Related papers: Geometry of currents, intersection theory and dyna…
This paper is a review on recently found connection between geodesically equivalent metrics and integrable geodesic flows. Suppose two different metrics on one manifold have the same geodesics. We show that then the geodesic flows of these…
Two dimensional electric potential maps based on voltage detection in conducting paper are common practice in many physics courses in college. Most frequently, students work on `capacitor-like' geometries with current flowing between two…
In this paper we study the ergodic theory and thermodynamic formalism of the geodesic flow on non-compact pinched negatively curved manifolds. We consider two notions of entropy at infinity, the topological and the measure theoretic entropy…
The holographic interpretation of the hydrodynamic entropy current is developed for the case of hydrodynamics with a conserved charge. This is carried out within a framework developed in earlier work, which showed how to associate entropy…
We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex…
From the work of Phong and Sturm in 2007, for a polarised projective manifold and an ample test configuration, one can associate the geodesic ray of plurisubharmonic metrics on the polarising line bundle using the solution of the…
We present some results concerning currents of integration on finite-dimensional analytic spaces in Hilbert spaces, using the setting of metric currents. In particular, we obtain the characterization of such currents as positive closed…
We introduce an operation that measures the self intersections of paths on a surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson…
We present all the possible solutions of a Josephson junction with bias current and magnetic field with both inline and overlap geometry, and examine their stability. We follow the bifurcation of new solutions as we increase the junction…
Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in…
The gauging of equations method, introduced in the preceding paper, is applied to the four-dimensional integral equations describing the strong interactions of three identical relativistic particles. In this way we obtain gauge invariant…
In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged…
In ultra-pure conductors, collective motion of charge carriers at relatively high temperatures may become hydrodynamic such that electronic transport may be described similarly to a viscous flow. In confined geometries (e.g., in ultra-high…
In this paper, we consider a class of continuous maps characterized by a singularity of order $x^{q/p}$ (with $p,q \in \mathbb{N}$, $p>q$, and $(p,q)=1$) on one side of the discontinuity boundary $\Sigma$ and a linear behaviour on the other…
Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction…
Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves…
A p-n junction, induced in graphene by gating, works to contrast the edge states of electrons and holes on each side of it. In a magnetic field those edge states carry two species of persistent current, which are intimately tied to the…
We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function satisfies a suitable convexity condition. These…
We equate dynamical properties (e.g., positive entropy, existence of a periodic curve) of complex projective surface automorphisms with properties of the pull-back actions of such automorphisms on line bundles. We use the properties of the…
The statistical-mechanical study of the equilibrium properties of fluids, starting from the knowledge of the interparticle interaction potential, is essential to understand the role that microscopic interaction between individual particles…