Related papers: Variations on murmurations
This short note is some obvious mathematical addendum to our papers on Wilson loops on polygon-like contours with circular edges \cite{Dorn:2020meb,Dorn:2020vzj}. Using the technique of osculating spheres and circles we identify the…
We give a short appreciation of Mumford's work on the moduli of varieties by putting it into historical context. By reviewing earlier works we highlight the innovations introduced by Mumford. Then we discuss recent developments whose…
We propose some conjectures on the integrality properties related to the variation of Mahler measures, inspired by the results in the elliptic curve case by Rodriguez Villegas, Stienstra and Zagier.
In this paper, we mainly investigate distortion and covering theorems on some classes of pluriharmonic mappings.
Short introduction to exotic differential structures on manifolds is given. The possible physical context of this mathematical curiosity is discussed. The topic is very interesting although speculative.
In this paper, we construct rotating frames for curves, including plane curves, space curves and curves on surfaces. Hence, the behaviour of an arbitrary moving point on a curve can be seen as the composite of linear motion and rotation.…
The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples…
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by…
We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.
The modularity of an elliptic curve $E/\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve…
We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.
We present a detailed review of some of the most recent developments on Eulerian and Lagrangian turbulence in homogeneous and isotropic statistics. In particular, we review phenomenological and numerical results concerning the issue of…
In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and…
$\mu$-ellipticity is a form of nonuniform ellipticity arising in various contexts from the calculus of variations. Understanding regularity properties of minimizers in the nonautonomous setting is a challenging task fostering the…
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical…
We study improved variants of L\"uscher's algorithm, including the cases of staggered fermions and finite density.
In this article we study polyharmonic curves of constant curvature where we mostly focus on the case of curves on the sphere. We classify polyharmonic curves of constant curvature in three-dimensional space forms and derive an explicit…
This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.
We find all the possible torsion groups of $\Q$-curves over quadratic fields and determine which groups appear finitely and which appear infinitely often.