## Finite descent and the Lawrence--Venkatesh method

If Y is a curve of genus at least 2 over a number field, then the finite descent obstruction cuts out a subset of the adelic points, which is

conjecturally equal to the set of rational points. In particular, we expect this set to be finite. In this talk, I will present ongoing work with Jakob

Stix proving that certain projections of the finite descent locus are finite, as predicted by this conjecture. The method we employ can be

loosely described as "Lawrence--Venkatesh for Grothendieck's section set".

## Algebraic spaces and Algebraic stacks

In this talk, after recalling the notion of a stack over an arbitrary site, I will introduce the notions of an algebraic space, an algebraic stack and a Deligne-Mumford stack. I will then introduce some examples and properties of these "generalized schemes".

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

## Online: Buildings, quaternions and Drinfeld-Manin solutions of Yang-Baxter equations

To be useful in theoretical physics, mathematical structure

## Definition and examples of stacks

As motivation, we will start by considering the category of continuous functions and illustrate some gluing properties that make it a stack over the category of topological spaces. We will then give the definition of a stack and explore some other examples of stacks coming from algebraic geometry.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

## Online: Noncommutative Weil conjectures

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

The Weil conjectures (proved by Deligne in the 70's) played a key role in the development of modern algebraic geometry. In this talk, making use of some recent topological "technology", I will extended the Weil conjectures from the realm of algebraic geometry to the broad noncommutative setting of differential graded categories. Moreover, I will prove the noncommutative Weil conjectures in some interesting cases.

## Online: Homotopy and Stratification in Arithmetic Geometry

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

## Online: L-series values of twists of elliptic curves

We consider certain families of sextic twists of the elliptic curve y^2=x^3+1 that are not defined over Q, but over Q[sqrt(-3)]. We compute a formula that relates the value of the L-function L(E_D, 1) to the square of a trace of a modular function at a CM point. Assuming the Birch and Swinnerton-Dyer conjecture, when the value above is non-zero, we recover the order of the Tate-Shafarevich group and show that its value is a square.

## Fibered categories (Part 2/2)

I will briefly recall the notion of fibered categories, and give the illustrative example of elliptic curves. I will then present some important results (foremost is Yoneda’s lemma for fibered categories), and conclude with a discussion of equivariant objects in a fibered category.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

## Modularity properties of false theta functions

Theta functions for positive definite and indefinite lattices are important examples of modular forms and mock-modular forms. Changing signs in the definition of theta functions breaks these modular symmetries and produces so-called false theta functions instead. By applying lessons from the study of indefinite theta functions, one can construct modular completions for these false theta functions, which I want to present in this talk.

## Fibered categories, I

In this talk, I will first introduce the notions of fibered categories over a category, pseudo-functors over a category and then give a correspondence between "fibered categories over a category C" and pseudo-functors over C. I will then give examples of fibered categories, in particular, the example of fibered category of quasi-coherent sheaves on Sch/S. I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. This is based on sections 3.1-3.4 of Vistoli's notes.

## Online: A geometric description of Reidemeister-Turaev Torsion

Zoom meeting ID: 919-9946-8404

Password: see email announcement or contact the seminar organisers:

Tobias Barthel (barthel.tobi[at]gmail.com)

David Gay (dgay[at]uga.edu)

Arunima Ray (aruray[at]mpim-bonn.mpg.de)

## Sites and sheaves

I will define sites, i.e. categories with a Grothendieck topology on them. I will give several examples of sites of topological spaces and of schemes. Sites are the right categorical context for sheaf theory, and I will explain how. Finally, I will sketch a proof of Grothendieck's result that representable functors are sheaves in the fpqc topology - and hence also in the fppf and étale topology. This is mostly based on Vistoli's notes, section 2.3.

## ONLINE: Zagier's polylogarithm conjecture and an explicit 4-ratio

Zoom Online Meeting ID: 919 6497 4060

For password see the email or contact Pieter Moree (moree@mpim...).

## ONLINE: When multiplicativity meets automaticity...

Automatic sequences - that is, sequences computable by finite automata,provide a basic model for computation. The main objective of the present talk is to show how a blend of ideas from number theory, combinatorics and ergodic theory can be used to characterize automatic sets with some

multiplicative properties.

Zoom Online Meeting ID: 919 6497 4060

For password see the email or contact Peter Moree (moree@mpim...).

## ONLINE: A glimpse on 3d modularity

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

## Online: Recap of scheme theory

I will recall the definition of sheaves and schemes and many of their properties, such as e.g. properness, smoothness, &c. This is all material from Hartshorne, parts II and III, with less of a focus on sheaf cohomology (already treated in the previous reading group, on DT invariants), and making use of the category theory background.

## Online: Category background for stacks

I will recall often-used categorical constructions, such as the Yoneda lemma, categorical limits, and adjunctions. Most examples will be algebraic or topological in nature, with more geometric examples coming in the next session. I will also introduce group objects and discrete group objects in a category.

## Computability of the Minimal Genus on Second Homology

Surface representatives of second homology classes can be used to give geometric invariants for second homology classes, the most prominent examples are the genus and the Euler characteristic. In this talk I will introduce the minimal genus problem, explain why determining the minimal genus of a given homology class is in general undecidable, and how to compute it for a large class of "negatively-curved" spaces including 2-dimensional CAT(-1)-complexes.

## ONLINE: A short journey through indefinite theta series

Online link will be send in e-mail announcement.

To start the summer season we give an informal introduction to the theory of indefinite theta series and their role in arithmetic and geometry. In particular, the talk will be colloquium style directed at the entire mathematical community of the MPI.

## ONLINE: Perfect points on abelian varieties

Let k be a field which is finitely generated over the algebraic closure of F_p, L be its perfection and let A be a k-abelian variety. The main goal of this talk is to provide some new result on the structure of the torsion free part of A(L). These results are motivated by their application to the "full" Mordell-Lang conjecture.The main tool is the study of various p-adic incarnation of certain 1-motives attached to L-rational points of A.

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

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