Welcome coffee

Differential transcendance and Galois theory

• Thomas Dreyfus (CNRS, Université Strasbourg)

In this talk we consider meromorphic solutions of difference equations and prove that very few among them satisfy an algebraic differential equation. The basic tool is the difference Galois theory of functional equations.

Lunch

"On solutions of Bessel equations"?

• Ksenia Fedosova (Freiburg)

In this talk, we concentrate on automorphic functions satisfying an inhomogeneous Laplace equation. We discuss their Fourier expansions and, in some set-ups, give explicit expressions for their coefficients. Interestingly enough, in the situations where we can provide explicit solutions, the latter belong to a certain Picard-Vessiot extension of the field of rational functions. Moreover, it turns out that these correspond exactly to the functions appearing in the graviton scattering in the string theory.

Coffee break

Logarithmic equidistribution in small heights

• Gerold Schefer (Basel)

Bilu's theorem about equidistribution of conjugates of algebraic numbers of small height is the prototype of many equidistribution results. It states that for any strict sequence of algebraic numbers $(\alpha_n)$ whose absolute logarithmic Weil height tends to zero, and any continuous and bounded function $f:\mathbb C^*\to \mathbb C$ the averages of the evaluations of $f$ in the conjugates of $\alpha_n$ converge to the integral of $f$ over the unit circle. In this talk we want to understand for which algebraic numbers $\kappa$ we still get convergence if we take the test function $x\mapsto \log|x-\kappa|$. Pineiro, Szpiro and Tucker conjectured convergence for rational $\kappa$. Autissier disproved it taking $\kappa=2$. Breuillard and Frey considered $\kappa=1$, for which convergence holds. Recently R. Baker and Masser showed that for every algebraic $\kappa\ne 0$ which does not lie on the unit circle, convergence fails. The focus will thus be on $\kappa$ lying on the unit circle, not a root of unity.