimera state in a Kuramoto oscillator network at ICMC-USP, Brazil.
- 12 January 2017, a talk on a chimera state in a Kuramoto oscillator network in the dynamical systems seminar at Utrecht University.
- 21 September 2016, a talk on nonholonomic dynamics as limit of infinite friction with two movies at ICMC-USP, Brazil.
- 30 August 2016, a talk on non-diffusion in families of hyperbolic Hamiltonian tori at the Billiards'16 meeting at ICMC-USP, Brazil.
- 1–5 July 2016, a talk on nonholonomic dynamics as limit of infinite friction at the AIMS dynamical systems conference in Orlando, USA.
- 5–6 May 2016, a poster on applying normal hyperbolicity to synchronization in (power-grid) networks, at the International Workshop in Industrial Mathematics at UNICAMP, Brazil.
- 6 January 2016, a talk on realizing nonholonomic dynamics, in the dynamical systems seminar at Utrecht University.
- 7–10 December 2015, a talk on nonholonomic dynamics as limit of infinite friction at the First Joint Meeting Brazil-Spain in Mathematics in Fortaleza.
- 29 October 2015, a talk on jetlet particles as symmetry reduced fluid dynamics model in the Rio joint Symplectic Seminar at PUC-Rio.
- 24 September 2015, a talk on synchronization in network dynamic at FGV EMAp in Rio de Janeiro.
- 3 September 2015, a talk on realizing nonholonomic dynamics through friction at ICMC-USP in São Carlos, Brazil.
- 3–7 August 2015, two lectures on the Chaplygin sleigh as example of nonholonomic dynamics as limit of friction at the 3rd winter school on Geometry, Mechanics, and Computation at PUC-Rio in Brazil.
- 15–19 June 2015, an invited talk on realizing nonholonomic dynamics through friction at the Lorentz Center workshop on Dynamics and Geometry in Leiden.
- 11 May 2015, a talk on jetlet particles as symmetry reduced fluid dynamics model, in the UvA–VU Dynamic Analysis Seminar in Amsterdam.
- 6 May 2015, a talk on jetlet particles as symmetry reduced fluid dynamics model, in the dynamical systems seminar at Utrecht University.
- 13 February 2015, a talk on realizing nonholonomic dynamics through friction, in the ANM seminar at the University of Bristol.
- October/November 2014, a "mini-course" on normal hyperbolicity and realisation of nonholonomic dynamics, in the informal DynamIC group meetings at Imperial College.
- 18 September 2014, a poster on biolocomotion, at the Advances in Applied Nonlinear Mathemathics workshop in Bristol.
- 8 July 2014, a talk on a model for biolocomotion using symmetry reduction and dissipation, at the AIMS 2014 conference in Madrid.
- 7 May 2014, a talk on a toy model for walking using symmetry reduction in the dynamical systems seminar at Utrecht University.
- 7–11 April 2014, a poster presentation on walking as a limit cycle at the Ibero-American conference on Geometric Mechanics & Control at IMPA, Brazil.
- 2 April 2014, a talk on NHIMs in a noncompact geometric setting in the Dynamical Systems seminar at IMPA, Brazil.
- 10 February 2014, a talk on a distance function for curve matching at the Analysis of Shape workshop at the MBI in Columbus, Ohio.
- 20 November 2013, a talk on synchronisation in networks in the dynamical systems seminar at Utrecht University.
- 7 May 2013, a talk in the ergodic theory and dynamical systems seminar at the University of Warwick.
- 18 October 2012, a talk in the dynamical systems seminar at Imperial College London.
- 26 April 2012, a short talk on normally hyperbolic invariant manifolds at the PhD days in analysis and dynamics in Lunteren, The Netherlands.
- 15 December 2011, a talk titled "Normally Hyperbolic Invariant Manifolds the noncompact way" in the Staff colloquium of the Mathematical Instutite at Utrecht University. (The slides comprise only the formal half of the presentation, these additional notes were used to explain things more informally.)
- May 2011, a poster presentation titled "Persistence of noncompact Normally Hyperbolic Invariant Manifolds" at the workshop Coherent Structures in Dynamical Systems in Leiden and at the SIAM DS11 conference in Snowbird, USA.
- 23 Februari 2011, a talk on the Perron method for normally hyperbolic invariant manifolds in the Dynamic Analysis Seminar at the Free University Amsterdam.
- 20 September 2010, a talk on persistence of noncompact normally hyperbolic invariant manifolds in the Dynamical Systems Seminar at the University of Groningen.
- 3 June 2010, a poster presentation titled "Persistence of non-compact Normally Hyperbolic Invariant Manifolds" at the DSPDEs'10 SIAM conference in Barcelona.
- 21 May 2010, a talk titled "Persistence of non-compact normally hyperbolic invariant manifolds" in the Dynamical Systems Seminar at the University of Amsterdam.
- 15 April 2010, a short talk on persistence of normally hyperbolic invariant manifolds at the NDNS+ workshop in Eindhoven.
- 17 March 2010, a talk titled "Persistence of normally hyperbolic invariant manifolds: a proof using the Perron method" in the Utrecht Dynamical Systems Seminar.
- 24 March 2009, a talk on proving the existence of invariant fibrations in normally hyperbolic systems using the Perron method, given in the dynamical systems seminar at Imperial College London.
- 3 November 2008, a talk on the implicit function theorem in Banach spaces, with an elegant proof, given in the Utrecht maths PhD students colloquium.
- 13 May 2008, a talk on reduction of dynamics in the Utrecht maths PhD students colloquium, giving an overview on how to achieve a reduction in dimension in (physical) dynamical systems, focussing specifically on non-holonomically constrained motion, and how to apply normally hyperbolic systems to that problem.