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Discrete mathematical modelling seminar
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Organizer(s) | Tetsuji Tokihiro, Ralph Willox |
|---|
Future seminars
2024/04/24
13:30-15:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Jaume Alonso (Technische Universität Berlin)
Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups (English)
Jaume Alonso (Technische Universität Berlin)
Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups (English)
[ Abstract ]
In this talk we propose a new method for the exact computation of the degree $\deg (f^n)$ of the iterates of a birational map $f:\mathbb{P}^n \dashrightarrow \mathbb{P}^n$. The method is based on two main ingredients. Firstly, the factorisation of a polynomial under the pull-back by $f$, based on local indices of a polynomial associated to blow-ups used to resolve the singularity. Secondly, the propagation of these indices along the orbits of $f$. We will illustrate the method in different examples, showing its flexibility, since it does not require the construction of an algebraically stable lift of $f$, unlike other methods based on the Picard group.
This is a joint work with Yuri Suris and Kangning Wei.
In this talk we propose a new method for the exact computation of the degree $\deg (f^n)$ of the iterates of a birational map $f:\mathbb{P}^n \dashrightarrow \mathbb{P}^n$. The method is based on two main ingredients. Firstly, the factorisation of a polynomial under the pull-back by $f$, based on local indices of a polynomial associated to blow-ups used to resolve the singularity. Secondly, the propagation of these indices along the orbits of $f$. We will illustrate the method in different examples, showing its flexibility, since it does not require the construction of an algebraically stable lift of $f$, unlike other methods based on the Picard group.
This is a joint work with Yuri Suris and Kangning Wei.
2024/05/01
13:00-15:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Jaume Alonso (Technische Universität Berlin)
Semitoric systems and their symplectic invariants (English)
Jaume Alonso (Technische Universität Berlin)
Semitoric systems and their symplectic invariants (English)
[ Abstract ]
Semitoric systems are a special class of completely integrable systems defined on four-dimensional symplectic manifolds. One of the reasons that make these systems interesting is their classification in terms of five symplectic invariants proposed by Pelayo and Vũ Ngọc. In the last years, many efforts have been made in order to extend this classification to broader settings, to generate more examples and to compute their invariants. In this talk we will discuss some of the most important properties of semitoric systems and introduce some families of systems with one and more focus-focus singularities. We will also show how the symplectic invariants of these systems change as we move the parameters of the families and how they can be computed using mathematical software.
This is a joint work with H. Dullin, S. Hohloch and J. Palmer.
Semitoric systems are a special class of completely integrable systems defined on four-dimensional symplectic manifolds. One of the reasons that make these systems interesting is their classification in terms of five symplectic invariants proposed by Pelayo and Vũ Ngọc. In the last years, many efforts have been made in order to extend this classification to broader settings, to generate more examples and to compute their invariants. In this talk we will discuss some of the most important properties of semitoric systems and introduce some families of systems with one and more focus-focus singularities. We will also show how the symplectic invariants of these systems change as we move the parameters of the families and how they can be computed using mathematical software.
This is a joint work with H. Dullin, S. Hohloch and J. Palmer.
