Vectorial free boundary problems.

*Bozhidar Velichkov*, Università di Pisa.

## Abstract

The vectorial Bernoulli problem is a variational free boundary problem involving the Dirichlet energy of a vector-valued function and the measure of its support. It is the vectorial counterpart of the classical one-phase Bernoulli problem, which was first studied by Alt and Caffarelli in 1981.

In this talk, we will discuss some results on the regularity of the vectorial free boundaries obtained in the last years by Caffarelli-Shahgholian-Yeressian, Kriventsov-Lin, Mazzoleni-Terracini-V., and Spolaor-V.. Finally, we will present some new results on the rectifiability of the singular set obtained in collaboration with Guido De Philippis, Max Engelstein and Luca Spolaor.

On naked singularities in Einstein equations.

*Igor Rodnianski*, Princeton University.

## Abstract

I will describe recent and ongoing work with Y. Shlapentokh-Rothman on a construction of solutions to the Einstein vacuum equations corresponding to a naked singularity forming from a regular past. The talk will focus on a new geometric phenomenon, corresponding to twisting of null geodesics, new type of self-similarity, dynamical approach to the problem, and on the comparisons with the naked singularity solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar field model.

Regularity of the optimal sets for the second Dirichlet eigenvalue.

*Dario Mazzoleni*, Università di Pavia.

## Abstract

First of all, we recall the basic notions and results concerning shape optimization problems for the eigenvalues of the Dirichlet Laplacian.

Then we focus on the study of the regularity of the optimal shapes and on the link with the regularity of related free boundary problems.

The main topic of the talk is the regularity of the optimal sets for a "degenerate'" functional, namely the second Dirichlet eigenvalue in a box. Given $D\subset \mathbb{R}^d$ an open and bounded set of class $C^{1,1}$, we consider the following shape optimization problem, for $\Lambda>0$,\begin{equation}\label{eq:main}\min{\Big\{\lambda_2(A)+\Lambda |A| : A\subset D,\text{ open}\Big\}},\end{equation}where $\lambda_2(A)$ denotes the second eigenvalue of the Dirichlet Laplacian on $A$.

In this talk we show that any optimal set $\Omega$ for \eqref{eq:main} is equivalent to the union of two disjoint open sets, $\Omega^\pm$, which are $C^{1,\alpha}$ regular up to a (possibly empty) closed singular set of Hausdorff dimension at most $d-5$, which is contained in the one-phase free boundaries.

In particular, we are able to prove that the set of two-phase points, that is, $\partial \Omega^+\cap \partial \Omega^-\cap D$, is contained in the regular set.

This is a joint work with Baptiste Trey and Bozhidar Velichkov.

Bulk-boundary correspondences with factorization algebras.

*Owen Gwilliam*, University of Massachusetts, Amherst.

## Abstract

Factorization algebras provide a flexible language for describing the observables of a perturbative QFT, as shown in joint work with Kevin Costello. Those constructions extend to a manifold with boundary for a special class of theories. I will discuss work with Eugene Rabinovich and Brian Williams that includes, as an example, a perturbative version of the correspondence between chiral ${\rm U}(1)$ currents on a Riemann surface and abelian Chern-Simons theory on a bulk 3-manifold, but also includes a systematic higher dimensional version for higher abelian CS theory on an oriented smooth manifold of dimension $4n+3$ with boundary a complex manifold of complex dimension $2n+1$. Given time, I will discuss how this framework leads to a concrete construction of the center of higher enveloping algebras of Lie algebras, in work with Greg Ginot and Brian Williams.

Exploring 4D topological physics in the laboratory.

*Hannah Price*, University of Birmingham.

## Abstract

Spatial dimensionality plays a key role in our understanding of topological physics, with different topological invariants needed to characterise systems with different numbers of spatial dimensions. In a 2D quantum Hall system, for example, a robust quantisation of the Hall response is related to the first Chern number: a 2D topological invariant of the electronic energy bands. Generalising to more spatial dimensions, it was shown that a new type of quantum Hall effect could emerge in four dimensions, but where the quantised response was related to a four-dimensional topological invariant, namely the second Chern number. While systems with four spatial dimensions may seem abstract, recent developments in ultracold atoms and photonics have opened the door to exploring such higher-dimensional topological physics experimentally. In this talk, I will introduce the theory of 4D topological phases of matter, before discussing recent experiments in cold atoms, photonics and electrical circuits that have begun to probe aspects of this physics in the laboratory.

To be announced.

*Leonor Godinho*, Instituto Superior Técnico and CAMGSD.

Modelling dependence between observed and simulated wind speed data using copulas.

*Patrícia de Zea Bermudez*, Departamento de Estatística e Investigação Operacional e CEAUL, Universidade de Lisboa.

## Abstract

In real applications, associations between variables are often non-linear and data commonly exhibit strong asymmetries and/or heavy tails. Copula models enable to create the joint distribution of vectors of random variables independently of their marginal distributions. This work aims to analyse and characterise the dependence between daily maximum wind speeds, X, observed in Portugal and simulated daily maximum wind speeds, Y, produced by a numerical-physical model. One of the major benefits of using simulated data is their availability at high spatial and temporal resolutions contrarily to observed data, which are commonly scarce. The main problem is that the simulated and the observed winds, in some stations, do not match well and tend to differ mostly in the right tail. Consequently, it is very important to understand the dependence between X and Y. The ultimate purpose is to calibrate the simulated data and bring it in line with observed data. That offers practitioners richer data sources. The results showed that, in the overall, Gamma and Lognormal are the most suitable marginal distributions for our data and Gumbel copula is the most adequate to model the dependence structure. Finally, the classical modelling is compared with a Bayesian approach.

