The art of mathematical physics

# The art of mathematical physics

Note: The poster above represents the state at the first round of invitations in October 2020. Please see the timetable below for the current list of speakers.

## Practical information

The conference will take place in Amphi Bloch at the Institut de Physique Théorique, CEA-Saclay.
We can arrange for the accommodation of speakers at the Hôtel Pierre Nicole in Paris, or at a hotel close to the conference venue (Bures or Gif-sur-Yvette).
To get to the conference venue from Paris, use the RER train service, line B southbound towards Orsay or Saint-Rémy-lès-Chevreuses and get off at station Le Guichet. When exiting the station turn right at the bar, then left at the school and follow the signs to the bus stop (200 metre walk). Finally catch the bus line 9 (there is one every 5 minutes) and get off at Ormes les Merisiers after a 10-minute ride. From the bus stop walk past a modern white building (number 714), turn left and walk all the way down the main alley towards the woods. Cross the parking lot and enter building 774, the conference venue.

## Timetable

Time Monday 20 Sept Tuesday 21 Sept Wednesday 22 Sept Thursday 23 Sept
9:00 Registration
Check of Covid-19 certificates
(F. David and J. Jacobsen)
J. Jacobsen F. Essler D. Bernard
9:40 J.-B. Zuber Y. He E. Boulat I. Kostov
10:20 Coffee break Coffee break Coffee break Coffee break
10:40 P. Fendley B. Duplantier J.-S. Caux O. Castro-Alvaredo
11:20 D. Ridout (online) S. Ribault S. Smirnov A. Gainutdinov
12:00 J. Dubail P. Calabrese (online) E. Granet E. Vernier
12:40 Buffet lunch Buffet lunch Buffet lunch Buffet lunch
14:00 V. Schomerus P. Martin (online) T. Yoshimura (online) V. Pasquier
14:40 B. Wehefritz-Kaufmann (online) A. Klümper N. Andrei (online) J.-M. Maillet (online)
15:20 Coffee break Coffee break Coffee break Coffee break
15:40 Y. Ikhlef G. Mussardo R. Konik (online) B. Pozsgay (online)
16:20 A. Roy S. Lukyanov R. Vasseur (online) B. Doyon (online)
17:00 N. Reshetikhin (online) Z. Wang (online) N. Read (online) P. Di Francesco (online)
17:40 J. Cardy (online) I. Affleck (online) P. Zanardi (online)
20:00 Banquet dinner

## Titles and abstracts

Click on the title of the presentation to download the corresponding slides.
Videos will be made available soon after post-production.

### Ian Affleck: The Majorana-Hubbard model in 1 dimension

It is predicted that a thin layer of type II superconductor in a magnetic field on top of a type 1 topological insulator will have a Majorana fermion at the centre of each vortex. We study this model including hopping and interactions. The hopping can be tuned to zero by varying the chemical potential on the topological insulator so the ratio of interaction strength to hopping can be tuned to infinity. I will present results on this model in 1 dimension with both signs of interaction and arbitrary ratio of hopping to interaction strength.

### Natan Andrei: Static and dynamic effects at the boundaries of a 1-dimensional topological charge conserving superconductors: Exact results

We study 1-D intrinsic superconductors with open boundary conditions and show that the spin triplet phase is topological and supports fractional spin zero-energy-modes. Twisting the boundary conditions away from the symmetry protected values we obtain the mid-gap modes and the full phase diagram. Attaching a Kondo impurity at the edge we solve the model for all values of superconducting and Kondo coupling and describe the system over a full range of the Kondo, shiba and local moments regimes. If time permits I'll discuss the fractionalization at the edge of the Heisenberg spin chain.

### Denis Bernard: Can the macroscopic fluctuation theory be quantized?

The macroscopic fluctuation theory (MFT) is an effective framework adapted to describe transports and their fluctuations in classical out-of-equilibrium diffusive systems. Whether the macroscopic fluctuation theory may be extended to the quantum realm and which form this extension may take is yet terra incognita but is a timely question. I will discuss possible questions that a quantum version of the macroscopic fluctuation theory aim at answering and how analysing quantum simple exclusion processes (Q-SSEP) yields pieces of answers to these questions.

