arXiv subjects

Duco van Straten

Publications and source records attributed to Duco van Straten.

At least 19 recordsLinked to original sources

arXiv2026

The platonic elliptic surfaces

We construct certain special rational elliptic surfaces with four reduced singular fibres that are naturally attached to the platonic solids and belong to a group of surfaces complementing the well-known semi-stable Beauville surfaces. We describe the basic algebraic geometric properties of these surfaces, their associated Picard-Fuchs operators, integer sequences, Laurent polynomial representations and Apéry constants. On the arithmetic side, we discover the need for a refinement of the Dwork-expansion used to find the Euler factors of the fibres of these fibrations. This is explained by comparing the $q$-coordinates coming from the elliptic curve and the $q$-coordinate of the Picard-Fuchs equation and is also directly reflected in the shape of the monodromy matrices in the Frobenius basis.

math.AG↗

arXiv2026

The Hamiltonian Normal Form

An important step in the proof of the Herman invariant tori conjecture was the introduction of a normal form with poles along the resonance loci, replacing the Birkhoff normal form, which we call the Hamiltonian normal form. This paper is extracted from previous versions (arXiv:1206.1245, arXiv:1909.06053) and aims to present this Hamiltonian normal form in its simplest form. It is expected that, not only theoretically but also in numerical computations, it will provide better approximations than the standard Birkhoff normal form.

math.DS↗

arXiv2026

A new quasi-analytic class

Spaces of quasi-analytic classes are defined by the existence and uniqueness of Taylor expansions, which are not necessarily convergent. First examples were given by Borel in his theory of monogenic functions, a generalisation of holomorphic functions defined on locally closed sets. Denjoy and Carleman then gave simpler examples of quasi-analytic classes which are now widely known. Unfortunately, in most examples coming from mathematical physics and number theory, the power series are neither of Borel nor Denjoy-Carleman's classes. In this paper we introduce a quasi-analytic class which is relevant to perturbation theory and especially to KAM theory and dynamical systems. Our theorems also explain geometrically the divergence of most perturbative expansions by the presence of accumulation points of poles.

math.CV↗

arXiv2026

Fixed point theorems for small divisors problems

In the seventies', Zehnder found a Nash-Moser type implicit function theorem in the analytic set-up. This theorem has found many applications in dynamical systems although its applications require, as a general rule, some efforts. We develop further the analytic theory and give fixed point theorems with direct applications to the study of dynamical systems.In practice, our theorems show that Bruno conditions are sufficient to ensure the existence of a positive measure set of invariant tori.

math.DS↗

arXiv2024

A $C^\ast$-algebraic view on the interaction of real- and reciprocal space topology in skyrmion crystals

Understanding the interaction of real- and reciprocal space topology in skyrmion crystals is an open problem. We approach it from the viewpoint of $C^\ast$-algebras and calculate all admissible Chern numbers of a strongly coupled tight-binding skyrmion system on a triangular lattice as a function of Fermi energy and texture parameters. Our analysis reveals the topological complexity of electronic states coupled to spin textures, and the failure of the adiabatic picture to account for it in terms of emergent electromagnetism. On the contrary, we explain the discontinuous jumps in the real-space winding number in terms of collective evolution in real-, reciprocal, and mixed space Chern numbers. Our work sets the stage for further research on topological dynamics in complex dynamic spin textures coupled to external fields.

cond-mat.mes-hall↗

arXiv2024

A hyperelliptic saga on a generating function of the squares of Legendre polynomials

We decompose the generating function $\sum_{n=0}^\infty\binom{2n}nP_n(y)^2z^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of $\textit{second}$ order differential equations. This is highly unusual since $\textit{four}$ is the expected order for genus 2. These second order equations are arithmetic and yet, surprisingly, their monodromy group is dense in $\operatorname{SL}_2(\mathbb{R})$. This suggests that they cannot be solved in terms of hypergeometric functions, which is novel for arithmetic second order differential equations that are $\textit{defined over}$ $\mathbb{Q}$, and also novel for a $\textit{family}$ of such equations. We complement our analysis with a recipe for constructing similar examples. Maple's support for the paper is available at https://www.math.fsu.edu/~hoeij/saga/.

math.NT↗

arXiv2024

Paramodular forms from Calabi-Yau Operators

In this note we report on the conjectural identification of paramodular forms from Calabi-Yau motives of Hodge type (1, 1, 1, 1) of moderately low conductor. We calculate Euler factors from Calabi-Yau operators from the AESZ database by the method described in P. Candelas, X. dela Ossa and D. van Straten, seek a fit with the tables provided by E. Assaf, W. Ladd, G. Rama, G. Tornaria, and J. Voight and for consistency check the approximate functional equation for the Euler product for primes < 1000 numerically, using the PARI implementation of T. Dokchitser's method.

math.NT↗

arXiv2024

Non degenerate Birkhoff functions

It is known after the works of Mahler, Kleinbock, Margulis, Sprindžuk and others that very well approximated numbers on a manifold form a zero measure set, assuming non-degeneracy conditions. These non-degeneracy conditions are, in many applications, difficult to check. We propose here a setting in which these are automatic. Rather than functions, we consider correspondences which have a better behaviour.

