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Discrete mathematical modelling seminar
Seminar information archive ～04/30｜Next seminar｜Future seminars 05/01～
| Organizer(s) | Tetsuji Tokihiro, Ralph Willox |
|---|
2024/04/10
13:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Room change: Room 470 → Room 056
Jaume Alonso (Technische Universität Berlin)
Integrable birational maps and a generalisation of QRT to 3D (English)
Room change: Room 470 → Room 056
Jaume Alonso (Technische Universität Berlin)
Integrable birational maps and a generalisation of QRT to 3D (English)
[ Abstract ]
When completely integrable Hamiltonian systems are discretised, the resulting discrete-time systems are often no longer integrable themselves. This is the so-called problem of integrable discretisation. Two known exceptions to this situation in 3D are the Kahan-Hirota-Kimura discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one non-zero linear parameter β, both birational maps of degree 3. The integrals of these systems define pencils of quadrics. By analysing the geometry of these pencils, we develop a framework that generalises QRT maps and QRT roots to 3D, which allows us to create new integrable maps as a composition of two involutions. We show that under certain geometric conditions, the new maps become of degree 3. We use these results to create new families of discrete integrable maps and we solve the problem of integrability of the Zhukovski-Volterra gyrostat with two β’s.
This is a joint work with Yuri Suris and Kangning Wei.
When completely integrable Hamiltonian systems are discretised, the resulting discrete-time systems are often no longer integrable themselves. This is the so-called problem of integrable discretisation. Two known exceptions to this situation in 3D are the Kahan-Hirota-Kimura discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one non-zero linear parameter β, both birational maps of degree 3. The integrals of these systems define pencils of quadrics. By analysing the geometry of these pencils, we develop a framework that generalises QRT maps and QRT roots to 3D, which allows us to create new integrable maps as a composition of two involutions. We show that under certain geometric conditions, the new maps become of degree 3. We use these results to create new families of discrete integrable maps and we solve the problem of integrability of the Zhukovski-Volterra gyrostat with two β’s.
This is a joint work with Yuri Suris and Kangning Wei.