### Edouard Boulat: Exact results in the out-of-equilibrium boundary sine-Gordon model

The Boundary sine-Gordon model has been explicitly solved out-of-equilibrium and at finite temperatures more than 20 years ago, for particular values of the sine-Gordon parameter where the scattering is diagonal. Yet, the finite temperature out-of-equilibrium properties for generic values of the sine-Gordon parameter are much more cumbersome to derive, due to off-diagonal scattering. In this talk, we show that the boundary scattering can be derived exactly within the so-called string basis and we describe an exact out-of-equilibrium, finite temperature solution for arbitrary rational sine-Gordon parameters. We discuss some of its surprising features.

### John Cardy: Some $T \bar{T}$-deformed mathematics

The so-called $T \bar{T}$ deformation of a 2d CFT leads to explicitly modular invariant partition functions. However we point out that the same construction may be extended to many of the objects of classical functional analysis, leading, for example, to deformed modular and Jacobi forms which retain some, if not all, of their modular properties.

### Olalla Castro Alvaredo: Generalized hydrodynamics of particle creation and decay

In this short talk I will review recent results obtained in collaboration with Cecilia De Fazio, Benjamin Doyon and Aleksandra Ziolkowska. We study the out-of-equilibrium properties of an integrable QFT which possesses two stable particles which can form a finitely lived (unstable) bound state. The presence of this unstable bound state leads to interesting new features in quantities such as the particle densities and effective velocities of the stable constituents. When the system is thermalized at a high temperature and then released into a cold environment, quantitative and qualitative evidence of the formation and subsequent decay of unstable particles can be found. If the environment is thermalized at a low but non-zero temperature, a new magnetic effect is observed whereby a finite density of unstable particles can persist for very long times.

### Pasquale Calabrese: Symmetry resolved entanglement

Entanglement and symmetries are two pillars of modern physics. Surprisingly, only in very recent times the interplay between these two fundamental concepts became the theme of an intense research activity merging together notions and ideas from quantum information, quantum field theory, quantum optics, holography, many-body condensed matter, and many more. In this talk, I will review some of the more interesting findings for symmetry resolved entanglement ranging from purely field theoretical ones to microscopical lattice models for disordered systems. The focus of the talk will be on the results and outlooks rather than on the the technical derivations.

### Jean-Sébastien Caux: Quench dynamics using renormalization from integrability

Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution; such overlaps are unavailable in most instances. This talk will present a numerical approach to preferentially generate the states with high overlaps for a generic quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model.

### Philippe Di Francesco: Triangular ice: Combinatorics and limit shapes

We consider the triangular lattice version of the two-dimensional ice model with domain-wall-like boundary conditions, leading to an integrable 20 Vertex (20V) model. We show a remarkable connection between the uniform 20V model and domino tilings of Aztec-like domains. Finally we determine the limit shape of large configurations, obtained by application of the so-called "Tangent Method". Based on work with E. Guitter and B. Debin.

### Benjamin Doyon: Correlation functions from hydrodynamics beyond the Boltzmann-Gibbs paradigm

The Euler-scale power-law asymptotics of space-time correlation functions in many-body systems, quantum and classical, can be obtained by projecting the observables onto the hydrodynamic modes admitted by the model and state. This is the Boltzmann-Gibbs principle; it works for integrable and non-integrable models alike. However, certain observables, such as some order parameters in thermal states or GGEs, do not couple to any hydrodynamic mode: the Boltzmann-Gibbs principle gives zero. After reviewing this principle, I will explain how hydrodynamics can give the leading exponential decay in space and time of order parameter correlation functions. This is related to full counting statistics (large-deviation theory), for which exact predictions are given by the ballistic fluctuation theory based on Euler hydrodynamics. I will explain the example of the quantum XX chain, reproducing results obtained previously by a more involved Fredholm determinant analysis and other techniques, and even giving a new formula for a parameter regime not hitherto studied. The hydrodynamic method is widely applicable, beyond the XX model.

### Bertrand Duplantier: Integral means spectrum, SLE and Riemann zeta function

The integral means spectrum is an essential tool in complex analysis for the multifractal analysis of the harmonic measure on random critical curves, such as the Schramm-Loewner Evolution (SLE). We discuss recent results in this case, as well as for the Riemann zeta function near the critical axis and the relationship of its integral means spectrum to Kraetzer's conjecture for the universal spectrum.
Joint works with D. Beliaev, E. Saksman and M. Zinsmeister.

