Given a finite-dimensional dynamical system $f: M \rightarrow M$ and an observable $g: M \rightarrow \mathbb{R}$ , the Birkhoff Ergodic Theorem states that the space average of $g$ is equal to the time average of $g$ for almost every point with respect to an ergodic invariant measure. This establishes a form of space-time equivalence, which becomes particularly important when the original dynamical system is unknown or too complex to model directly. In the context of modern AI, we pose the following question: given data from a single observable $g$, is it possible to recover the underlying dynamical system? For instance, with a large dataset of positional observations from an $n$-body system, can we predict its future motion without resorting to Newtonian mechanics? Surprisingly, the answer is yes for almost any typical observable. We introduce the principle of space-time swap: the absence of spatial information in a dynamical system can be compensated by leveraging temporal information. This principle is grounded in Takens' Embedding Theorem (building upon Whitney's embedding theorem). We believe this idea has broad potential for applications in the analysis and prediction of complex systems.

Minimizing methods have been successfully applied to construct various types of periodic solutions for the n-body and n-center problems during the past two decades. Majority of relevant researches were endeavored to understand qualitative features such as existence, uniqueness, and stability. In this talk we discuss a topic with relatively less attention — quantitative estimates for action values and mutual distances for action minimizing solutions. We will demonstrate some simple but nontrivial bounds. These estimates will facilitate numerical explorations to effectively locate and search new orbits.

In 1987, Lions firstly proposed the homogenization for Hamilton-Jacobi equations, which revealed the significance of effective Hamiltonian in controlling the large time behavior of solutions. The quantitative estimate of such a homogenization was studied in recent years, which mainly answers the convergence rate for compact case. In this talk, we will introduce a novel quasi-periodic approach, which reveals the relation between the smoothness of effective Hamiltonian and the convergence rate.

In this talk, we will explore Hamiltonian dynamics from a quantitative perspective, focusing on Floer theories. We delve into the large-scale geometric characterization of the Hamiltonian diffeomorphism group, under the spectral norm - a measurement that has garnered significant attention recently. Specifically, we prove that for any unit co-disk bundle, the Hamiltonian diffeomorphism group encompasses an arbitrarily large Euclidean space. This underscores the inherent complexity of Hamiltonian dynamics on unit co-disk bundles. Intriguingly, the proof is connected to certain rigidity properties of subsets, which can be examined via symplectic cohomology. This talk is based on joint work with Qi Feng.

In this talk, we first introduce the trace formulas for both the linear Hamiltonian systems and Sturm–Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic relative equilibrium, we give a quantitative estimation of the stability for the Lagrange, Euler and the regular (1 + n)-gon ERE with any e∈ [0,1).

In this talk, we examine the homotopy classes of positive loops in $\Sp(2n)$. We demonstrate that two positive loops are homotopic if and only if they are homotopic through positive loops. This provides a positive answer to a long standing conjecture raised by McDuff. As a consequence, we extend several results of McDuff and Chance to higher dimensional symplectic manifolds without dimensional restrictions. This is a joint work with Jian Wang.

In this talk, we give a proof of the famous conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that any tight Reeb flow on $S^3$ has either precisely two or infinitely many geometrically distinct periodic orbits. As its application, we also prove a longstanding conjecture of Bangert-Long that every irreversible Finsler metric on $S^2$ has either precisely two or infinitely many geometrically distinct closed geodesics. This talk is based on my joint work with Cristofaro-Gardiner, Hryniewicz and Hutchings.

In this talk, we will discuss the construction of Birkhoff-Gustvason normal form of a two-degree-of-freedom Hamiltonian system near an equilibrium point. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like energy surface that sufficiently close to the equilibrium contains at least two periodic orbits forming a Hopf link due to Weinstein. Then based on a certain non-resonance condition established by Hryniewitz, Momin and Salomao for the Hopf link, we will provide some sufficient conditions on the normal form that imply the existence of infinitely many periodic orbits. Finally, we will discuss the spatial isosceles three-body problem, Hill's lunar problem and the Henon-Heiles problem as the main applications. Joint with C. Grotta-Ragazzo and P. A. S. Salomao.

In this talk, we will discuss that there are abundant new periodic orbits near relative equilibria of the planar N-body problem. All of these periodic orbits lie on a $2d$-dimensional central manifold of the planar N-body problem, and generically the relative measure of the closure of the set of periodic orbits near relative equilibria on the central manifold is close to 1.

In 2001, Mohnke found a proof of the Arnold chord conjecture for the contact boundary of displacable domains. In this talk, I will explain an alternative proof based on Lagrangian Rabinowitz Floer cohomology. The use of such algebraic gadget allows various generalizations, including the Maurer–Cartan deformation as well as linearization w.r.t. algebraic augmentations. This allows to prove several new cases of the Arnold chord conjecture including prequantization bundles, cosphere bundles of torus etc.