C3

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.

A landmark result in the study of locally compact, abelian groups is the Pontryagin duality. In simple terms, it says that for a given locally compact, abelian group G, one can uniquely associate another locally compact, abelian group called the Pontryagin dual of G. In the realm of C*-algebras, whenever such an abelian group G acts on a C*-algebra A, there is a canonical action of the dual group of G on the crossed product of A by G. In particular, it is natural to ask to what extent one can relate properties of the given G-action to those of the dual action. 

In this talk, I will first introduce a property for actions of locally compact abelian groups called the abelian Rokhlin property and then state a duality type result for this property. While the abelian Rokhlin property is in general weaker than the known Rokhlin property, these two properties coincide in the case of the acting group being the real numbers. Using the duality result mentioned above, I will give new examples of continuous actions of the real numbers which satisfy the Rokhlin property. Part of this talk is based on joint work with Johannes Christensen and Gábor Szabó.

Quantized universal enveloping algebras admit an intriguing class of (unbounded) Hilbert space representations obtained via their cluster structure. In these so-called positive representations the standard generators act by (essentially self-adjoint) positive operators. 

The aim of this talk is to discuss some analytical questions arising in this context, and in particular to what extent these representations can be understood using the theory of locally compact quantum groups in the sense of Kustermans and Vaes. I will focus on the simplest case in rank 1, where many of the key features (and difficulties) are already visible. (Based on work in progress with Kenny De Commer, Gus Schrader and Alexander Shapiro). 

Building on the concept of diagonal dimension introduced by Li, Liao, and Winter in 2023, we propose a topological dimension for an inclusion pair of C*-algebras. This new framework allows for finite values in cases of Cartan inclusions that are not diagonal. In this talk, we present calculations for both upper and lower bounds concerning the inclusion of the unitization of c_0(\mathbb{N}) into the Toeplitz algebra. This work is a collaboration with W. Winter.

The landmark completion of the Elliott classification program for unital separable simple nuclear C*-algebras saw three regularity properties rise to prominence: Z-stability, a C*-algebraic analogue of von Neumann algebras' McDuffness; finite nuclear dimension, an operator algebraic version of having finite Lebesgue dimension; and strict comparison, a generalization of tracial comparison in II_1 factors. Given their relevance to classification, most of the investigations into their interplay have focused on the simple nuclear case.

 The purpose of this talk is to advertise the general study of these properties and discuss their applications both within and outside operator algebras. Concretely, I will explain how characterizing when certain twisted group C*-algebras are Z-stable can provide new partial solutions to a well-known problem in generalized time-frequency analysis; this is joint work with U. Enstad. If time allows, I will also briefly discuss how a different incarnation of tracial comparison (finite radius of comparison) for non-commutative tori relates to the existence of smooth Gabor frames; this last part is joint work with U. Enstad and also H. Thiel.

Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension for an étale groupoid that can be used to bound the nuclear dimension of its groupoid C*-algebra.  To have finite dynamic asymptotic dimension, the isotropy subgroups of the groupoid must be locally finite.  I will discuss 1) how to use similar ideas to bound the nuclear dimension of the C*-algebra of a groupoid with `large' isotropy subgroups and 2) the limitations of that approach. In an application to the C*-algebra of a directed graph,  if the C*-algebra is stably finite, then its nuclear dimension is at most 1.  This is joint work with Dana Williams. 

Kasparov's bivariant K-theory (or KK-theory) is an extremely powerful invariant for both C*-algebras and C*-dynamical systems, which was originally motivated for a tool to solve classical problems coming from topology and geometry. Its paramount importance for classification theory was discovered soon after, impressively demonstrated within the Kirchberg-Phillips theorem to classify simple nuclear and purely infinite C*-algebras. Since then, it can be said that every methodological novelty about extracting information from KK-theory brought along some new breakthrough in classification theory. Perhaps the most important example of this is the Lin-Dadarlat-Eilers stable uniqueness theorem, which forms the technical basis behind many of the most important articles written over the past decade. In the recent landmark paper of Carrion et al, it was demonstrated how the stable uniqueness theorem can be upgraded to a uniqueness theorem of sorts under extra assumptions. It was then posed as an open problem whether the statement of a desired "KK-uniqueness theorem" always holds.

In this talk I want to present the affirmative answer to this question: If A and B are separable C*-algebras and (f,g) is a Cuntz pair of absorbing representations whose induced class in KK(A,B) vanishes, then f and g are strongly asymptotically unitarily equivalent. The talk shall focus on the main conceptual ideas towards this theorem, and I plan to discuss variants of the theorem if time permits. It turns out that the analogous KK-uniqueness theorem is true in a much more general context, which covers equivariant and/or ideal-related and/or nuclear KK-theory.