# Titles and abstracts

## Renate Loll, IMAPP, Radboud University

TItle: Quantum Ricci curvature in action

Abstract: Our ability to understand the nonperturbative dynamics of four-dimensional quantum gravity at the Planck scale depends crucially on identifying and measuring observables. A well-known example is the spectral dimension of quantum spacetime, which in Causal Dynamical Triangulations (CDT) was found to exhibit a "Planckian fingerprint" at short distances. Importantly, in a field too often dominated by prima facie arguments, this has stimulated a computational effort across approaches to reproduce this result. While "dimensions" are an important tool, more intrinsically geometric observables are needed to be able to relate the properties of quantum geometry to those of classical spacetimes in general relativity. I will report on very promising progress that has been made using the new quantum Ricci curvature, a notion of Ricci curvature applicable on nonsmooth (ensembles of) geometries. It has been extensively tested and explored in (C)DT models of quantum gravity, and opens exciting new opportunities for bridging the gap with "real" early-universe physics.

## Ryszard Nest, University of Copenhagen

Title: Twisting quantum groups, torsion and Baum Connes assembly map.”

Abstract: We will describe the notions of projective representations of quantum groups and the corresponding twisted quantum group C*-algebras. In particular, for the case of compact quantum groups (which correspond to discrete classical groups), we will study the torsion phenomena and their appearance in the construction of Baum-Connes assembly map for quantum groups.

## Martin Loebl, Charles University, Prague

(joint work with Anetta Jedličková and David Sychrovsky)

Title: A critical Distribution System

Abstract: A distribution crisis is characterized by a large disparity of supply and demand for certain goods, such as respirators.

These are extreme situations where the supply is so small that organizations will realistically not have even the necessary minimum available to ensure operation. Such a situation has two basic characteristics from a distribution point of view:

1. The ethical fairness of the distribution of supply of critical goods is important. According to the rules of fairness, which are declared in advance, it is possible to hypothetically distribute the supply of a critical commodity fairly among organizations.

In other words, let's imagine that buyers have rights to purchase a fair amount of the commodity in question.

1. The existence of a market for the critical commodity is another essential aspect of the distribution. Lack leads to a significant increase in price, organizations try to obtain more than a fair amount to cover their needs, and this is always at the expense of other participants. A free market leads to narrowing of distribution.

Experience shows that centrally controlled distribution is also not socially advantageous, for instance distributors do not have sufficient advantage in such an environment and try to leave it.

We propose a hybrid model based on an autonomous behaviour of the participants.

## Søren Eilers, University of Copenhagen

Title: The statistical mechanics of the $1\times 2$ LEGO brick

Abstract: Almost 20 years ago, I found myself wondering if the number $b_n$ of contiguous buildings constructible with $n$ LEGO bricks of the same shape would grow superexponentially or have finite entropy. I asked several of my Copenhagen colleauges and got diverging answers, but Bergfinnur very quickly backed up his claim that it would have to be finite with a concrete estimate, starting a very enjoyable collaborative effort on what we ended up calling the entropy of LEGO bricks. I will provide more details on this somewhat old story, and supplement with newer data that may say something interesting about what happens for the smallest nontrivial LEGO brick with one side of length two and the rest of lenght one. I am particularly keen on understanding what happens when one varies the dimensionality of the problem.

## John Wheater, University of Oxford

Title: Hard dimers on CDT

Abstract:  We discuss the full model of hard dimers coupled to two-dimensional causal dynamical triangulations (CDT), and its equivalence to a labelled tree model which can be solved subject to one restriction. Depending on the dimer weights there are, in addition to the usual gravity phase of CDT, two tri-critical and two dense dimer phases. We establish the properties of these phases, and discuss how they are related to continuum models.

## Jakob Björnberg, University of Gothenburg

Title: Stable shredded spheres and causal random maps with large faces

Abstract: We consider a model of causal random planar maps where the probability measure is chosen so that large faces are forced to appear. We show that there arises an interesting scaling limit which we call the stable shredded sphere. I will define the stable shredded sphere, describe some of its properties and explain briefly the key ingredients in the proof of the scaling limit result. This is joint work with Nicolas Curien and Sigurður Örn Stefánsson.

## Thordur Jonsson, University of Iceland

Title: Quantum walk on a comb

Abstract: We study continuous time quantum walk on a comb with infinite teeth and show that the return probability to the starting point decays with time $t$ as $t^{-1}$.    We analyse the diffusion along the spine and into the teeth and show that the walk can escape into the teeth with a finite probability and goes to infinity along the spinewith a finite probability.  The walk along the spine and into the teeth behaves qualitatively as quantum walk on the discrete line.