### Fabian Essler: Yang-Baxter integrable open quantum systems

Recently it has been realised that certain open quantum systems described by Lindblad equations are Yang-Baxter integrable. One example is the quantum version of the asymmetric exclusion process (Q-ASEP), which exhibits a novel integrable structure based on the fragmentation of operator space. I exhibit the implication of this structure for a related model that can be solved by using free fermion methods.

### Paul Fendley: Free fermions and parafermions

Free fermions are ubiquitous in theoretical physics. Essentially all such models are found by expressing the Hamiltonian and/or action as a sum over bilinears of local fermionic operators or fields, sometimes requiring a Jordan-Wigner transformation. I describe models that become free fermionic only after under a much subtler transformation that is both non-local and non-linear in the original interacting fermions. Including also the usual fermion bilinears breaks the solvability but allows a simple lattice analog of an interacting conformal field theory, very useful for numerical analysis. I will also give a brief overview of how free-parafermion chains can be solved in a similar fashion.

### Azat Gainutdinov: Web models as generalisations of statistical loop models

This talk is about higher-rank generalisations of statistical loop models that we introduced recently with Augustin Lafay and Jesper Jacobsen. It is well known that the local formulation of the loop models has underlying quantum group symmetry $U_q sl(2)$ which is manifest in Temperley-Lieb diagrams as elementary blocks of the transfer matrix. In our work we use invariant theory of $U_q sl(n)$ for $n>2$ where the Temperley-Lieb diagrams are replaced by (coloured) spider diagrams of Kuperberg (for $n=3$) and Cautis-Kamnitzer-Morrison for higher values of $n$. These spider diagrams form elementary blocks in our web models defined on the honeycomb lattice - a statistical model of closed, cubic graphs with certain non-local Boltzmann weights that are computed from spider relations. We show that the web model possesses a particular point, at $q=e^{i \pi/(n+1)}$, where the partition function is proportional to that of a $Z_n$-symmetric chiral spin model on the dual lattice. Moreover, under this equivalence, the graphs given by the configurations of the web model are in bijection with the domain walls of the spin model.

### Etienne Granet: On strongly coupled bosons in one dimension

I will present two approaches to compute equilibrium and out-of-equilibrium correlation functions in the strongly coupled delta Bose gas. The first approach relies on a strong-weak and boson-fermion duality that is made manifest by a new regularisation of point-like potentials in quantum mechanics. The second approach consists in developing a machinery to evaluate spectral sums (Lehmann representations) perturbatively in the inverse of the coupling constant, using Bethe ansatz results.

### Yifei He: Four-point cluster connectivities in 2d percolation and LCFT

A signature example of 2d geometric phase transition is the critical percolation, and the fundamental observables involve cluster connectivities which can be related to the correlation functions of spin operators in the Potts CFT. In this talk I will consider the four-point cluster connectivities and describe recent results on determining these quantities using the conformal bootstrap approach together with lattice algebraic consideration. These results further allow analyzing the identity module at $c=0$ of percolation and polymers LCFT which involves a rank-3 Jordan cell with identical Virasoro structure for both theories.

### Yacine Ikhlef: Connectivity operators of 2d loop models

In 2d loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. Their scaling exponents are well known, but some finer properties (typically, geometrical correlations, like percolation probabilities) are harder to derive, as they involve the knowledge of higher correlation functions. In this talk, I will present some advances on this problem for the O(n) loop model, namely : (1) the determination, by analytically solving the conformal bootstrap, of three-point amplitudes for some restricted families of connectivity operators, and (2) the construction of periodic Temperley-Lieb representations which encode the fusion of connectivity operators in the lattice setting. This talk is based on joint work with B. Estienne, Th. Dupic, J. Jacobsen, A. Morin-Duchesne and H. Saleur.

### Jesper Lykke Jacobsen: Rejoyce, the artist comes of age

A few reminiscences and a review of some papers of Hubert in the period 2001-2021.

