Beyond Multiple Zeta Values:
finite versions, q-analogues, and multiple Eisenstein series

Schedule

Tuesday 27th August 

• 14:00 - 15:00 Benjamin Brindle: Various new aspects in the structure of q-MZVs
  Abstract: We focus on Bachmann's conjecture about the structure of q-MZVs (2015) and several new partial results. Furthermore, we will discuss some ideas of how the big conjecture could be broken down into smaller problems and how they could be proven.

• 15:15 - 16:15 Jan-Willem van Ittersum: Integer partitions detect the primes' and MacMahon-like q-series
  Abstract: We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. For example, an integer n ≥ 2 is prime if and only if (n^2 − 3n + 2) M_1(n) − 8 M_2(n) ≥ 0, where M_a(n) are MacMahon’s well-studied partition functions. These functions are examples of qMZVs. Further, we discuss several generalizations of MacMahon's q-series and explain the prime-detecting properties of their coefficients, their limiting behaviour as well as their modular completions. 

• 16:30 - 18:00 Short talks and discussions
  Henrik Bachmann: Conjectures for (formal) multiple Eisenstein series  (Slides)
  Benjamin Brindle: How to invert a matrix

• ~18:30 Dinner at Altes Mädchen 

Wednesday 28th August 

• 10:30 - 11:30 Niclas Confurius: Towards a Hoffman basis for formal multiple zeta values
  Abstract: F. Brown proved the Hoffman basis conjecture for motivic multiple zeta values. A key ingredient in his proof is the Goncharov coproduct. We will revisit the proof in the context of formal multiple zeta values. We show that the Hoffman basis conjecture in this context holds, if we assume that the double shuffle Lie algebra dm_0 is a free Lie algebra with one generator in each odd degree ≥ 3. This talk is the first in a series of three talks. In upcoming talks by U. Kühn and A. Burmester, recent approaches to constructing an analogue to Goncharov's coproduct for multiple q-zeta values by means of post-Lie algebras will be discussed.

• 11:30 - 12:30 Short talks and discussions
  Jan-Willem van Ittersum: Modular forms as products of at most two Eisenstein series
  Steven Charlton: Depth 2 polylogs and high level coloured MZV’s

• Lunch

• 14:00 - 15:00 Ulf Kühn: Post Lie algebras with application to multiple (q-)zeta values
  Abtract: Post Lie algebras have recently gained prominence as a key tool in the study of multiple zeta values (MZVs) and their q-analogues, multiple q-zeta values (qMZVs). This talk introduces the fundamental concepts of Post Lie algebras and explores their application to the algebraic structures underlying MZVs and qMZVs. We will discuss the construction of analogues to the Goncharov coproduct for qMZVs using Post Lie algebras, highlighting their role in addressing problems such as the 123-basis conjecture.
  (Slides are available on request)

• 15:15 - 16:15 Annika Burmester: How to derive a coproduct on Z_q?
  Abstract: The algebra Z_q is spanned by various models of multiple q-zeta values as well as a rational version of multiple Eisenstein series. One way to get a deeper understanding of the algebraic structure of these objects is to follow Racinet’s work on multiple zeta values and obtain a Lie algebra to a formal version of Z_q. As an initial step, we present a spanning set of Z_q, which satisfies the particular simple index shuffle and hence gives a very useful formalization of Z_q. Then, we provide an explicit description of the corresponding affine scheme using non-commutative power series. We expect the linearized space to be equipped with a Lie structure coming from a post-Lie bracket. We discuss ongoing progress and challenges related to this conjecture. Finally, we explain how Racinet’s double shuffle Lie algebra fits into this framework.

• 16:30 - 18:00 Short talks and discussions
  Khalef Yaddaden: The twisted Magnus product of multiple zeta maps
  Anna Johannsen: Overview of Okounkov’s Conjecture and Recent Developments
  Julius Mann: A Glimpse into Prismatic Cohomology

• ~18:30 Grillen im Schanzenpark

Thursday 29th August 

• 10:30 - 11:30 Henrik Bachmann: Finite multiple zeta values and their variants (Slides)
  Abstract: Finite multiple zeta values, introduced by Kaneko and Zagier around a decade ago, are variants of multiple zeta values. Although their definitions share some similarities, they are fundamentally different. Kaneko and Zagier have conjectured a deep relationship between these two. Like multiple zeta values, finite multiple zeta values are governed by a conjectured set of relations that are believed to encompass all possible relations among them. This naturally leads to the definition of formal finite multiple zeta values, in analogy to formal multiple zeta values. In this talk, we give an introduction to finite multiple zeta values, present the Kaneko-Zagier conjecture, and discuss supporting evidence based on the "BTT-philosophy". Additionally, we introduce the algebra of formal finite multiple zeta values and propose a formal analogue of the Kaneko-Zagier conjecture. This talk is based on joint work with Tasaka (Osaka) and Takeyama (Tsukuba), as well as ongoing work with Risan (Nagoya).
  References: Lecture notes on q-analogues and finite multiple zeta values

• Lunch

• 14:00 - 15:00 Khalef Yaddaden: Double shuffle torsor of cyclotomic MZVs and stabilizers of de Rham and Betti coproducts
  Abstract: Racinet described the double shuffle and regularization relations between multiple polylogarithm values at Nth roots of unity through a Q-scheme called DMR^iota. Here, iota is a group embedding from a finite cyclic group G of order N into the multiplicative group of nonzero complex numbers. Enriquez and Furusho later proved, for the case N=1, that a subscheme DMR^iota_x is a torsor of isomorphisms between Betti and de Rham objects. In this talk, we establish a cyclotomic generalization of this result for N greater than or equal to 1.
  First, we explicitly describe the torsor structure of DMR^iota_x. We then introduce the appropriate de Rham and Betti objects in this context: the de Rham side arises from a crossed product algebra, which allows for a reformulation of Racinet's harmonic coproduct that better aligns with the formalism introduced by Enriquez and Furusho. The Betti side, on the other hand, arises from a group algebra of the orbifold fundamental group of the space (C^* minus mu_N) divided by mu_N, where mu_N is the group of Nth roots of unity.
  Finally, we demonstrate the existence of Betti coalgebra and Hopf algebra coproducts such that DMR^iota_x is a torsor of isomorphisms linking these Betti coproducts to their de Rham counterparts.

• 15:15 - 16:15 Steven Charlton:  Depth reductions of multiple polylogarithms in weight 6
  Abstract: One of the main challenges in the study of multiple polylogarithms revolves around understanding how on many variables a multiple polylogarithm function (or `interesting' combinations thereof) actually depend (``the depth''), as for example Li_{1,1} can already be expressed via Li_2. Goncharov gave a conjectural criterion (``the Depth Conjecture'') for determining this, using the motivic coproduct, as part of his programme to investigate Zagier's Polylogarithm Conjecture on values of the Dedekind zeta function zeta_F(m).
  I will give an overview of Goncharov's Depth Conjecture, and its implications. I will discuss what is currently known, including recent results in weight 6, in particular: my proof of the depth reduction of a weight 6 depth 3 function under the dilogarithm symmetries x \mapsto 1-x, 1/x, and Matveiakin-Rudenko's proof of depth reduction of this function under the 5-term relation (modulo the symmetries).