arXiv research

The paper outlines key geometry issues in Siegel-Jacobi space.

Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.

Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.

We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial …

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.

Equations of motion for linear Hamiltonians in the real Jacobi group

Let $\mathcal{N}$ be the space of Gaussian distribution functions over $\mathbb{R}$, regarded as a 2-dimensional statistical manifold parameterized by the mean $μ$ and the deviation $σ$. In this paper we show that the tangent bundle of $\mathcal{N}$, endowed with its natural Kähler structure, is the Siegel-Jacobi space…

We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.

In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential oper…

Extended metric defined on Siegel-Jacobi space using invariant forms.

We find the homogenous Kähler isomorphism $FC$ which expresses the Kähler two-form on the Siegel-Jacobi domain $\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1$ as the sum of the Kähler two-form on $\mathbb{C}$ and the one on the Siegel ball $\mathcal{D}_1$. The classical motion and quantum evolution on $\mathcal{D}^J_1$…

We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in $\mathbb{C}^n$. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk $\mathcal{D}^J_1$, which is a parti…

We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.

Geodesics on extended Siegel-Jacobi upper half-plane determined.

This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.

Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.

The coherent state representation of the Jacobi group $G^J_1$ is indexed with two parameters, $μ(=\frac{1}{\hbar})$, describing the part coming from the Heisenberg group, and $k$, characterizing the positive discrete series representation of $\text{SU}(1,1)$. The Ricci form, the scalar curvature and the geodesics of th…

We show that the Wei-Norman method applied to describe the evolution on the Siegel-Jacobi disk $\mathcal{D}^J_1=\mathcal{D}_1\times\mathbb{C}^1$, where $\mathcal{D}_1$ denotes the Siegel disk, determined by a hermitian Hamiltonian linear in the generators of the Jacobi group $G^J_1$ and Berezin's scheme using coherent …

Researchers explore geometric dualities in statistical manifolds.

We determine the homogeneous Kähler diffeomorphism $FC$ which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermiti…

Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.

Abstract: Study of metrics on line bundles over complex varieties.

The real Jacobi group $G^J_1(\mathbb{R})$, defined as the semi-direct product of the group ${\rm SL}(2,\mathbb{R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group ${\rm Sp}(2,\mathbb{R})$. The left-invariant one-forms on $G^J_1(\mathbb{R})$ and their dual orthogonal left-i…

The paper extends Siegel-Veech formula to convex flat cone spheres.

New geometry based on Siegel upper half-space with volume formula.

Proposes Siegel neural networks for improved classification tasks.

Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.

Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.

Dynamics of four-dimensional massless fields of all spins is formulated in the Siegel space of complex $4\times 4$ symmetric matrices. It is shown that the unfolded equations of free massless fields, that have a form of multidimensional Schrodinger equations, naturally distinguish between positive- and negative-frequen…

The study predicts large genus behavior of quadratic differential volumes and constants.

Sharp inequalities on Siegel domains and complex hyperbolic spaces established.

We show that a locally symmetric space of noncompact type and with finite volume is quasi-isometric to the euclidean cone over a finite simplicial complex. A detailed analysis of metric properties yields a proof of a conjecture of Siegel.

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.

Study refines Siegel-Veech constants for abelian differentials.

A new model explains relative spreads between economies using dynamic Nelson-Siegel and functional regression.

In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient c…

The paper decomposes spectral functions on marked tori strata.

Quasimodular forms were first studied in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veec…

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show…

Two Kähler structures are PCR equivalent in the Siegel domain.

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

We extend asymptotic formulas for saddle connections on translation surfaces.

Computes constants for specific geometric structures.

We relate trimmed sums of twists in cylinders along a typical Teichmuller geodesic to the area Siegel-Veech constant.

Genetic Algorithm improves Nelson-Siegel-Svensson model calibration for interest rates.

Estimates Bergman kernel for Siegel varieties, focusing on geodesic distances.