# Events for 07/08/2020 from all calendars

## Groups and Dynamics Seminar

Time: 12:00PM - 1:00PM

Location: 940 9667 3668

Speaker: Rachel Skipper, OSU

Title: Generating lamplighter groups with bireversible automata

Abstract: We use the language of formal power series to construct finite state automata generating groups of the form A \wr Z, where A is the additive group of a finite commutative ring and Z is the integers. We then provide conditions on the units of the ring and the power series which make automata bireversible. This is a joint work with Benjamin Steinberg.

## Noncommutative Geometry Seminar

Time: 1:00PM - 2:00PM

Location: Zoom 942810031

Speaker: Aurélien Sagnier, John Hopkins University

Title: Towards arithmetic sites at some places

Abstract: Adèle class spaces for number fields are, as A. Connes stressed it in 1996, very important for the spectral interpretation of zeroes of Hecke $L$-functions (and so in particular for the Riemann zeta function or for the Dedekind zeta function of a number field). The semiringed topos $\left(̂\widehat{\N^{\times}},(\Z∪{−∞},\max,+)\right)$ called by A. Connes and C. Consani the arithmetic site and introduced by them in 2014 provides a algebro-geometric background for the adèle class space of $\Q$. Thanks to this algebro-geometric backbground, one could hope in the long term to transfer and adapt to the context of number fields (and in A. Connes' and C. Consani's case $\Q$) ideas coming from Weil's proof of the analogue of the Riemann hypothesis in the function field case. In my PhD thesis, I introduced for the number field $\Q(\imath)$ a semiringed topos $\left(\widehat{\mathcal{\Z[\imath]}},(\Z[\imath]^{\text{conv}},\text{Conv}(\cup),+)\right)$ similar to the one introduced by A. Connes and C. Consani and which is linked to the adèle class space of $\Q(\imath)$ and consequently linked to the Dedekind zeta function of $\Q(\imath)$ and a family of Hecke $L$-functions. However the question of naturality of the choice of $\Z[\imath]^{\text{conv}}$ as structure sheaf remained unanswered. In this lecture, I will show that $\Z[\imath]^{\text{conv}}$ is the solution to an universal problem coming from the hyperaddition on $\Z[\imath]/\{\pm 1,\pm\imath\}$ and so that this choice of structure sheaf was natural after all. This universal problem coming from an hyperaddition will help us to get hints on how to define arithmetic sites for other number fields and for example for $\Q(\sqrt{2})$.