Geometry and Quantum Theory (GQT)


Abstracts GQT Conference

A. V.  Kiselev (RUG)

The differential graded Lie algebra of unoriented graphs

Abstract: Like matrices or vector fields, unoriented graphs with wedge ordering of edges form a Lie algebra. We recall this intuitive construction (designed by Kontsevich in 1993-4 in the context of mirror symmetry) and establish that it is well defined [arXiv:1811.10638]. Indeed, a graph can equal minus itself so that the entire calculus goes modulo zero graphs. A structure of dgLa on the vector space of graphs modulo equivalence relation from the edge count is provided by the vertex-expanding differential [•−•, ·]. The study of cocycles located on and near the ray (#V, #E) = (n, 2n−2) is a domain of active research (Willwacher et al., Merkulov et al., Kontsevich) because countably many cocycles on the ray (n, 2n−2) stem from the generators of Grothendieck–Teichmüller Lie algebra grt (introduced by Drinfeld). Every such cocycle at n = (2ℓ+1)+1 contains a (2ℓ + 1)-gon wheel with nonzero coefficient. Knowing these cocycles (and their iterated commutators, which generate a free Lie algebra) is important: the graph orientation morphism [arXiv:1811.07878] takes them to universal infinitesimal symmetries of Poisson brackets on arbitrary finite-dimensional affine Poisson manifolds. We illustrate the calculus of graphs by using the tetrahedron (ℓ = 1), pentagon-wheel cocycle (ℓ = 2), and heptagon-wheel cocycle (ℓ = 3), see [arXiv:1710.00658]. Examples and substantiation proofs are joint work with R. Buring and N. J. Rutten. (This material can be used as student exercises in general algebra courses, e.g., group theory.)

Irakli Patchkoria (Aberdeen)

On polynomial maps and Witt vectors

Abstract: Witt vectors are a generalization of p-adic numbers and show up in computations in topology. Motivated by those calculations, this talk will discuss a new structure on Witt vectors which is functoriality in certain polynomial maps. We will start by introducing Witt vectors and polynomial maps. Along the way we will focus on explicit examples. Then we will explain the main functoriality result. Finally, we will mention applications in algebra and topology. This is joint work with E. Dotto and K. Moi.

Wadim Zudilin (Nijmegen)

q-Deformation of modular forms

Abstract: Modular forms form a unique enterprise with applications in diverse areas of mathematics. Their Fourier coefficients represent an important information about the algebraic count of points on algebraic varieties over finite fields. In this talk I will discuss a “motivated” problem of $q$-deformation of these coefficients and include some explicit pointers to the recent works.

Margarida Melo (Roma III)

Tropicalizing the moduli space of stable spin curves and applications

Abstract: In recent years, the combinatorial systematic treatment of degenerations of classical linear series within the theory of tropical linear series has seen spectacular developments and has led to many important results on algebraic curves. On the other hand, the introduction and study of a number of tropical moduli spaces of curves along with its realization as skeletons of their classical (compactified) counterparts allows for a deeper understanding of combinatorial aspects of moduli spaces and in particular of their compactifications. In this talk, which is based on joint work with Lucia Caporaso and Marco Pacini, I will explore this principle for Cornalba’s moduli space of spin spin curves. In particular, I will describe a stratification of this moduli space and introduce a tropical interpretation for the skeleton of its analytification. Time permitting, I will mention a number of interesting connections and applications of our work.

Valentina di Proietto (Exeter)

A crystalline incarnation of Berthelot’s conjecture and Kunneth formula for the crystalline fundamental group

Abstract: In this talk I will introduce the crystalline site of a variety in positive characteristic, discuss the notion of crystals and isocrystals over it and define its crystalline fundamental group. I will then discuss Berthelot’s conjecture and its relation with Kunneth formula for the crystalline fundamental group. This is a joint work with Fabio Tonini and Lei Zhang.

Johannes Schmitt (ETH Zürich)

Admissible cover cycles in the moduli space of stable curves

Abstract: Inside the moduli space of stable curves there are closed
subsets defined by the condition that the curve C admits a finite cover
of a second curve D with specified ramification behavior. I will show
how these sets can be parametrized by nice smooth and proper moduli
spaces. In many cases, this parametrization can be used to compute the
fundamental class of such admissible cover loci in the cohomology group
of the moduli space. This is joint work with Jason van Zelm.

Teresa Monteiro Fernandes (Universidade de Lisboa)

Relative Riemann-Hilbert correspondence for a projection

Abstract: Let X×S be a product of complex manifolds where S is a complex line, and let p: X×S→S be the projection.  In this talk we present an overview of the construction of the relative Riemann-Hilbert functor as a right quasi-inverse to the solution functor for regular relative holonomic modules.

This is a summary of joint work with Claude Sabbah which aimed to study the case of modules underlying a mixed twistor D-module. If the dimension of X is 1, the situation is simpler to deal with, and we proved that, in a generic sense, any relative holonomic module admits a coherent restriction to any divisor of the form Y×S, which, in turn, allowed us to prove that, in a generic sense, the Riemann-Hilbert functor is also a left quasi-inverse functor. As a consequence of work in progress with Luisa Fiorot, the condition of genericity can be withdrawn.

Marcello Seri (Groningen)

Can you hear the shape of a “broken” drum?

Abstract: Sub-Riemannian manifolds are a generalisation of Riemannian manifolds where the geodesic flow is subject to some restrictions on the velocities and the metrics can become degenerate. In this talk I would like to motivate the interest in developing a spectral geometry of sub-Riemannian manifolds, the challenges that we are facing in doing it and the main results and ideas.

Javier Gutiérrez (Barcelona)

Morita homotopy theory for simplicial operads

Abstract: In operad theory, it is interesting to characterize the morphisms of operads that induce an equivalence between the corresponding categories of algebras. In this talk, I will present a model structure on the category of simplicial operads, called the Morita model structure, whose weak equivalences are precisely the morphisms with that property. Since the cofibrant resolution of every operad gives a model for the corresponding notion of homotopy invariant algebraic structure, the Morita model structure provides a model for a homotopy theory of homotopy invariant algebraic structures. This is joint work with G. Caviglia.