Spring 2026 • Weekly Seminars
For information on the related DUT Differential Equations Seminar (DDES), please check here.
In this talk, I will explain the Poisson bracket estimates in the paper Kolmogorov Invariant Torus Theorem for Weakly Interacting Particles I: Full Dimensional Tori. This part plays a key role in the functional setting of the infinite-dimensional KAM iteration. The main goal is to show that the Poisson bracket is well controlled in the analytic Banach spaces $\mathcal Y_{\beta,\rho,\sigma,O}$. More precisely, if two Hamiltonians $h$ and $g$ belong to suitable spaces, then their Poisson bracket $\{h,g\}$ still belongs to the same type of space, but on a slightly smaller analytic domain. The corresponding estimate is of the form \[ \|\{h,g\}\|_{\beta,\rho-r,\sigma-s,O} \lesssim C(r,s,\rho,\sigma,\rho_*,\sigma_*) \|h\|_{\beta,\rho,\sigma,O} \|g\|_{\beta,\rho_*,\sigma_*,O}. \] This estimate is important because the KAM scheme repeatedly uses Hamiltonian flows and Lie transforms, whose expansions involve Poisson brackets. In the long-range setting considered in this paper, the Hamiltonian vector field does not decay with respect to the particle index, so the usual short-range estimates are not sufficient. Therefore, the Poisson bracket estimate provides the analytic control needed to close the inductive KAM step.
The Poincaré–Melnikov method gives a classical criterion for detecting transverse homoclinic intersections in small time-periodic perturbations of planar systems. The starting point is an autonomous system with a hyperbolic saddle and a homoclinic orbit. In many mechanical Hamiltonian examples, such as the pendulum, this orbit appears as a separatrix loop contained in the critical energy level of the saddle. In this talk, we first discuss how these separatrices arise in one degree of freedom Hamiltonian systems, emphasizing the pendulum as the guiding example. We then introduce a small time-periodic perturbation and the associated stroboscopic Poincaré map. The stable and unstable manifolds of the perturbed saddle generally split, and the Melnikov function gives the first-order signed distance between them. Finally, we explain why a simple zero of the Melnikov function implies a transverse homoclinic intersection and, consequently, horseshoe dynamics.
We study the localization bounds for quantum many-body systems at positive spatial density in disordered media. Specifically, we consider dynamics of interacting bosons in the mean-field regime, subjected to a disordered potential which is either random or quasi-periodic. We prove that starting from a factorized and spatially localized initial wave function, the corresponding time-evolution propagates with a small velocity due to the disorder. This provides an example of a disordered quantum many-body system with provably slow transport behavior in any spatial dimension. The main technical novelty in the proof is an interaction picture analysis relative to the localization bounds of the associated one-body dynamics.
The Smale horseshoe provides one of the clearest geometric mechanisms by which deterministic systems generate chaotic dynamics. Beginning with a simple process of stretching, contracting and folding a region of the phase space, the horseshoe map produces an invariant Cantor set on which the dynamics is conjugate to the full shift of two symbols. This symbolic description reveals the essential features of chaos: sensitive dependence on initial conditions, dense periodic orbits, and topological transitivity. In the context of Hamiltonian systems, the horseshoe is especially important because it models the dynamics created near transverse intersections of stable and unstable manifolds of hyperbolic saddles. Such intersections generate homoclinic tangles, which are the geometric source of chaotic motion in many conservative systems. Thus, the horseshoe map serves as a bridge between local hyperbolic manifold geometry and chaos. This talk introduces the horseshoe construction as a foundational example before studying how similar structures arise in Hamiltonian systems through homoclinic intersections and separatrix splitting.
A connected sum glues connected topological or smooth manifolds of the same dimension together by removing a disc in both and identifying the boundary spheres with each other. In two dimensions, it can be used to fully classify closed oriented surfaces, which are all connected sums of two-dimensional tori. In three dimensions, it can also be used in the handle body decomposition. In four dimensions, it seems that no classification exists, although there are many hints. It's assumed, although without even a concrete conjecture formulated yet, that symplectic 4-manifolds play a crucial role as the fundamental building blocks of smooth 4-manifolds. A main hint was obtained by Clifford Taubes from Seiberg-Witten theory: It claims, that the connected sum of symplectic 4-manifolds, hence which posess the non-trivial symplectic 2-form, but both additionally allow non-trivial self-dual 2-forms, cannot be symplectic as well. Hence these 4-manifolds cannot be decomposed in a non-trivial way including others of them again.
