Related papers: Numerical Methods in Poisson Geometry and their Ap…
We develop a framework for Poisson geometry on loop spaces of low regularity, extending Mokhov's classical constructions from smooth loops to weak Sobolev spaces $W^{s,p}(\mathbb{S^1},\mathbb{R}^m)$ with $o < s \frac{1}{2}$ and $1 < p <…
This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants…
We provide an introduction to the old-standing problem of isometric immersions. We combine a historical account of its multifaceted advances, which have fascinated geometers and analysts alike, with some of the applications in the…
The aim of this study is to show that harmonic geometric polynomials can be represented in terms of geometric polynomials. This problem was first considered by Keller [14]; however, the corresponding coefficients were not fully determined.…
Starting from a contact Hamiltonian description of Li\'enard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we…
Poisson's equation plays an important role in modeling many physical systems. In electrostatic self-consistent low-temperature plasma (LTP) simulations, Poisson's equation is solved at each simulation time step, which can amount to a…
Automorphism, isomorphism, and embedding problems are investigated for a family of Nambu-Poisson algebras (or $n$-Lie Poisson algebras) using Poisson valuations.
Systems with dissipation can be described using contact geometry. We introduce the concepts of symmetries and dissipation laws for contact Hamiltonian systems and study the relation between them. This is an ongoing collaboration with Xavier…
An exposition of the basic geometry of twistor integrals, intended for mathematicians.
We discuss how the integrators used for the Hybrid Monte Carlo (HMC) algorithm not only approximately conserve some Hamiltonian $H$ but exactly conserve a nearby shadow Hamiltonian (\tilde H), and how the difference $\Delta H \equiv \tilde…
Some cohomology classes associated with an ideal in a Lie algebra, a Poisson structure on the basic functions algebra of contact structure, its Poisson cohomology and geometric (pre)quantization are considered from the algebraic point of…
In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these…
Poisson algebraic structures on current manifolds (of maps from a finite dimensional Riemannian manifold into a 2-dimensional manifold) are investigated in terms of symplectic geometry. It is shown that there is a one to one correspondence…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the {\em computationally dense\/} ones) are seen to be the ones…
We report a few sumerical tests comparing some newly defined energy-preserving integrators and symplectic methods, using either constant and variable stepsize.
We consider geometric numerical integration algorithms for differential equations evolving on symmetric spaces. The integrators are constructed from canonical operations on the symmetric space, its Lie triple system (LTS), and the…
Some problems related to the structure of higher terms of the epsilon-expansion of Feynman diagrams are discussed.
We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and…
This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.