Related papers: Topological Recursion, Airy structures in the spac…
Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We…
A Laplace transform that maps the topological recursion (TR) wavefunction to its $x$-$y$ swap dual is defined. This transform is then applied to the construction of quantum curves. General results are obtained, including a formula for the…
The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A…
The goal of this work is to extend Dirac-type tensor equations to a curved space. We take four 1-forms (a tetrad) as a unique structure, which determines a geometry of space-time.
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…
Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…
Amorphous solids remain outside of the classification and systematic discovery of new topological materials, partially due to the lack of realistic models that are analytically tractable. Here we introduce the topological Weaire-Thorpe…
We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of…
On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a…
We derive recurrence relations between phase space expressions in different dimensions by confining some of the coordinates to tori or spheres of radius $R$ and taking the limit as $R \to \infty$. These relations take the form of mass…
We present a unified approach to holomorphic anomaly equations and some well-known quantum spectral curves. We develop a formalism of abstract quantum field theory based on the diagrammatics of the Deligne-Mumford moduli spaces…
Recurrence plots exhibit line structures which represent typical behaviour of the investigated system. The local slope of these line structures is connected with a specific transformation of the time scales of different segments of the…
This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent…
For the hypergeometric spectral curve and its confluent degenerations (spectral curves of "hypergeometric type"), we obtain a simple formula expressing the topological recursion free energies as a sum over BPS states (degenerate spectral…
In this note, using some regular triangular tilings of the sphere, the Euclidean plane and the hyperbolic plane, we examine the potential relationship between their discrete Bakry - Emery curvatures and the smooth curvatures of their…
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever…
We present a computational pipe aiming at recovery of the topology of the underlying phase space from observation of an output function along a sample of trajectories of a dynamical system.
We classify 1-dimensional connected dually flat manifolds $M$ that are toric in the sense of [Molitor, arXiv:2109.04839], and show that the corresponding torifications are complex space forms. Special emphasis is put on the case where M is…
Smectic order on arbitrary curved substrate can be described by a differential form of rank one (1-form), whose geometric meaning is the differential of the local phase field of the density modulation. The exterior derivative of 1-form is…