The rotating Kepler problem, which arises as a limiting case of the circular restricted three-body problem, is one of the fundamental systems in celestial mechanics. Owing to its complete integrability, its orbits can be computed explicitly.
In this talk, based on my thesis and my previous work published this year, I will describe a classification of the orbits of the rotating Kepler problem in terms of the angular momentum and the Laplace-Runge-Lenz vector. I will then present the computation of the Conley–Zehnder indices of all periodic orbits below the critical energy level. These results reveal, in particular, the relationship between the types of bifurcations occurring in the rotating Kepler problem and the corresponding Conley-Zehnder indices.
We consider the breathing circle billiard, namely the free motion of a point particle inside a disk whose radius varies periodically in time, with elastic reflections at the moving boundary. Since angular momentum is preserved, fixing a value $c$ reduces the dynamics to a two-dimensional exact symplectic map on a cylinder.
In the high-energy regime, the corresponding map is generated by a diagonally periodic twist generating function $h_c$. We study the small angular momentum regime as a perturbation of the limiting case $c = 0$, which corresponds to the Fermi-Ulam dynamics along a diameter. Using this perturbative structure and a quantitative version of Mather converse-KAM criterion, we exclude invariant Lipschitz graphs for suitable rotation numbers. Combined with Aubry-Mather theory and Forni’s theorem, this yields positive topological entropy for sufficiently small $c\geq 0$. Our result gives an essential improvement of a previous similar results obtained via the standard Mather converse-KAM criterion.
The Smale horseshoe provides one of the clearest geometric mechanisms by which deterministic systems generate chaotic dynamics. Beginning with a simple process of stretching, contracting and folding a region of the phase space, the horseshoe map produces an invariant Cantor set on which the dynamics is conjugate to the full shift of two symbols. This symbolic description reveals the essential features of chaos: sensitive dependence on initial conditions, dense periodic orbits, and topological transitivity. In the context of Hamiltonian systems, the horseshoe is especially important because it models the dynamics created near transverse intersections of stable and unstable manifolds of hyperbolic saddles. Such intersections generate homoclinic tangles, which are the geometric source of chaotic motion in many conservative systems. Thus, the horseshoe map serves as a bridge between local hyperbolic manifold geometry and chaos. This talk introduces the horseshoe construction as a foundational example before studying how similar structures arise in Hamiltonian systems through homoclinic intersections and separatrix splitting.
In this talk, we will present a stochastically perturbed Kepler problem, which is based on a result of A. Saha (https://link.springer.com/article/10.1007/s10569-025-10265-z). We will first introduce the stochastic Hamilton equations. Next, we consider a stochastic Kepler problem perturbed by a Hamiltonian noise affecting the angular momentum vector. We show that the angular momentum and the Laplace–Runge–Lenz vectors are conserved in magnitude and as a consequence, the distance and speed of the particle follow deterministic dynamics. If time permits, we will also show the Moser regularization, whereby orbits for a fixed negative energy level are transformed to the geodesic flow on the 3-sphere.
Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. Under suitable conditions, the fast variable can be “averaged out” to produce an averaged system, which is easier for analysis and governs the evolution over a long time scale. In this talk, we consider the averaging principle for SDEs.
This talk will give a brief introduction to the regularization theory, with a particular focus on the problems from celestial mechanics, such as the Kepler problem and the restricted three-body problem. Starting with the Kepler problem, we will demonstrate how to manipulate its equations of motion to regularize the singularity. Next, we will show the regularization method from the Hamiltonian perspective. After that, the planar circular restricted three-body problem and its regularization will be introduced. We will understand why this is important for both theoretical and numerical studies. If time permits, the elliptic problem will also be mentioned.
This talk continues our discussion of ergodic theory on horseshoes. We begin with a brief review of the horseshoe map, a classic example in differentiable dynamical systems, and its relationship with symbolic systems. We then recall some basic concepts in ergodic theory, including invariant measures, ergodic measures, and entropy. Finally, we investigate intermediate value properties and multifractal analysis on horseshoes.
This talk begins with a brief overview of KAM theory for Hamiltonian PDEs and the challenges posed by derivative nonlinearities. We then discuss recent progress on full-dimensional invariant tori for a 1D derivative nonlinear wave equation. To control frequency shifts while preserving non-resonance conditions through KAM iteration, we introduce a modified quasi-Töplitz framework unifying Töplitz-Lipschitz and quasi-Töplitz techniques. This yields linearly stable invariant tori with sub-exponentially decaying amplitudes under mild parameter assumptions.
In this talk， we will discuss the existence of full dimensional tori for Hamiltonian PDEs by KAM theory for infinite dimensional Hamiltonian systems.