Journal

On 27 June 2017 I will be giving a talk at the JBI Mathematics Colloquium. The topic of my talk is Nontrivial Topology and Symmetry in Integrable Hamiltonian Systems.

The Chemical Engineering students in Groningen have selected me as Teacher of the Year 2016-2017 for my course Calculus for Chemistry. I am thankful to the students, and deeply honored, for selecting me as Teacher of the Year, and to the student association De Chemische Binding for organizing the election event.

I will be presenting a talk on Fractional Hamiltonian monodromy and circle actions at the conference Finite dimensional integrable systems in geometry and mathematical physics (FDIS 2017), which takes place at the CRM in Barcelona from 3 to 7 July 2017.

I will be presenting a talk on Fractional Hamiltonian monodromy at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), which takes place in Prague from 6 to 10 June 2017. The talk is part of the special session on Spectral Asymptotics of Quantum Integrable Systems that is being organized by John Toth, San Vũ Ngọc, and Steve Zelditch.

Pulse coupled oscillator networks with delay are a source of interesting dynamics, for example, unstable attractors. In recent work with Pan Li and Wei Lin we found that such networks exhibit isochronous regions―subsets of the phase space filled with periodic orbits having the same period. We studied such isochronous regions both analytically and numerically, giving a proof of their existence and a detailed description of their properties.

Together with Andrea Giacobbe and Tudor Ratiu we are organizing a special session on Geometry and Dynamics at the 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications (AIMS 2018) which takes place in Taipei, Taiwan from 5 to 9 July 2018.

On 2 December 2016 I give a talk on Fractional Hamiltonian Monodromy and Circle Actions at the GQT colloquium. The organizers of the GQT school and colloquium are Raf Bocklandt, Gil Cavalcanti, and Maarten Solleveld.

Several years ago I had written here about a Mathematica implementation of the algorithm for computing normal forms for Hamiltonian systems. That post was written for people who are already familiar how the algorithm works and could figure out what functions like rangePart are inverseAd are doing. To make that implementation more accessible I give here an example of how to use it to compute the normal form for the Hénon-Heiles Hamiltonian given by $$ H = \frac12 (p_1^2 + p_2^2 + q_1^2 + q_2^2) + q_1^2 q_2 - \frac13 q_2^3.

On 19 October 2016 I give a talk on Fractional Hamiltonian Monodromy and Circle Actions at the Symposium on advances in semi-classical methods in mathematics and physics. The symposium takes place in Groningen to honor H. J. Groenewold’s seminal 1946 paper On the principles of elementary quantum mechanics and the 70 years since its publication.

In this work, together with Nikolay Martynchuk, we investigated fractional monodromy in two-degree of freedom integrable Hamiltonian systems. In such systems, Seifert manifolds naturally appear by taking closed paths in the image of the integral map and considering the preimage of such paths in the phase space. Fractional monodromy is then defined through the parallel transport of homology cycles along Seifert manifolds, expanding on an idea from Uncovering fractional monodromy.

I first learned how to compute monodromy of torus bundles in integrable Hamiltonian systems from Cushman’s and Bates’s “Global Aspects of Classical Integrable Systems”, now at its second edition. The computation of monodromy there effectively boils down to the computation of the variation of the rotation number along properly chosen closed paths in the system.

I am giving a presentation on “Pulse Coupled Oscillator Networks” at the 7th Shanghai International Symposium on Nonlinear Sciences and Applications that takes place in Shanghai and Hunan between 13-19 July 2016.

I am giving an invited presentation on the “Bifurcations and Geometry in 1:1:-2 Resonant Hamiltonian Systems” at the conference Computational perturbative methods for Hamiltonian systems that takes place in Athens between 11-13 July 2016.

I gave an invited talk on “Generalized Hamiltonian Monodromy and Circle Actions” at the conference on Integrable Hamiltonian Systems that took place in Ascona from 19 to 24 June 2016. The conference was organized by Anton Alekseev, Sonja Hohloch, and Tudor Ratiu.

Monodromy in Hamiltonian systems has been traditionally associated to singularities of the integrable Hamiltonian fibration and, in particular, to singularities of focus-focus type. In a recent work with Nikolay Martynchuk, we showed that in 2 degree of freedom Hamiltonian systems with a circle action, to understand monodromy, one should look at the fixed point of the circle action.

At the end of October I will be in Chicago. On 26 October I am giving a seminar on recent results on monodromy at the Geometry, Topology and Dynamics Seminar at the University of Illinois at Chicago. From 25-30 October I am attending Vis2015 where we present our most recent work on particle selection.

A few years ago together with Lingyun Yu and Petra and Tobias Isenberg we presented CloudLasso, a method for selecting clusters in 3D particle data. We now introduce the CAST methods, novel context-aware algorithms for particle selection. For example, with PointCast one can select a whole particle cluster with a single mouse-click or touch; the algorithm infers an appropriate selection that can then be fine-tuned.

Filippov systems are dynamical systems with discontinuities. They are being used to model the dynamics in cases where there is very rapid transition between different dynamical regimes. For example, if in a predator-prey system the number of prey crosses a threshold then the predation rate can increase discontinuously. As the simplest case of a Filippov system one can consider a two-dimensional dynamical system where the phase space is separated into two regions.

From mid-April until mid-May 2015 I will be at Fudan University as a Senior Visiting Scientist at the Shanghai Key Laboratory of Contemporary Applied Mathematics. There I will be working with Prof. Lin Wei on the dynamics of pulse coupled oscillator networks.

I wrote a short article for the Nieuw Archief voor Wiskunde that appeared in the March 2015 issue. The article offers a bird’s eye view of my research, experiences, and aspirations. The pictures of the Hopf fibration and the curled torus look much better in color; you can see them below.