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This article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of its analytic higher order perturbations. We…
Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a…
Let $M$ be a $2n$-dimensional closed symplectic manifold admitting a Hamiltonian circle action with isolated fixed points. We show that if $M$ contains an $S^1$-invariant symplectic hypersurface $D$ such that $M\setminus D$ is a homology…
We classify all contact projective spaces with contact surgery number one. In particular, this implies that there exist infinitely many non-isotopic contact structures on the real projective 3-space which cannot be obtained by a single…
We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and…
We continue the study of engineered complete intersections (ECI) -- an umbrella generality for a number of important objects in combinatoiral and applied algebraic geometry (such as nondegenerate toric complete intersections, critical loci…
Precontact manifolds extend contact geometry by weakening the maximal non-integrability condition of the defining $1$-form. We clarify the geometric foundations of this structure by studying general pairs of a $1$-form and a $2$-form under…
By considering the most general metric which can occur on a contractable two dimensional symplectic manifold, we find the most general Hamiltonians on a two dimensional phase space to which equivariant localization formulas for the…
This is an expository paper, which provides a first introduction to geometric structures on $TM\oplus T^*M$. The paper contains definitions and characteristic properties of generalized complex, generalized Kaehler, generalized (normal,…
Preliminary results toward the analysis of the Hamiltonian structure of multifield theories describing complex materials are mustered: we involve the invariance under the action of a general Lie group of the balance of substructural…
We give a combinatorial description of the embedded contact complex (ECC) of a certain family of contact toric lens spaces that we call concave lens spaces. We also define a notion of a concave toric domain that generalizes the usual…
This paper is devoted to a systematic study of the geometry of nondegenerate $\bbR^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems…
We propose a generalization of Heisenberg picture quantum mechanics in which a Lagrangian and Hamiltonian dynamics is formulated directly for dynamical systems on a manifold with non--commuting coordinates, which act as operators on an…
We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical…
We demonstrate that the functorial properties of the symplectic field theory under strong cobordisms and surgery cobordisms can produce finite algebraic (planar) torsions from simple examples, which gives a unified treatment of most of the…
Moduli spaces of polygons have been studied since the nineties for their topological and symplectic properties. Under generic assumptions, these are symplectic manifolds with natural global action-angle coordinates. This paper is concerned…
In this work we extend the notion of universal quantum Hamiltonians to the setting of translationally-invariant systems. We present a construction that allows a two-dimensional spin lattice with nearest-neighbour interactions, open…
In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coK\"ahler structures, in the same way as K-contact…
We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e.,…
An effective mathematical framework based on Presymplectic Geometry for dealing with the "phase space picture" of timeless dynamics in General Relativity is presented. In General Relativity, the presence of the scalar Hamiltonian constraint…