Related papers: Coleman Integration on Modular Curves

Nonlinear ordinary differential equations can rarely be solved analytically. Koopman operator theory provides a way to solve nonlinear systems by mapping nonlinear dynamics to a linear space using eigenfunctions. Unfortunately, finding such…

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…

We use a new idea to construct a theory of iterated Coleman functions in higher dimensions than 1. A Coleman function in this theory consists of a unipotent differential equation, a section on the underlying bundle and a solution to the…

Let $K$ be a number field and $O_K$ the ring of integers of $K$. In the spirit of Siegel's theorem on integral points on affine algebraic curves, the plane Jacobian conjecture over $K$ is equivalent to the following statement: if $P,Q\in…

We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.

The cohomology theory TMF of topological modular forms is a derived algebro-geometric interpretation of the classical ring of complex modular forms from number theory. In this article, we refine the classical Adams operations, Hecke…

Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package \cde for…

We investigate the Hilbert scheme of points on curves with n-fold singularities, that is curves that look locally around their singular points as the axis in an affine space. We describe the structure and number of its irreducible…

Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in…

A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.

We find a natural $L_{\omega_1,\omega}$-axiomatisation $\Sigma$ of a structure on the upper half-plane $\mathbb{H}$ as the covering space of modular curves. The main theorem states that $\Sigma$ has a unique model in every uncountable…

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…

Koopman analysis of a general dynamics system provides a linear Koopman operator and an embedded eigenfunction space, enabling the application of standard techniques from linear analysis. However, in practice, deriving exact operators and…

A novel high-order numerical scheme is proposed to compute the covariant derivative, particularly for divergence and curl, on any curved surface. The proposed scheme does not require the construction of a curved axis or metric tensor, which…

Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools.…

Using lower bounds for linear forms in elliptic logarithms we determine the integral points of the modular curve associated to the normalizer of a non-split Cartan group of level 11. As an application we obtain a new solution of the class…

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these…

We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference…

This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that…

The main goal of this thesis is to develop the integration theory of curved homotopy Lie algebras. In the first chapter, we develop the operadic calculus needed: we encode non-necessarily conilpotent coalgebras with operads and introduce…