math.DS↗

arXiv2024

Product formulas for the Higher Bessel functions

We consider the generating function $Φ^{(N)}$ for the reciprocals $N$-th power of factorials. We show a connection of product formulas for such series with the periods for certain families of algebraic hypersurfaces. For these families we describe their singular loci. We show that these singular loci are given by zeros of the Buchstaber-Rees polynomials, which define $N$-valued group laws. We describe a generalized Frobenius method and use it to obtain special expansions for multiplication kernels in the sense of Kontsevich. Using these expansions we provide some experimental results that connect $N$-Bessel kernels and the hierarchies of the palindromic unimodal polynomials. We study the properties of such polynomials and conjecture positivity of their roots. We also discuss the connection with Kloosterman motives as a version of the mirror duality.

math.AG↗

arXiv2024

A remarkable eelliptic curve

We describe a system of plane algebraic curves defined over \Z, attached naturally to the exponential function. On of these is a remarkable curve of degree 6 that has genus equal to 1. As the sectic curve has rational points, it is an elliptic curveand can be tranformed over \Q into the curve 1584.j1 of the LMFDB. One is left to wonder what the number 11, appearing in the factorisation 1584=2^4.3^2.11 has to do with the exponential function.

math.HO↗

arXiv2023

A Category of Banach Space Functors

We introduce a sheaf theoretic viewpoint on functional analysis designed for infinite dimensional Lie group actions. We develop functional calculus for Banach valued functors and, in particular, prove the existence of an exponential map for a certain class of operators that generalise first order partial differential operators. This algebraic framework can be used in dynamical systems and KAM theory to provide normal forms and versal deformations. it is used in our papers on the subject such as the Herman conjecture paper, the versal deformation theorem for vector fields etc.

math.CV↗

arXiv2022

The Herman invariant tori conjecture

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori conjecture namely the existence of a positive measure set of invariant tori near the critical point. This paper is an update of the first 2012 proof of the author. The functional analytic arguments have been simplified using Banach functors, minor points have been clarified. A series of videos is available on the webpage https://www.agtz.mathematik.uni-mainz.de/category/alg-geom/

math.DS↗

arXiv2021

Local Zeta Functions From Calabi-Yau Differential Equations

The zeta-function of a manifold is closely related to, and sometimes can be calculated completely, in terms of its periods. We report here on a practical and computationally rapid implementation of this procedure for families of Calabi-Yau manifolds with one complex structure parameter phi. Although partly conjectural, it turns out to be possible to compute the matrix of the Frobenius map on the third cohomology group of X(phi) directly from the Picard-Fuchs differential operator of the family. To illustrate our method, we compute tables of the quartic numerators of the zeta-functions for six manifolds of increasing complexity as the parameter phi varies in Fp. For four of these manifolds, we do this for the 500 primes p=5,7,...,3583, while for two manifolds we extend the calculation to 1000 primes. The tables for 5 <= p <= 97 are part of this article while the remaining tables are attached in electronic form. Interest attaches to the cases for which the numerators factorise. Some of these factorisations can be associated with parameter values for which the underlying manifold becomes singular. For the cases we consider here, the singularities are all of conifold or hyperconifold type. In these cases the numerator degenerates to a cubic and this factorises into the product of a linear and a quadratic factor. The quadratic term contains a coefficient that is the p'th coefficient of a modular form. Some of our examples have singularities when the parameter satisfies a polynomial equation that does not factorise over Q. When this happens, the corresponding forms are modular forms with neben type or Hilbert modular forms. The numerator can also factorise into two quadrics. This happens when the Hodge structure of the manifold splits, sometimes this happens for algebraic values of the parameter and we identify, in this way, attractor points of rank two of the parameter space.

hep-th↗

arXiv2021

Calabi-Yau operators of degree two

We show that the solutions to the equations defining the so-called Calabi-Yau condition for fourth order operators of degree two defines a variety that consists of ten irreducible components. These can be described completely in parametric form, but only two of the components seem to admit arithmetically interesting operators. We include a description of the 69 essentially distinct fourth order Calabi-Yau operators of degree two that are presently known to us.

math.AG↗

arXiv2020

Versal deformations of vector field singularities

When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a family of invariant cylinders. In his thesis, Stolovitch established a new type of normal form and proved that for a simple resonance and under arithmetic conditions the cone is (the germ of) an analytic variety. In this paper, we prove a versal deformation theorem for analytic vector fields with an isolated singularity over Cantor sets. Our result implies that, under arithmetic conditions, the resonant cone is the degeneration of a set of invariant manifolds like in Pyartli's example. For the multi-Hopf bifurcation, that is for the $(-1,1)^d$-resonance, this implies the existence of vanishing tori carrying quasi-periodic motions generalising previous results of Chenciner and Li.

math.DS↗

arXiv2020

Quasi-periodic motions on symplectic tori

The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifolds in Hamiltonian systems with $n$ degrees of freedom $n \leq k < 2n $. In this paper, we extend this result to the case of a quasi-periodic motion on symplectic tori $k = 2n$.

math.DS↗

arXiv2020

The Spectrum of Hypersurface Singularities

This text is the write-up of a series of lectures on the asymptotic mixed Hodge theory of isolated hypersurface singularities, held at the Third Latin American school on Algebraic Geometry and its applications (ELGA 3) in Guanajuato, Mexico, in august 2017. Its focus is on the classical application of the semi-continuity of the spectrum due to Varchenko and Steenbrink to the problem of bounding the possible singularities on a projective hypersurface.

math.AG↗

arXiv2019

A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over Q this modularity is assured by the proof of the Serre Conjecture. We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over Q, and identify, for all three cases, the associated modular groups. We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the L-functions of the modular groups. Thus the critical L-values enter into the calculation of physical quantities such as the area of the black hole in the 4D spacetime. In our search for additional rank two attractor points, we perform a statistical analysis of the numerator of the zeta-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.

hep-th↗