### Andreas Klümper: Exact solution of the spin-1/2 XXX chain with off-diagonal boundary fields

The spin-1/2 Heisenberg chain with periodic boundary conditions is a seminal model of integrable resp. exactly solvable systems. It is known that the Heisenberg chain with arbitrary boundary fields is still integrable, but so far defied an explicit solution for the case of off-diagonal fields which break the $U(1)$ symmetry. As the magnetization is no longer a good quantum number, the direct application of the Bethe ansatz fails. Here we show how the problem can be solved by a set of non-linear integral equations (NLIEs). Instead of two NLIEs as in the case of the periodically closed chain, we find a set of three NLIEs from which the eigenvalues of the Hamiltonian can be obtained.

### Robert Konik: Time dependent Renyi entropies in quantum quenches of Sine-Gordon

The growth of Renyi entropies after the injection of energy into a correlated system provides a window upon the dynamics of its entanglement properties. We provide here a scheme by which this growth can be determined in Luttinger liquids systems with arbitrary interactions, even those introducing gaps into the liquid. This scheme introduces the notion of a generalized mixed state Renyi entropy. We show that these generalized Renyi entropies can be computed and provide analytic expressions thereof. Using these generalized Renyi entropies, we provide analytic expressions for the short time growth of the second and third Renyi entropy after a quantum quench of the coupling strength between two Luttinger liquids, relevant for the study of the dynamics of cold atomic systems. For longer times, we use truncated spectrum methods to evaluate the post-quench Renyi entropy growth.

### Ivan Kostov: Effective quantum field theory for (boundary) thermodynamical Bethe Ansatz

An effective Quantum Field Theory is constructed for the wrapping effects in $1+1$ dimensional models of factorised scattering with periodic and open boundary conditions. The effective QFT involves both bosonic and fermionic fields and possesses a symmetry which makes it one-loop exact. The corresponding path integral localises to a critical point determined by the TBA equation. For periodic boundary conditions the one-loop effects cancel completely while for open boundary conditions they give the universal part of the boundary entropy.

### Sergei Lukyanov: On the scaling limit of the inhomogeneous six-vertex model

The inhomogeneous six-vertex model is a multi-parametric integrable 2D statistical system. With the anistropy parameter $|q|=1$, the model is critical and covers a variety of interesting universality classes. The goal of the talk is to describe a conjecture from the recent work arXiv:2106.01238. It predicts the scaling limit of the inhomogeneous six-vertex model and its higher spin generalizations in a certain regime of the anisotropy $\arg(q)$.

### Jean-Michel Maillet: Bethe algebras and quantum separation of variables

The construction of "complete" families of commuting conserved charges (Bethe algebras) is instrumental in the resolution of quantum integrable models. In this talk I'll try to explain what "complete" means in this context and how the knowledge of resulting commutative Bethe algebras structure constants determines the full spectrum of these models (eigenvalues and eigenvectors) through an effective construction of separation of variables bases. I'll take the gl(n) quantum integrable models as my main examples. The role of Baxter's T-Q relation (and of its higher rank generalized form as quantum spectral curve) will be described in this framework.

### Paul P. Martin: Spin-chain braid representations

A braid representation is a monoidal functor from the braid category B. A rank-$N$ charge-conserving representation (or spin-chain representation) is a monoidal functor from a monoidal category to the category of rank-$N$ charge-conserving matrices (see talk for definition). In this work we construct all such braid representations up to isomorphism. And celebrate our friend Hubert Saleur.

### Giuseppe Mussardo: Generalised Riemann hypothesis and brownian motion

If Number Theory is arguably one of the most fascinating subjects in Mathematics, Theoretical Physics adds to it the standard of clarity, beauty and deepness which have helped us to shape our understanding of the laws of Nature: together, these two subjects present a fascinating story worth telling, one of those vital, wonderful and superb narrative of enquires often found in science. From this point of view, the seminar presents the main features of the Riemann Hypothesis and discusses its generalisation to an infinite class of complex functions, the so-called Dirichlet L-functions, regarded as quantum partition functions on the prime numbers. The position of the infinite number of zeros of all the Dirichlet L-functions along the axis with real part equal to 1/2 finds a very natural explanation in terms of one of the most basic phenomena in Statistical Physics, alias the Brownian motion. We present the probabilistic arguments which lead to this conclusion and we also discuss a battery of highly non-trivial tests which support with an extremely high confidence the validity of this result.

