## q-Series, their Modularity, and Nahm’s Conjecture [Promotionskolloquium]

## Picard groups of quotient ring spectra

In classical algebra, the Picard group of a commutative ring R is invariant under quotient by nilpotent elements. In joint work in progress with Ishan Levy and Guchuan Li, we study Picard groups of some quotient ring spectra. Under a vanishing condition, we prove that Pic(R/v^{n+1}) --> Pic(R/v^n) is injective for a ring spectrum R such that R/v is an E_1-R-algebra. This allows us to show Picard groups of quotients of Morava E-theory by a regular sequence in its π_0 are always ℤ/2.

## $\infty$-Category of higher geometric groupoids

## Construction of the approximation maps

## Galois cohomology of O_C (continued)

## tba [NT seminar]

## Averages of class numbers [NT seminar]

In the 1950s Erdos developed a method to give upper and lower

bounds of the correct order of magnitude for $d(P(n))$ where $d$ is the

divisor function and $P$ is a polynomial. This was greatly extended by

Nair and Tenenbaum to a wide class of multiplicative functions and

sequences.

## On the size and structure of dynamical Galois groups.

Following an analogy with Serre's open image theorem, dynamical Galois groups are expected to be large and complicated, unless the underlying rational function is "special" in that it features some exceptional amount of extra-structure (the amount depending on the property at hand). I will review conjectures and expectations along these lines and present past and ongoing joint work with Andrea Ferraguti, where the property at hand for the dynamical Galois group is "being abelian". Here "special" has been explicitly conjectured by Andrews--Petsche.

## 6 torsion of quadratic number fields [NT seminar]

I will discuss some new results on averages of multiplicative functions over integer sequences from https://arxiv.org/abs/2403.13359. I will then give applications to Cohen-Lenstra conjecture.

Joint work with Chan, Koymans and Pagano.

## Hidden structure in moduli spaces of graphs

A natural setting for studying the $n$-loop contribution to a Feynman amplitude is the appropriate moduli space of graphs. I will explain how geometric group theory methods can be used to explore the structure of such spaces, and then describe some features we can detect indirectly but have not yet found.

## Hopf algebra structures in the cohomology of moduli spaces

I will discuss two spectral sequences of Hopf algebras, a morphism between them, and some applications. One spectral sequence is related to the algebraic K-theory of the integers and was introduced, without Hopf algebra structure, by Quillen in his proof that algebraic K-groups are finitely generated. The other spectral sequence is related to the Grothendieck-Teichmüller group and to Kontsevich's graph complexes. This is joint work with Francis Brown, Melody Chan, and Sam Payne, cf. arXiv:2405.11528.

## Connes—Kreimer—Feynman type bialgebras

Bialgebras built on the model of Connes and Kreimer are ubiquitous in mathematics. One explanation is that they stem from a categorical construction using Feynman categories. This opens bridges between combinatorics, algebra and topology as we explain. Examples include (compactifications of) moduli spaces of Riemann surfaces on one end, and inclusion/exclusion algebras on the other. In between, these are factorization systems. Using this framework, statements that things that seem analogous or similar can be made rigorous for instance in terms of functors.

## Tubing expansions of Dyson—Schwinger equations

I will give an overview of different occurrences of tubings of rooted trees in combinatorics and physics and then explain how one kind of tubing of rooted trees can be used to give series solutions of Dyson—Schwinger equations in quantum field theory.

## Goncharov's programme, and depth reductions of multiple polylogarithms (in weight 6)

Multiple polylogarithms $Li_{k_1,\ldots,k_d}(x_1,\ldots,x_d)$ are a class of multi-variable special functions appearing in connection with K-theory, hyperbolic geometry, values of zeta functions/L-functions/Mahler measures, mixed Tate motives, and in high-energy physics.

## Resurgent Extrapolation and QFT: Mining Perturbation Theory for Non-perturbative Information

Resurgence implies that a wealth of non-perturbative information is generically encoded in perturbative data. It also motivates improved approaches to the problem of decoding this information. I describe some recent developments in the search for optimal methods of extrapolation and analytic continuation, based on ideas from the theory of resurgent asymptotics. I will illustrate some of the ideas with applications to examples in quantum field theory.

## Multisymplectic structure and homotopy reduction of General Relativity

In General Relativity, the Hamiltonian formulation of the dynamics and the Poisson bracket of observables depend on the choice of a codimension 1 submanifold as initial time-slice. But such a submanifold is not invariant under diffeomorphisms. As a consequence, Noether's first theorem, which associates to a symmetry a conserved momentum, does not define a homomorphism of Lie algebras. Starting in the 1960s, this issue has led to the development of multisymplectic geometry, eventually replacing the Poisson algebra of momenta with an $L_\infty$-algebra of currents. I will show that:

## On the Euler characteristic of the commutative graph complex

I will present recent results on the asymptotic growth rate of the Euler characteristic of Kontsevich's commutative graph complex. These results imply the same asymptotic growth rate for the top-weight Euler characteristic of $\mathcal{M}_g$, the moduli space of curves, due to a theorem by Chan, Galatius and Payne. Further, the results establish the existence of large amounts of unexplained cohomology in this graph complex and many related cohomologies.

## Resurgence with quasi-modularity: outperforming Ramanujan

I describe recent progress on two intriguing problems. The first concerns improvements on Ramuajan's evaluation of odd zeta values, by expansion in $\exp (-2 \pi) < 1/535$. The second concerns a study of resurgence in the asymptotic expansion of Lambert series, where easily computable perturbative terms are accompanied by non-perturbative corrections in $\exp (-1/x)$ at small $x$. I shall explain how the pioneering work of Spencer Bloch, Pierre Vanhove and Matt Kerr, on elliptic polylogarithms from Feynman integrals, led to progress on both problems, by exploiting quasi-modularity.

## On some arithmetic statistics for integer matrices

We consider the set $\mathcal{M}_n(\mathbb{Z}; H))$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb{Z}; H)$ with a given characteristic polynomial $f \in\mathbb{Z}[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible.

## An introduction to the BPS cohomolgy of 2-Calabi—Yau categories

BPS cohomology is a cohomological invariant of sufficently geometric abelian categories of homological dimension 1,2, or 3 and has its origins in the enumerative geometry of Calabi—Yau threefolds. Nowadays there are many applications of BPS cohomology, especially to (geometric) representation theory.

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