Ausführliche Liste aller demnächst stattfindenden Vorträge und Seminare. Für eine Übersicht konsultieren Sie bitte auch den Kalender.

## Seminar on Kac-Moody algebras and related topics

## On a certain class of linear relations among the multiple zeta values arising from the theory of iterated integrals

In this talk, we consider iterated integrals on a projective line minus generic four points and introduce a new class of linear relations among the MZVs, which we call confluent relations. We start with Goncharov’s notation for iterated integrals, review some basic notions and properties of iterated integrals, and define a class of relations among iterated integrals, which naturally arise as “solving differential equations step by step”. Confluent relations are defined as the limit of these relations when merging two out of the four punctured points.

## 4d TFTs, Categorical bialgebras, and 2-Segal spaces

In the 90s Crane-Frenkel proprosed the construction of a 4d TQFT from a bimonoidal category, that is, a category equipped with both a monoidal and comonoidal structure so that the data defining the comonoidal structure is itself monoidal. In this talk I will explain a conceptual reason for the connection between bialgebras and TFTs. Secondly, I will describe how to extend the classical construction of the Hall algebra of an abelian category to produce new examples of bimonoidal categories: the Hall bimonoidal categories.

## On linear relations between L-values and arithmetic functions

In 1975 Cohen constructed a series of modular forms of half-integral weights. Its q-coefficients contain special values of Dirichlet functions and were used by Cohen to create various equations of them with arithmetic functions. The modular forms are called Cohen-Eisenstein series and were later generalized to the case for Hilbert modular forms.

## On a mean value result for a product of $L$-functions

The asymptotic behaviour of moments of $L$-functions is of special interest to number theorists and there are conjectures that predict the shape of the moments for families of $L$-functions of a given symmetry type. However, only some results for the first few moments are known. In this talk we will consider the asymptotic behaviour of the first moment of the product of a Hecke $L$-function and a symmetric square L-function. This is joint work with O. Balkanova, G. Bhowmik, D. Frolenkov.

## On the structure of mixed weight Hilbert modular forms

In this talk we discuss joint work with Sho Takemori on Hilbert modular forms

over the real quadratic field of discriminant 5, with respect to its full modular group.

The graded ring of all Hilbert modular forms of parallel weight were determined by Gundlach.

By using his result and some elementary technique, we establish a structure theorem on mixed

Hilbert modular forms.

## Computing weight 1 forms -- a p-adic approach

The computation of Hecke-eigenforms of weight at least 2 is readily accomplished through the

theory of modular symbols as these Hecke-eigensystems occur in the cohomology of modular

curves. However, the same is not true for weight 1 modular forms which makes computing

the dimensions of such spaces difficult let alone the actual system of Hecke-eigenvalues.

Recently effective methods for computing such spaces have been introduced building on an

algorithm of Kevin Buzzard. In this talk, we present a different, p-adic approach towards

## Algebra up to homotopy

## A p-adic family of Saito-Kurokawa lifts for a Coleman family and the Bloch-Kato conjecture

We will construct a $p$-adic family of Saito-Kurokawa lifts for a Coleman family and extend the result of Agarwal and Brown on the Bloch-Kato conjecture for elliptic modular forms of low weights to higher weights. More precisely, we will prove that the $p$-valuation of the order of the Selmer group of a Coleman deformation is bounded below by the $p$-valuation of the algebraic part of the critical $L$-value attached to the initial Hecke eigenform of a Coleman family satisfying some reasonable assumptions given by Agarwal and Brown.