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One of Hill's main contributions?to celestial mechanics was his discovery of a planar direct periodic orbit with a period of one synodic lunar month in a limiting case of the circular restricted three-body problem (CR3BP) when the mass ratio of the two primaries approches zero.

Since then, the Hill problem has played a fundamental role in multi-body gravitational systems.

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In this talk I discuss two applications of the Conley--Zehnder index to periodic orbits within the framework of Hill's modification of the CR3BP:

(1) I demonstrate how to express the anomalistic and draconitic lunar months in terms of Conley--Zehnder indices and Floquet multipliers associated to Hill's lunar orbit.

(2) I show how to use the Conley--Zehnder index to analyze the network structure of symmetric periodic orbit families of the Hill problem. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite $g$, $f$, the libration (Lyapunov) $a$, $c$, halo and collision $\mathcal{B}_0$ families. Since the Conley--Zehnder index leads to a grading on a topological bifurcation invariant, the local Floer homology and its Euler characteristic, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among periodic orbit families.

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Furthermore, I present recent results related to the Distant Retrograde Orbit (DRO) family within the Earth--Moon CR3BP.

The Hofer–Zehnder capacity is a symplectic notion of size defined through Hamiltonian dynamics. For convex domains in Euclidean space, it is given by the minimal action of a Reeb orbit on the boundary; we call this quantity the systole. For codisk bundles, however, the situation is more subtle, and only special examples have been computed. If the base is simply connected and carries a metric making it a symmetric space, then in many cases — including spheres, Grassmannians, quadrics, and compact Lie groups — the capacity is indeed the systole. But once the metric is perturbed, it is much less clear what the capacity should detect.

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In this talk, I will discuss what happens under explicit deformations of the round metric on the two-sphere, focusing on ellipsoids and Katok-type Finsler examples. In these cases the capacity remains spectral: it is still the length of a closed geodesic, but not necessarily the shortest one. For ellipsoids, the relevant closed geodesic is selected from a finer part of the length spectrum, involving either twice the systole or a distinguished closed geodesic of Morse index three. The lower bound comes from billiard dynamics, while the upper bounds use pseudoholomorphic curves and symplectic homology.

The goal of this talk is to define and study a new morphism on the fundamental group of the space of Lagrangians. We will discuss relationships with the topology of that space and applications to computing which possible values the Lagrangian flux can take on paths and loops of Lagrangians. This is joint work with Jean-Philippe Chassé and Rémi Leclercq.?

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We introduce skein valued curve counts and survey applications to skein lifting maps, wall crossing formulas, and counts of U(1)-invariant associatives in G2-manifolds.

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In this talk I will explain the way to construct A infinity category? and open-closed closed-open maps? linear over Novikov ring with rational numbers as the ground ring? keeping all the symmetry including cyclic symmetry.

(Such structure was constructed over the real numbers? in a joint paper with Abouzaid-Oh-Ohta-Ono.) I will explain how it could be related to the Q structure in? non-commutative Hodge theory and Mirror symmetry.

Motivated by the question whether one can extract the Gromov-Witten invariants from a symplectic manifold from its Fukaya category, I will introduce an algebraic structures based on moduli space of stable maps with Lagrangian boundary condition. Called an open-closed Deligne-Mumford field theory, it provides a bridge between the Fukaya algebra of a Lagrangian and the Gromov-Witten theory of the ambient symplectic manifold. I will explain how to deal with curvature and the usual transversality issues. This is joint work with Kai Hugtenburg.

The talk traces the development of Floer theory while portraying the life of mathematician Andreas Floer (1956–1991), whose ideas transformed geometry and topology.

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It reconstructs the existing ideas, and introduces the main characters, describing their interactions, their judgements and attempts, wrong or right.

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Chaos is often regarded as the ultimate obstruction to prediction. Since the pioneering work of Stephen Smale, the horseshoe has become the canonical signature of chaos: a simple topological mechanism that generates sensitive dependence on initial conditions, symbolic dynamics, and infinitely many periodic orbits.

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But is chaos truly the final frontier of unpredictability?

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Recent developments point to a deeper phenomenon: undecidability in dynamics. Beyond systems whose long-term behaviour is merely hard to predict, there are systems for which certain natural questions are algorithmically impossible to answer. No computer program can decide, in full generality, whether a trajectory reaches a given region, whether a prescribed behaviour occurs, or whether a qualitative event will ever happen.

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In this talk, I will present recent results showing how computational universality emerges in dynamical systems including billiards and fluid flows. Many of these examples lie naturally within symplectic and contact topology, with dynamics given by Reeb and Hamiltonian flows. Undecidability thus appears as a new layer beyond classical wild dynamics. I will explain how symbolic mechanisms reminiscent of horseshoes can be enhanced to encode full computation, turning classical phase spaces into machines capable of simulating arbitrary algorithms. This viewpoint opens new connections between computational complexity, Floer-theoretic methods, and Hamiltonian dynamics.

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I will also discuss implications for celestial mechanics. The three-body problem, one of the historic cradles of chaos, naturally exhibits horseshoes and intricate symbolic dynamics. This raises a fundamental question: can gravitational systems also display undecidability? Might the three-body problem be not only chaotic, but algorithmically unpredictable in a deeper sense?

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This suggests a hierarchy of complexity: from integrability, to chaos, to universality, and finally to undecidability. If the horseshoe is the shape of chaos, what then is the shape of the undecidable?

I will describe recent advances at the intersection of the study of symplectic geometry and Anosov flows, and in particular how one can associate symplectic invariants to Anosov flows. I will also describe some of my joint work with Kai Cieliebak, Oleg Lazarev and Thomas Massoni, which is aimed at computing some of these symplectic invariants in particular cases.

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