arXiv research

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

New findings on isospectral tori and harmonic maps between flat tori.

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…

Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta func…

Quantum modularity proven for specific theta series.

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 …

Proves resurgence properties for Habiro elements from radial limits of theta series.

New techniques prove quantum modularity for various functions.

The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynom…

New formulas connect knot invariants with theta functions.

We derive a formula for the global gravitational anomaly of the self-dual field theory on an arbitrary compact oriented Riemannian manifold. Along the way, we uncover interesting links between the theory of determinant line bundles of Dirac operators, Siegel theta functions and a functor constructed by Hopkins and Sing…

New formula and properties of inverted Habiro series derived from GM series.

In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced by Dufresne to computing the negative moments of Yor's accumulation processes. For these he has given functional recursion rules whose probabilistic structure has been the object of intensive recent studies of Yor an…

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…

New indefinite false theta functions match homological blocks for a specific 3-manifold.

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 construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conj…

Hikami observed a discontinuity in a WRT invariant at roots of unity.

We study generalized special cycles on Hermitian locally symmetric spaces $Γ\backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R})$ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally symmetric spaces associated to subgroups of $G$ which are of the same type. Using oscillator…

The paper calculates colored Jones polynomials for specific link configurations.

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…

The paper decomposes spectral functions on marked tori strata.

The tail of a sequence $\{P_n(q)\}_{n \in \mathbb{N}}$ of formal power series in $\mathbb{Z}[[q]]$ is the formal power series whose first $n$ coefficients agree up to a common sign with the first $n$ coefficients of $P_n$. This paper studies the tail of a sequence of admissible trivalent graphs with edges colored $n$ o…

New resurgent analysis reveals dual $q$-series for Chern-Simons theory crossing natural boundaries.

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

A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…

Meta-learning predicts optimal ensemble size and methods for time series forecasting.

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 use superconnections to define and study some natural differential forms on period domains $\mathbb{D}$ that parametrize polarized Hodge structures of given type on a rational quadratic vector space $V$. These forms depend on a choice of vectors $v_1,\ldots,v_r \in V$ and have a Gaussian shape that peaks on the locu…

Study refines Siegel-Veech constants for abelian differentials.

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

In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymp…

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.

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

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this short paper, we propose the basic problems in the geometry of the Siegel-Jacobi space.

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.

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…

Study theta-curves on torus in 3-sphere, classifying them.

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

Two Kähler structures are PCR equivalent in the Siegel domain.

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

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.

We extend asymptotic formulas for saddle connections on translation surfaces.

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

ModelRadar evaluates forecasting models across multiple aspects.

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 $\mathcal{D}^J_1$…

Determinants of theta curves and symmetric graphs are studied.