## Peter Orland, the City University of New York

Title: From Gauge Theories to Form Factors and Back

Abstract: Yang-Mills theories can be thought of as asymptotically-free principal chiral models (PCM) coupled together, as was first noticed by Durhuus and Froehlich in 1980. This connection motivated us to study SU(N) PCM correlation functions. Calculating the large-N limit of correlation functions is possible, because all form factors can be determined.

Remarkably, after summing exponentially-decaying contributions, logarithmic behavior emerges in the short-distance limit, in exact agreement with the renormalization group. Correlation functions of the scaling field, the current and the stress-energy tensor can be studied over many length scales. I briefly discuss applications to lower-dimensional SU(N) Yang-Mills theories.

## Anna Beliakova, University of Zürich

Title: Algebraisation of low-dimensional Topology

Abstract: Categories of n-cobordisms (for n=2,3 and 4) are the most studied objects in low dimensional topology.

For n=2 we know that 2Cob is a monoidal category freely generated by its commutative Frobenius algebra object: the circle.

This result classifies all TQFT functors on 2Cob. In this talk, I will present similar classification results for n=3 and n=4 obtained in collaboration with Marco De Renzi and based on the work of  Bobtcheva and Piergallini.

The role of the Frobenius algebra is taken in these cases by a braided Hopf algebra.

I plan to finish by relating our results with the famous problem in combinatorial group theory — the Andrews–Curtis conjecture.

## Philippe Chassaing, Université de Lorraine

Title: Pascal’s formulas and vector fields

Abstract: We study some examples of combinatorial triangles (e.g. Pascal's triangle, Stirling's triangles of both types, Euler's triangle): each of their Pascal's formulas define a vector field, and its field lines, that turn out to be the limits of sample paths of well known Markov chains. As an application to the asymptotic study of complete accessible automata, we provide a new derivation of a formula due to Korsunov. This is a joint work with Jules Flin and Alexis Zevio.

## Niels Obers, University of Copenhagen

Title: Non-Lorentzian geometries in gravity and string theory

Abstract: I will review recent developments on the non-relativistic corner of gravity and string theory, in which the speed of light is very large.

Central to these adances has been the geometric formulation in terms of novel versions of Newton-Cartan (NC) geometry, originally discovered by Cartan in the 20s to geometriz Newton’s law of gravity. I will discuss the underlying algebraic structure of NC, its torsional generalization as well as the torsional  string Newton-Cartan geometry that enters the sigma model of non-relativistic strings probing a non-relativisic target spacetime.  Finally, I will discuss recent insights into the complementary case when the speed of light goes to zero, which is known as the Carroll or ultra-local expansion of gravity, in which case the underlying geometric structure is Carroll geometry.

Title: From the cluster expansion to chromatic polynomials

Abstract: In the cluster expansion of the partition function of a polymer model with hard-core interactions (also known as Mayer expansion of a lattice gas), an important role is played by the so-called Ursell functions. These can be related to a particular graph invariant: the linear coefficient of the chromatic polynomial of the graph. Therefore understanding the behavior these coefficients is crucial both to prove convergence of the cluster expansion.

In this talk, I will present a set of recursion relations for the coefficients of the chromatic polynomial, which provide an easy way to show interesting facts about them. Moreover, the proof can be adapted to cover the case of hypergraphs.

## Jan Ambjørn, University of Copenhagen

Title: Scaling or no scaling of the string tension of random surfaces?

Abstract: It is shown how it is possible to understand both scaling and non-scaling of the string tension of the bosonic string. While one in a continuum formalism is able define the two limits, it is less clear how to obtain the scaling limit in the case of strings on a hypercubic lattice or  strings defined via Dynamical Triangulations.

## George Savvidy, Institute of Nuclear and Particle Physics, NCSR Demokritos, Athens, GreeceA.I. Alikhanyan, National Science Laboratory, Yerevan, Armenia

Title: Stability of Yang Mills Vacuum State.
Gauge Field Theory Vacuum and Cosmological Inflation.

Abstract:  We examine the phenomena of the chromomagnetic gluon condensation in Yang-Mills theory and the problem of stability of the vacuum state. The stability of the vacuum state is analysed in the nonlinear regime.  It is shown that an apparent instability of the Yang Mills vacuum is a result of quadratic approximation.   In the case of (anti)self-dual fields the interaction of chromomagnetic modes of the quantised field in the direction of zero modes is calculated by using a new method of infrared regularisation as well as by the integration over the collective variables of self-interacting  zero modes.  The deformation of (anti)self-dual fields is also considered in the nonlinear regime by the integration, in this case,  over the collective variables of self-interacting unstable modes.    All these vacuum field configurations are stable and indicate that the vacuum is stable and is a superposition of many states.

