C3

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A famous theorem of Selberg asserts that $\log|\zeta(\tfrac12+it)|$ is approximately a normal distribution with mean $0$ and variance $\tfrac12\log\log T$, when we sample $t\in [T,2T]$ uniformly. This extends in a natural way to a plethora of other $L$-functions, one of them being Dirichlet $L$-functions $L(s,\chi)$ with $\chi$ a primitive Dirichlet character. Viewing $\zeta(\tfrac12+it)$ and $L(\tfrac12+it,\chi)$ as normal variables, we expect indepedence between them, meaning that for fixed $V_1,V_2 \in \mathbb{R}$: $$\textrm{meas}_{t \in [T,2T]} \left\{\frac{\log|\zeta(\tfrac12+it)|}{\sqrt{\tfrac12 \log\log T}}\geq V_1 \text{   and   } \frac{\log|L(\tfrac12+it,\chi)|}{\sqrt{\tfrac12 \log\log T}}\geq V_2\right\} \sim \prod_{j=1}^2 \int_{V_j}^\infty e^{-x^2/2} \frac{\textrm{d}x}{\sqrt{2\pi}}.$$
    When $V_j\asymp \sqrt{\log\log T}$, i.e. we are considering values of order of the variance, the asymptotic above breaks down, but the Gaussian behaviour is still believed to hold to order. For such $V_j$ the behaviour of the joint distribution is decided by the moments $$I_{k,\ell}(T)=\int_T^{2T} |\zeta(\tfrac12+it)|^{2k}|L(\tfrac12+it,\chi)|^{2\ell}\, dt.$$ We establish that $I_{k,\ell}(T)\asymp T(\log T)^{k^2+\ell^2}$ for $0<k,\ell \leq 1$. The lower bound holds for all $k,\ell >0$. This allows us to decide the order of the joint distribution when $V_j =\alpha_j\sqrt{\log\log T}$ for $\alpha_j \in (0,\sqrt{2}]$. Other corollaries include sharp moment bounds for Dedekind zeta functions of quadratic number fields, and Hurwitz zeta functions with rational parameter. 
    

Let G be an infinite discrete group; e.g. free group, surface groups, or hyperbolic 3-manifold group.

Finite dimensional unitary representations of G of fixed dimension are usually very hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps. 

The talk is a broadly accessible discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas, R. van Handel.

Roe algebras were introduced in the late 1990's in the study of indices of elliptic operators on (locally compact) Riemannian manifolds. Roe was particularly interested in coarse equivalences of metric spaces, which is a weaker notion than that of quasi-isometry. In fact, soon thereafter it was realized that the isomorphism class of these class of C*-algebras did not depend on the coarse equivalence class of the manifold. The converse, that is, whether this class is a complete invariant, became known as the 'Rigidity Problem for Roe algebras'. In this talk we will discuss an affirmative answer to this question, and how to approach its proof. This is based on joint work with Federico Vigolo.