### Vincent Pasquier: On the XXZ chain at roots of unity

I will review Hubert and I's work on the XXZ spin chain at the roots of unity and its posterity over the years.

### Balázs Pozsgay: Integrable spin chains with medium range interactions

We consider spin chains where the Hamiltonian has a finite interaction range which is bigger than 2. This is a family of models for which there was no general theory so far. We embed such models into the formalism of the Quantum Inverse Scattering Approach, by constructing the proper Lax operators and commuting transfer matrices. We derive a new integrability condition for such spin chains, which can be used to classify such models. We also show concrete examples for medium range models.

### Nicholas Read: Some recent results in spin glass theory

The nature of the ordered phase of finite-dimensional classical spin glasses (such as Ising models with random interactions with mean zero) is a long-standing and fundamental problem of statistical mechanics; in particular the question of whether there are many ordered or "pure" states at low temperature, or only those that reflect the presence of spontaneous breaking of a symmetry. I will review mostly rigorous work on finite-range disordered classical spin systems, in particular, the meaning and structure of Gibbs states, pure states, and metastates, including quantitative results and bounds for exponents.

### Nicolai Reshetikhin: Superintegrable systems

Superintegrability is a feature in classical Hamiltonian mechanics that is in certain sense stronger than Loiville integrability. This talk will start with an overview of superintegrability and with some examples. It will end with a description of superintegrable systems on moduli spaces of flat connections.

### Sylvain Ribault: On the spectrum of the critical $O(n)$ model in two dimensions

I will propose a conjecture for the spectrum of the $O(n)$ model as a combination of irreducible representations of the group $O(n)$, and indecomposable representations of the conformal algebra.

### David Ridout: Inverse quantum hamiltonian reduction

Quantum hamiltonian reduction is a fundamental construction in conformal field theory. Procedures that roughly amount to its inversion have been discussed before, but Adamović's recent work indicates that inverse quantum hamiltonian reduction seems to be a very powerful tool to investigate certain logarithmic conformal field theories. I will give an overview of what this is, precisely, and how it works and what we hope it can do.

### Ananda Roy: Integrable quantum field theories and quantum electronic circuits

In this talk, I will describe realizations of several strongly interacting integrable quantum field theories (QFTs) using mesoscopic quantum electronic circuit lattices. The tunable, robust and dispersive Josephson nonlinearity gives rise to the nonlinear interactions in these QFTs. I will concentrate on the free, compactified boson conformal QFT and its integrable perturbation: the quantum sine-Gordon model in two space-time dimensions. In particular, I will show that superconducting quantum circuits provide a robust platform for probing permeable interfaces of free boson conformal QFT and scattering of topological excitations of the quantum sine-Gordon model. Furthermore, I will show that the quantum-circuit-based lattice models of the aforementioned QFTs are amenable to numerical analysis using the density matrix renormalization group technique, which opens a new window into the rich entanglement properties of strongly interacting QFTs.

### Volker Schomerus: Supergroup Chern-Simons theory

3D Chern-Simons theory plays a very fundamental role for physics and mathematics, with a very wide range of applications. In this talk I will describe how to extend the combinatorial quantization procedure for Hamiltonian Chern-Simons theory to gauge supergroups. The general construction of observables and states is illustrated at the example of $GL(1|1)$. In this context I will be able to make contact with previous work on the associated WZW model and 3-manifold invariants.
The talk is based on work in progress with N. Aghaei, A. Gainutdinov and M. Pawelkiewicz.

### Stanislav Smirnov: Of crystals and corals

There are many real-world processes exhibiting fractal growing shapes - from mineral deposition and coral growth to lightning strikes, and in many of them growth is related to diffusion properties. We will discuss two seminal models: Diffusion Limited Aggregation was introduced by Witten and Sanders in 1981 and was generalized to Dielectric Breakdown Model by Niemayer et al shortly afterwards. Numerically they approximate very well a wide range of physical phenomena. However, despite a very simple definition (DLA cluster grows by attaching particles undergoing Brownian motion when they hit the aggregate), very little is understood today, and even less is known rigorously - essentially, only the famous Harry Kesten upper bound on the DLA growth. We will try to show the flavor of these models and present some new results.