The deep interrelation between elementary particle physics and cosmology manifests itself when one considers the contribution of quantum fluctuations of vacuum fields to the dark energy and the effective cosmological constant. The contribution of zero-point energy exceeds by many orders of magnitude the observational cosmological upper bound on the energy density of the universe. Therefore it seems natural to expect that vacuum fluctuations of the fundamental fields would influence the cosmological evolution in any way.  Our aim in this review article is to describe a recent investigation of the influence of the Yang-Mills vacuum polarisation and of the chromomagnetic condensation on the evolution of Friedmann cosmology, on inflation and on primordial gravitational waves.

## François David IPhT Saclay, France

Abstract: Isoradial triangulations are examples of critical planar graphs, on which discrete analyticity, integrability, discrete and continuous conformal invariance can be defined and studied for many models. I present some results on the deformations of such triangulations, which break integrability, and their effect on the critical Laplacian and some of its extensions, and for their conformal properties. The relevance and the consequence of these results for some models of two-dimensional quantum gravity will be discussed.
(joint work with J. Scott)

## Antti Kupiainen, University of Helsinki

Title: Integrability of the Liouville theory

Abstract: Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas.

## Andrzej Sitarz, the Jagiellonian University, Krakow

Title: The spectral Einstein tensor.

Abstract: It is well known that the volume of a compact Riemannian manifold, as well as its scalar curvature, can be retrieved using Wodzicki residue applied to negative powers of the Laplace operator. We extend these results to obtain much finer geometric objects, tensors, recovering the Einstein tensor. We provide a suitable noncommutative generalizations and prove that the noncommutative 2-torus has a vanishing Einstein tensor. based on a joint work with L.Dabrowski and P.Zalecki.

## Jakob Yngvason, University of Vienna

Title: Quantum Hall Physics in Higher Landau Levels

Abstract: The quantum states of charged particles moving in a plane orthogonal to a homogeneous magnetic field are naturally grouped into Landau levels which correspond to quantization of the cyclotron motion of the particles  in the magnetic field. The interplay of this motion with external electric fields and  impurities as well as Coulomb interactions between the particles is the subject of Quantum Hall Physics which has in the past 40 years developed into a major subfield of condensed matter physics.

In the talk, I shall review one particular theoretical aspect of Quantum Hall Physics, namely the relations between states and effective Hamiltonians in different Landau levels. The emphasis is on the role of the noncommutative guiding center variables which in a strong magnetic field replace the commuting position variables.

## Jürg Froehlich, ETH Zürich

Title: The Classical Periphery of Quantum Mechanics –Emergence of Particle Tracks in Detectors

Abstract: In this talk, I discuss regimes of Quantum Mechanics that can be described in terms of classical physical theories. The general ideas underlying my analysis are illustrated by a study of particle tracks close to classical point-particle trajectories made visible in detectors.

Some general remarks about the notion of "events" in Quantum Mechanics and their role in understanding measurements will be presented.

## Des Johnston, Heriot-Watt University, Edinburgh

Title: From Subdivision Invariant Random Surfaces to Fractons

Abstract: Some (many?) years ago Bergfinnur (and others in the audience.....) dabbled with subdivision invariant random surfaces, also dubbed "gonihedric" random surfaces.
We trace a line from these through plaquette Ising models to fractons, which have generated a lot of recent interest/papers.

## Sigurdur Stefansson, University of Iceland

Title: Random decorated trees

Abstract: Many combinatorial objects may be decomposed in a natural way into an underlying tree whose vertices are identified with structures which we will call decorations. We refer to such objects as decorated trees. A simple example is an ordered tree, whose decorations are linear orderings of size equal to the degree of their corresponding vertex. Another less trivial example are so called looptrees, introduced by Curien and Kortchemski, where the decorations are circle graphs of length equal to the degree of their corresponding vertex.

I will introduce a very general model of random decorated trees where the underlying tree is a size conditioned branching process whose offspring distribution has infinite variance. This implies that in large trees, there will be many vertices which have a large degree. Under some suitable conditions on the decorations, the decorated tree will have a scaling limit and due to the presence of vertices of large degree the decorations will be present in the limit.

Such random decorated trees appear naturally in statistical physics models on random 2D triangulations (and maps in general). Curien and Kortchemski showed that random looptrees describe the boundary between components in critical percolation on uniform triangulations and there is evidence that this holds for more general models of maps and matter.

Joint work with Delphin Sénizergues and Benedikt Stufler: https://arxiv.org/abs/2205.02968