Fernando Chamizo

On the Sphere Problem

One of the oldest problems in analytic number theory consists of counting points with integer coordinates in the d-dimensional ball. It is very easy to find a main term for the counting function, but the size of the error term is difficult to estimate (...).

Automorphic forms and differentiability properties

We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.

Non-Euclidean visibility problems

We consider the analog of visibility problems in hyperbolic plane (represented by Poincaré half-plane model ℍ), replacing the standard lattice ℤ × ℤ by the orbitz = i under the full modular group SL2(ℤ). We prove a visibility criterion and study orchard problem and the cardinality of visible points in large circles.

Tachyonic instabilities in 2 + 1 dimensional Yang–Mills theory and its connection to number theory

We consider the

Hyperbolic lattice point problems

2+1

Irreducibility and the distribution of some exponential sums

dimensional Yang-Mills theory with gauge group

Trends in Number Theory

\text{SU}(N)

A path integral approach to lattice point problems

on a flat 2-torus under twisted boundary conditions. We study the possibility of phase transitions (tachyonic instabilities) when

Lattice points in bodies of revolution

N

Lattice points in rational ellipsoids

and the volume vary and certain chromomagnetic flux associated to the topology of the bundle can be adjusted. Under natural assumptions about how to match the perturbative regime and the expected confinement, we prove that the absence of tachyonic instabilities is related to some problems in number theory, namely the Diophantine approximation of irreducible fractions by other fractions of smaller denominator.