### Romain Vasseur: Measurement-induced criticality and $c=0$ LCFTs

An open quantum system is continuously "monitored" by its environment, so its dynamics consists of two competing processes: unitary evolution, which generates entanglement and generically leads to chaotic dynamics, and non-unitary operations resulting from measurements and noisy couplings to the environment, that tend to irreversibly destroy quantum information by revealing it. A minimal model that captures these competing processes consists of a quantum circuit made up of random unitary gates interlaced with local projective measurements. Remarkably, this minimal model undergoes a dynamical phase transition as the rate of measurements is increased. In this talk, I will introduce an exact replica statistical mechanics approach to this problem, and show that the transition is described by a CFT with central charge $c=0$. I will also present numerical results on the spectrum of this theory using a transfer matrix approach, that indicate multifractal scaling of correlation functions at the critical point, reflected in a continuous spectrum of scaling dimensions.

### Eric Vernier: Hard rod deformed spin chains - the simplest interacting integrable models?

Several models have emerged in the recent literature which could be interpreted as "hard rod deformation" of non-interacting models, obtained from the latter by assigning a finite width to particles. While genuinely interacting, such models are more easily amenable to exact calculations than the more traditional ones (for instance the XXZ spin chain), an important example being the computation of the non-equilibrium dynamics of certain local observables following a quantum quench. Therefore, they are sometimes advertised as "the simplest interacting integrable models". In this talk I will present the so-called "folded XXZ model" which emerges naturally in the $\Delta\to\infty$ limit of the XXZ chain, and which can be interpreted as a hard rod deformation of the XX spin chain, in addition to possessing several interesting features including Hilbert space fragmentation and fracton like behaviour. I will then investigate the possibility of constructing systematically hard-rod deformations from a given integrable spin chain. The main focus will be on the recently uncovered "hard rod deformed XXZ" chain (see also the talk by Balázs Pozsgay), for which I will try to explain some of its integrability properties.
Based on Pozsgay, Gombor, Hutsalyuk, Jiang, Pristyák, EV arXiv:2105.02252, as well as on some ongoing work.

### Zhenghan Wang: In and around Koo-Saleur formulas

Koo-Saleur formulas provide lattice version of Virasoro generators in quantum spin chains that are conjectured to converge to the Virasoro generators in the scaling limit. Motivated by the quantum Church-Turing thesis, we reexamine and modify the Koo-Saleur formulas for anyonic chains with an eye towards an efficient simulation of unitary rational two dimensional conformal field theories by quantum computers.

### Birgit Wehefritz-Kaufmann: Theory of materials formed as complements of triply periodic CMC surfaces

We discuss the classical and non-commutative geometry of wire systems which are the complement of triply periodic surfaces. Our predominant example is the gyroid surface and its complementary wire network which can be fabricated on a nanoscale, but we also worked on wire networks stemming from other CMC surfaces. For all these geometries, we study the Harper Hamiltonian and its band structure in the commutative and non-commutative cases. In the latter, an ambient magnetic field destroys the commutativity of the relevant algebras. We have successfully applied methods from singularity theory, representation theory and topological invariants to these materials. In this setting the gyroid geometry can be seen as the 3d generalization of graphene.

### Takato Yoshimura: Macroscopic fluctuation theory for integrable systems

Macroscopic fluctuation theory (MFT) has been developed as a universal framework to study large deviation properties of many-body systems. I will explain how the idea of MFT can also be applied to compute the large deviation of currents in integrable systems, reproducing some of the results that were obtained before using a different method.

### Paolo Zanardi: Quantum information scrambling over bipartitions

In recent years, the out-of-time-order correlator (OTOC) has emerged as a diagnostic tool for information scrambling in quantum many-body systems. Here, we present exact analytical results for the OTOC, and their long-times averages, for a typical pair of random local operators supported over two regions of a bipartition. We show that this "bipartite OTOC" is equal to the operator entanglement of the evolution and provide further operational significance to it by showing its intimate connections with average entropy production and scrambling of information at the level of quantum channels.

### Jean-Bernard Zuber: A portrait of the artist as a young man

A few reminiscences and a review of some papers of Hubert in the period 1985-2000.

## Online participation

Online participation is open to anybody who wishes to attend.