Alternatively have a look at the program.

## The unipotent mixing conjecture

Low-lying horocycles are known to equidistribute on the modular curve. Here we consider the joint distribution of two low-lying horocycles of different speeds in the product of two modular curves and show equidistribution under certain necessary diophantine conditions. The techniques involve a combination of arithmetic, automorphic and ergodic techniques. This is joint work with Philippe Michel.

## Quantum modularity of additive twists of Maass forms L values

Given the sequence $a(n)$ of Fourier coefficients of a Maass form for a Fuchsian group $\Gamma$ and a rational $x$, we consider the additively twisted central $L$-value

$$ L(x) = \sum_{n\geq 1} a(n) e^{2\pi i n x} / n^{1/2} $$

## Ergodic proof of an equidistribution result of Ferrero-Washington

We reprove the main equidistribution instance in the Ferrero-Washington proof of the vanishing of cyclotomic Iwasawa $\mu$-invariant, based on the ergodicity of a certain $p$-adic skew-product extension dynamical system that can be identified with Bernoulli shift (joint with Bharathwaj Palvannan).

## On the density hypothesis in higher rank

The non-tempered part of the $L^2$-spectrum of (finite volume) locally symmetric spaces plays an exceptional role in many applications. However, this part of the spectrum (besides the constant function) is very difficult to understand and is the object of some deep conjectures in the field of automorphic forms, such as the (generalized) Ramanujan conjecture. Sometimes good quantitative estimates on the *size* of the non-tempered spectrum are sufficient in practice. This leads to Sarnak's density hypothesis in higher rank.

## Statistics of roots of congruences

The roots of quadratic congruences are a classical object of study, and largely due to applications in analytic number theory, their statistical distribution is well-studied. On the other hand, very little is known about roots of higher degree congruences beyond their equidistribution. We discuss some recent results (joint with Jens Marklof and Zonglin Li) in the quadratic case and some results related to the cubic case.

## Distributions of Manin’s iterated integrals

We recall the definition of Manin’s iterated integrals of a given length. We then explain how these generalise modular symbols and certain aspects of the theory of multiple zeta-values. In length one and two we determine the limiting distribution of these iterated integrals. Maybe surprisingly, even if we can compute all moments also in higher length we cannot in general determine a distribution for length three or higher. This is joint work with N. Matthes.

## Continuity and value distribution of quantum modular forms

Quantum modular forms are functions $f$ defined on the rationals whose period functions, such as $\psi(x):= f(x) - x^{-k} f(-1/x)$ (for level 1), satisfy some continuity properties. In the case of $k=0$, $f$ can be interpreted as a Birkhoff sum associated with the Gauss map. In particular, under mild hypotheses on $\psi$, one can show convergence to a stable law. If $k$ is non-zero, the situation is rather different and we can show that mild conditions on $\psi$ imply that $f$ itself has to exhibit some continuity property.

## Automorphic functions twisted by representations, unitary and non-unitary

Automorphic functions that are twisted by unitary or

## Sparse equidistribution of hyperbolic orbifolds associated to real quadratic fields

Associated to class groups of real quadratic fields, Duke, Imamoglu, and Tóth have defined certain hyperbolic orbifolds and showed that they equidistribute when projected to the modular curve. In this talk I will describe (and show pictures of) an extension of their construction and results to modular curves of higher level as well as sparse subsets of the class groups.

This is joint work with Peter Humphries.

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