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This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion…

Reparametrization invariant Lagrangian theories with higher derivatives are considered. We investigate the geometric structures behind these theories and construct the Hamiltonian formalism in a geometric way. The Legendre transformation…

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the…

The toughness of graph $G$, denoted by $\tau(G)$, is $\tau(G)=\min\{\frac{|S|}{c(G-S)}:S\subseteq V(G),c(G-S)\geq2\}$ for every vertex cut $S$ of $V(G)$ and the number of components of $G$ is denoted by $c(G)$. Bondy in 1973, suggested the…

We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H\subset SO(n)$, on any connected and oriented $n$-manifold…

By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its…

We construct an example of a Hamiltonian flow $f^t$ on a $4$-dimensional smooth manifold $\mathcal{M}$ which after being restricted to an energy surface $\mathcal{M}_e$ demonstrates essential coexistence of regular and chaotic dynamics that…

For regular one-dimensional variational problems, Ball and Nadirashvilli introduced the notion of the universal singular set of a Lagrangian L and established its topological negligibility. This set is defined to be the set of all points in…

In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary…

First we survey and explain the strategy of some recent results that construct holomorphic $\text{sl}(2, \mathbb C)$-differential systems over some Riemann surfaces $\Sigma_g$ of genus $g\geq 2$, satisfying the condition that the image of…

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler,…

Lie-Poisson gauge formalism provides a semiclassical description of noncommutative $U(1)$ gauge theory with Lie algebra type noncommutativity. Using the Dirac approach to constrained Hamiltonian systems, we focus on a class of Lie-Poisson…

Let $(M,g)$ be a compact, smooth Riemannian manifold and $\{u_h\}$ be a sequence of $L^2$-normalized Laplace eigenfunctions that has a localized defect measure $\mu$ in the sense that $ M \setminus \text{supp}(\pi_* \mu) \neq \emptyset$…

We establish a cosymplectic counterpart of Banyaga's theorem by proving that the group of weakly Hamiltonian diffeomorphisms, $\Ham_{\eta,\omega}(M)$, is simple on any closed cosymplectic manifold. A key structural result, derived from Lie…

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras…

Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $M\xi$ in terms of characteristic numbers (indexed by quasi-symmetric functions) for…

By virtue of the well-known theorem, a structure Lie group K of a principal bundle $P$ is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K. In gauge theory, such sections are treated as Higgs…

As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h-Lagrangian submersions from almost quaternionic Hermitian manifolds onto…

We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative…

Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincar\'{e} and Lie-Poisson…