Related papers: An enlargement of some symplectic objects
This is a survey about certain "almost homomorphisms" and "almost linear" functionals (called quasi-morphisms and quasi-states) in symplectic topology and their applications to Hamiltonian dynamics, functional-theoretic properties of…
Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect…
We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…
We discuss $C^0$-continuous homogeneous quasi-morphisms on the identity component of the group of compactly supported symplectomorphisms of a symplectic manifold. Such quasi-morphisms extend to the $C^0$-closure of this group inside the…
This manuscript introduces novel approaches to three phenomena. First, we extend the algebraic formulation of kinetic theory within the contact framework by making explicit the gauge freedom, thereby obtaining a formulation in which the…
We describe a symplectic approach towards thermodynamics in which thermodynamic transformations are described by (symplectic) Hamiltonian dynamics. Upon identifying the spaces of equilibrium states with Lagrangian submanifolds of a…
Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection,…
We show that a topological symplectic manifold has a canonically associated bi-Lipschitz structure. As a corollary, we obtain the first examples of non-existence and non-uniqueness for topological symplectic structures. Our arguments hold…
We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics "at infinity" for representations of discrete groups into Lie groups.
This article consists in applications of [arXiv:2511.14232] in the case of homemomorphisms of higher genus surfaces whose homological rotation set is big enough -- a class of dynamics that is open. We first prove a structure theorem for the…
The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such…
We detect, by using symplectic topology, invariant measures with large rotation vectors for a class of Hamiltonian flows.
We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine…
An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also…
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…
Following Thurston's geometrisation picture in dimension three, we study geometric manifolds in a more general setting in arbitrary dimensions, with respect to the following problems: (i) The existence of maps of non-zero degree (domination…
The purpose of this article is to give an exposition of topological properties of spaces of homomorphisms from certain finitely generated discrete groups to Lie groups $G$, and to describe their connections to classical representation…
We argue that discrete dynamics has natural links to the theory of analytic functions. Most important, bifurcations and chaotic dynamical properties are related to intersections of algebraic varieties. This paves the way to identification…
A topological dynamical system induces two natural systems, one is on the hyperspace and the other one is on the probability space. The connection among some dynamical properties on the original space and on the induced spaces are…
Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and…