Related papers: Low-dimensional topology and symplectic dynamics
This paper revisits building machine learning algorithms that involve interactions between entities, such as those between financial assets in an actively managed portfolio, or interactions between users in a social network. Our goal is to…
This note aims to bring attention to a simple class of discrete dynamical systems exhibiting some complex behaviour. Each of these systems is defined as a self-mapping of the unit square and is obtained by coupling two families of…
The aim of this note is to present a construction of symplectic structures on orientable globally hyperbolic 4-dimensional lorentzian manifolds. Said structures are defined on the manifold itself, not on its cotangent bundle. It also…
Dynamists have been studying Hamiltonian systems for a long time. However, many physical systems are dissipative and do not preserve a symplectic form. This is the case, for example, with systems involving friction, which multiply the…
We obtain a correspondence between the group of symplectic diffeomorphisms of a 4-dimensional real torus and the vanishing locus of a certain hyperK\"ahler moment map. This observation gives rise to a new flow, called the modified moment…
In this set of five lectures we present a basic toolbox to discuss the dynamics of four dimensional supersymmetric quantum field theories. In particular we overview the program of geometrically engineering the four dimensional…
The study of Weyl points in electronic systems has inspired many recent researches in classical systems such as photonic and acoustic lattices. Here we show how the Weyl physics can also inspire the design of novel elastic structures. We…
The paper gives a review of very recent results related to the Poncelet Theorem, on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of Geometry, recalling for the general mathematical…
Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have…
These lecture notes can be read in two ways. The first two Sections contain a review of the phenomenology of several physical systems with slow nonequilibrium dynamics. In the Conclusions we summarize the scenario derived from the solution…
Finding interesting symmetrical topological structures in high-dimensional systems is an important problem in statistical machine learning. Limited amount of available high-dimensional data and its sensitivity to noise pose computational…
New classes of Lie-Hamilton systems are obtained from the six-dimensional fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(6,\mathbb{R})$. The ansatz is based on a recently proposed procedure for constructing…
We consider relativistic hydrodynamics in the limit where the number of spatial dimensions is very large. We show that under certain restrictions, the resulting equations of motion simplify significantly. Holographic theories in a large…
A new procedure for the construction of higher-dimensional Lie-Hamilton systems is proposed. This method is based on techniques belonging to the representation theory of Lie algebras and their realization by vector fields. The notion of…
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between…
We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been…
The mathematical theory underlying Hamiltonian mechanics is called symplectic geometry. So symplectic geometry arose from the roots of mechanics and is seen as one of the most valuable links between physics and mathematics today. Symplectic…
In this note, the geography problem in dimension four is reviewed and then its extension to dimension six for the symplectic case is explained. Finally some examples in dimension six are provided.
We highlight some recent developments that widen the scope and reach of mesoscopic thin-film (or long-wave) hydrodynamic models employed to describe the dynamics of thin films, drops and contact lines of simple and complex liquids on solid…
Understanding the topological structure of phase space for dynamical systems in higher dimensions is critical for numerous applications, including the computation of chemical reaction rates and transport of objects in the solar system. Many…