报告题目 (Title)：Dynamics on and off elliptic curves: from Newton to Okamoto: I, II（基于椭圆曲线的动力学：从Newton 到 Okamoto）
报告人 (Speaker)：Nalini Joshi 教授 (悉尼大学，澳大利亚）
报告时间 (Time)：I. 2024年09月29日（周日）10:00-11:30
II. 2024年09月30日（周一）10:00-11:30
报告地点 (Place)：校本部GJ303
邀请人(Inviter)：张大军
主办部门：理学院数学系
报告摘要：
Lecture 1: Elliptic curves are well known have a canonical cubic form, parametrized by Weierstrass elliptic functions, with points on each curve moving along the curve by due to the parametrization: $(\wp(t-t_0), \wp(t-t_0))$. Elliptic curves also arise as autonomous cases of polynomial Hamiltonians, given by Okamoto (1981), for each Painlev\’e equation. However, the evolution of a solution of a Painlev\’e equation changes the elliptic curve and so points move from one such curve to another under its time evolution. One of the only ways in which we can describe their solutions, the Painlev\’e transcendents, is to consider asymptotic limits. I will give a toy model to describe how Boutroux (1913) initiated such asymptotic descriptions.
Lecture 2: I will describe 3 classes of solutions arising in the toy model from the last lecture in the limit as the independent variable $|z|\to\infty$, which are analogous to those seen in the Painlev\’e equations. Next, we give an overview of asymptotic results for the first Painlev\’e equation before describing how we deduced global results from a geometric description of the regularized projective space of initial values. The latter were carried out in collaboration with many co-authors: Duistermaat and Joshi (2011), Howes and Joshi (2014), Joshi and Radnovic (2016-2019), and Heu, Joshi and Radnovic (2023).
