The upcoming talk
At the next session of the TCS,
on the 12th of March 2025,
at 2:15 – 4:15 p.m. CET / 3:15 – 5:15 p.m. EET
Asier López-Gordón
(IM PAN)
shall talk about
“Homogeneous symplectic manifolds and integrable contact systems“.
Abstract
Contact dynamics has gained popularity as a geometric framework for modelling certain dynamical systems which are not Hamiltonian in the usual sense, including some dissipative systems, as well as some thermodynamic systems or cosmological models. On the other hand, integrable systems have fascinated geometers and mathematical physicists for decades. This has motivated us to initiate a program to extend the theory of integrable systems to the realm of contact geometry.
The category of contact manifolds is equivalent to the category of 1-homogeneous symplectic manifolds. Moreover, contact Hamiltonian vector fields are in one-to-one correspondence with (symplectic) Hamiltonian vector fields of 1-homogeneous functions. These equivalences unveil a way to study the integrability of contact dynamics, namely, by considering integrable systems whose structures are homogeneous of an appropriate degree. It is noteworthy that this approach simplifies the problem substantially but does not render it trivial, as results in the category of symplectic manifolds may fall outside the category of homogeneous symplectic manifolds if homogeneity is not properly accounted for.
We have proven a Liouville–Arnol’d theorem for homogeneous functions on homogenous symplectic manifolds, which shows the existence of homogeneous action-angle coordinates, and deduced from it a counterpart for contact manifolds. Moreover, we have shown that, unlike in the symplectic-Poisson case, the recursion operator relating two compatible Jacobi structures cannot produce a maximal set of functions in involution. Nevertheless, bi-Hamiltonian structures can still be used to obtain a maximal set of functions in involution on a contact manifold. At the end of the talk, I will outline different directions for further research on integrable contact systems.
This is a joint work with L. Colombo, M. de León, M. Lainz, and M. E. Eyrea Irazú.
(All seminar talks are given on the online zoom platform. The web link for every session is circulated via the TCS mailing list. In order to be enlisted, please contact the Organisers.)
Calendar 2024/2025
| Date | Speaker (affiliation) | Title (with a link to the slides, whenever available) |
| October 9th, 2024 | Thomas Strobl (ICJ, Université Claude Bernard Lyon 1) | From Gauge Theories to the octonionic Lie groupoid |
| October 16th, 2024 | Calin I. Lazaroiu (IFIN-HH, București & UNED, Madrid) | Consistency conditions and fiducial 2-field models for SRRT inflation |
| October 23rd, 2024 | Alexander Schmeding (NTNU, Trondheim) | Linking infinite-dimensional and higher differential geometry |
| October 30th, 2024 | Konrad Waldorf (Universität Greifswald) | Buscher rules in general topology |
| November 6th, 2024 | Mikołaj Rotkiewicz (MIMUW, Warsaw) | Exploring the structure of higher algebroids |
| November 13th, 2024 | No seminar on this date. | – |
| November 20th, 2024 | Participation in an online lecture by Alain Connes (Collège de France & IHES, Paris) at IM PAS | From class field theory to ζ spectral triples |
| November 27th, 2024 | Marian Aprodu (IMAR, București) | Resonance and vector bundles |
| December 4th, 2024 | Carlos Shahbazi (UNED, Madrid) | A class of differential-geometric problems in supergravity |
| December 11th, 2024 | No seminar on this date. | – |
| December 18th, 2024 | Anton Alekseev (Université de Genève) | Jackiw-Teitelboim gravity, Teichmüller spaces, and Virasoro algebra |
| January 8th, 2025 | Ingo Runkel (Universität Hamburg) | Topological symmetries and their gaugings in 2dCFT and 3dTFT |
| January 15th, 2025 | No seminar on this date. | – |
| January 22nd, 2025 | Peter Kristel (Universität Greifswald) | An Eclectic Excursion into 2-Categorical Geometry |
| February 26th, 2025 | No seminar on this date. | – |
| March 5th, 2025 | Katarzyna Grabowska (KMMF WFUW) | Dirac algebroids in Hamiltonian and Lagrangian dynamics – theory and examples |
| March 12th, 2025 | Asier López-Gordón (IM PAN) | Homogeneous symplectic manifolds and integrable contact systems |
| March 19th, 2025 | Leonid Ryvkin (ICJ, Université Claude Bernard Lyon 1) | Introduction to singular foliations |
| March 26th, 2025 | Witold Respondek (INSA Rouen Normandie) | TBA |
| April 2nd, 2025 | Bartosz Prech | TBA |
| April 9th, 2025 | Rafał R. Suszek (KMMF WFUW) | TBA |
| April 16th, 2025 | Tilmann Wurzbacher (IECL Metz) | TBA |
| April 23rd, 2025 | ||
| April 30th, 2025 | ||
| May 7th, 2025 | ||
| May 14th, 2025 | ||
| May 21st, 2025 | Nils Carqueville (Universität Wien) | Higher structures in topological quantum field theory |
| May 28th, 2025 | ||
| June 4th, 2025 | ||
| June 11th, 2025 |