Pedro Salomão（南方科技大学、幾何学分野）による数学・数理科学グローバル特別講義7が下記の要領で開催されます。出席希望の方は下記のGoogleフォームのから申込をお願いします。
日時：11月 4日（火） 13:30～14:30、15:00～16:00
11月５日（水） 13:30～14:30、15:00～16:00
11月６日（木） 13:30～14:30、15:00～16:00
11月７日（金） 10:00～12:00、14:00～16:00
会場：理学研究科3号館108室
タイトル：Pseudo-holomorphic curves, Reeb dynamics, and applications to Celestial Mechanics
概要：This lecture series will explore the deep interplay between pseudo-holomorphic curve theory and Reeb dynamics in symplectic and contact topology, with a focus on recent applications to classical problems in Celestial Mechanics. We will develop the theory of finite-energy pseudo-holomorphic curves in symplectizations, discuss their relationship to periodic orbits of Reeb flows, and present finite energy foliations in various settings.
A central theme will be the role of convexity—both strong and weak—in shaping the qualitative dynamics of Hamiltonian systems. The lectures will cover classical and modern results, including the foundational works of Hofer, Wysocki, and Zehnder, as well as new constructions of finite energy foliations in non-convex energy levels.
We will then move to concrete applications. These include the Hénon–Heiles system, the Euler problem of two fixed centers, and, especially, the circular planar restricted three-body problem. In particular, we will discuss new results establishing the existence of global finite-energy foliations that organize the dynamics, and allow for a detailed qualitative analysis of the dynamics in these models.
The aim is to introduce a powerful toolkit for studying Hamiltonian dynamics via holomorphic curve techniques and to show its relevance in problems with rich physical and geometric content. We will conclude with a discussion of open problems and possible directions for further exploration.
Lecture 1 - Introduction to Pseudo-holomorphic Curves in Symplectizations
We begin with an introduction to pseudo-holomorphic curves in the context of symplectic and contact geometry. After reviewing the notion of contact structures and Reeb vector fields, we introduce symplectizations and the setup for finite-energy pseudo-holomorphic curves. The lecture will cover basic analytical properties, energy bounds, and bubbling-off phenomena, drawing from the foundational work of Hofer.
We also explain the link between pseudo-holomorphic curves and Reeb orbits, emphasizing the geometric intuition behind the use of holomorphic curves to study Hamiltonian dynamics.
Lecture 2 - Finite-energy Foliations and the Work of Hofer–Wysocki–Zehnder
This lecture will focus on finite-energy foliations in symplectizations. We present the main existence results by Hofer, Wysocki, and Zehnder under convexity assumptions, and explain the geometric meaning of these foliations in terms of Reeb flows. Topics include asymptotic behavior of curves near punctures, global surfaces of section, and the use of holomorphic curves to analyze the topology of energy levels.
We also introduce the notion of weakly convex energy levels, setting the stage for later applications in non-convex settings like the restricted three-body problem.
Lecture 3 - Construction of Finite-energy Foliations in Celestial Models
We move to the construction of finite-energy foliations in physically relevant Hamiltonian systems. After briefly reviewing the relevant dynamical systems (Hénon–Heiles, Euler’s two-center problem), we describe how these models can be lifted to contact manifolds and how their Reeb dynamics reflect the original Hamiltonian dynamics.
We present joint work on constructing finite-energy foliations for these systems, even in the absence of strict convexity. These foliations provide a powerful tool for global analysis, yielding information about periodic orbits, homoclinic and heteroclinic orbits, escape dynamics, etc.
Lecture 4 - Finite-energy Foliations in the Restricted Three-Body Problem
This lecture will be devoted to recent results on the circular planar restricted three-body problem (CPR3BP). We explain how to regularize collision singularities using elliptic coordinates and how to lift the system to a contact-geometric setting where holomorphic curve techniques become applicable.
We construct a global finite-energy foliation for certain energy levels of the CR3BP slightly above the first Lagrange value, showing how it encodes the global dynamics and connects with classical families of periodic orbits. This work opens the door to a holomorphic curve approach to classical problems in celestial mechanics, including capture, escape, and transition dynamics.
Lecture 5 - Open Problems, Physical Implications, and Future Directions
In the final lecture, we reflect on the broader implications of the theory developed so far. We discuss physical and mathematical motivations for studying foliation structures in Hamiltonian systems, including connections with transition state theory in chemistry and black hole dynamics in general relativity.
We survey open problems related to the existence and classification of finite-energy foliations, especially in non-convex and degenerate settings. We also discuss potential generalizations: foliations in higher dimensions, relations with Floer homology and symplectic field theory, and the challenge of understanding chaotic regimes via holomorphic methods.
要申込：講演を受講希望者は、Googleフォームにて申込みを行って下さい。QRコードまたは下記URLからアクセスしてください。
URL：https://forms.gle/6yzLzTfjkFGorVL27
締切日： 10月30日 （木）
※数学・数理科学グローバル講義Ⅳは数学・数理科学イノベーション人材育成強化コースにおける
中核科目です。
※数学・数理科学グローバル講義Ⅳを履修するにはKULASIS での履修登録が必要です。
前期科目の履修登録期間は10 月10 日（金）～14 日（月）。
※履修登録していなくても聴講可（本学学生に限る。Googleフォーム申込みは必要）。
各特別講義のアブストラクト等の詳細は、当コースのホームページ
https://www.math.kyoto-u.ac.jp/ja/ktgu/courses をご覧ください。