Journal
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. It turns out that fractional monodromy is determined by the exceptional fibers of the Seifert manifold and by the fixed points of the circle action. This result, in the same spirit as results for monodromy of torus bundles, provides a deep understanding of fractional monodromy and its origins. This is the first time that fractional monodromy has been associated to a certain type of singularities in integrable Hamiltonian systems (fixed points of a circle action). Details can be found in our paper which is now on the arXiv.
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.
Recently, together with Andrea Giacobbe, Pavao Mardešić, and Dominique Sugny we needed to better understand this computation. Studying the problem led us to a much better appreciation of the subtleties involved and revealed the deep connection between this analytical computation and the geometry of the fibration.
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. This result follows from the fact that, in such systems, a 2-torus bundle
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.
After Chicago I fly to Japan. First, to Kobe for SIGGRAPH ASIA 2015 and then, on 6 November, I will visit Kyoto University where I am giving an extended talk at the Kyoto Dynamical Systems Seminar.
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. For more details, check the paper and video. The paper will be presented at Visweek'15 in Chicago.
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. A smooth dynamical system is defined in each region but the two systems do not match in a continuous way along the discontinuity boundary.
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.