Modelling dependence between observed and simulated wind speed data using copulas.

*Patrícia de Zea Bermudez*, Departamento de Estatística e Investigação Operacional e CEAUL, Universidade de Lisboa.

## Abstract

In real applications, associations between variables are often non-linear and data commonly exhibit strong asymmetries and/or heavy tails. Copula models enable to create the joint distribution of vectors of random variables independently of their marginal distributions. This work aims to analyse and characterise the dependence between daily maximum wind speeds, X, observed in Portugal and simulated daily maximum wind speeds, Y, produced by a numerical-physical model. One of the major benefits of using simulated data is their availability at high spatial and temporal resolutions contrarily to observed data, which are commonly scarce. The main problem is that the simulated and the observed winds, in some stations, do not match well and tend to differ mostly in the right tail. Consequently, it is very important to understand the dependence between X and Y. The ultimate purpose is to calibrate the simulated data and bring it in line with observed data. That offers practitioners richer data sources. The results showed that, in the overall, Gamma and Lognormal are the most suitable marginal distributions for our data and Gumbel copula is the most adequate to model the dependence structure. Finally, the classical modelling is compared with a Bayesian approach.

Machine Learning of Robot Skills.

*Jan Peters*, Technische Universitaet Darmstadt.

Towers of bubbles for Yamabe-type equations in dimensions larger than 7.

*Bruno Premoselli*, Université Libre de Bruxelles.

## Abstract

In this talk we consider perturbations of Yamabe-type equations on closed Riemannian manifolds. In dimensions larger than 7 and on locally conformally flat manifolds we construct blowing-up solutions that behave like towers of bubbles concentrating at a critical point of the mass function. Our result does not assume any symmetry on the underlying manifold.

We perform our construction by combining finite-dimensional reduction methods with a linear blow-up analysis in order to sharply control the remainder of the construction in strong spaces. Our approach works both in the positive and sign-changing case. As an application we prove the existence, on a generic bounded open set of $\mathbb{R}^n$, of blowing-up solutions of the Brézis-Nirenberg equation that behave like towers of bubbles of alternating signs.

Skein Lasagna modules of 2-handlebodies.

*Ikshu Neithalath*, UCLA, California.

## Abstract

Morrison, Walker and Wedrich recently defined a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. We will discuss recent joint work with Ciprian Manolescu on computing the "skein lasagna module," a basic part of MWW's invariant, for a certain class of 4-manifolds.

Double Field Theory and Geometric Quantisation.

*David Berman*, Queen Mary University of London.

## Abstract

We examine various properties of double field theory and the doubled string sigma model in the context of geometric quantisation. In particular we look at T-duality as the symplectic transformation related to an alternative choice of polarisation in the construction of the quantum bundle for the string. Following this perspective we adopt a variety of techniques from geometric quantisation to study the doubled space. One application is the construction of the double coherent state that provides the shortest distance in any duality frame and a stringy deformed Fourier transform.

To be announced.

*Anatoli Polkovnikov*, Boston University.

To be announced.

*Giulia Saccà*, Columbia University.

Information-theoretic bounds on quantum advantage in machine learning.

*Hsin Yuan Huang, (Robert)*, Caltech.

## Abstract

We compare the complexity of training classical and quantum machine learning (ML) models for predicting outcomes of physical experiments. The experiments depend on an input parameter x and involve the execution of a (possibly unknown) quantum process $E$. Our figure of merit is the number of runs of $E$ needed during training, disregarding other measures of complexity. A classical ML performs a measurement and records the classical outcome after each run of $E$, while a quantum ML can access $E$ coherently to acquire quantum data; the classical or quantum data is then used to predict outcomes of future experiments. We prove that, for any input distribution $D(x)$, a classical ML can provide accurate predictions on average by accessing $E$ a number of times comparable to the optimal quantum ML. In contrast, for achieving accurate prediction on all inputs, we show that exponential quantum advantage exists in certain tasks. For example, to predict expectation values of all Pauli observables in an $n-$qubit system, we present a quantum ML using only $O(n)$ data and prove that a classical ML requires $2^{\Omega(n)}$ data.

To be announced.

*Gabriele Benomio*, Princeton University.

To be announced.

*Shrish Parmeshwar*, University of Bath.

Quantum many-body dynamics in two dimensions with artificial neural networks.

*Markus Heyl*, Max-Planck Institute for the Physics of Complex Systems, Dresden.

## Abstract

In the last two decades the field of nonequilibrium quantum many-body physics has seen a rapid development driven, in particular, by the remarkable progress in quantum simulators, which today provide access to dynamics in quantum matter with an unprecedented control. However, the efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter still remains a key challenge for current computational methods especially beyond one spatial dimension. In this talk I will present a versatile and efficient machine learning inspired approach. I will first introduce the general idea of encoding quantum many-body wave functions into artificial neural networks. I will then identify and resolve key challenges for the simulation of real-time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, I will consider the dynamics of the paradigmatic two-dimensional transverse field Ising model, where we observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached time scales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.

To be announced.

*Lorenzo Foscolo*, University College London.

To be announced.

*Yukihiko Nakata*, Aoyama Gakuin University, Tokyo.

Entanglement and Complexity in Topological Quantum Field Theories.

*Dmitry Melnikov*, ITEP Moscow.

## Abstract

One of the attractive ideas of building a quantum computer is based on the topological properties of matter. In such a realization, the Topological Quantum Field Theories (TQFT) become the main language to describe the functioning of the quantum computer. In my talk I will discuss some basic elements of the topological quantum computing. I will start from a description of TQFTs as instances of quantum mechanics in terms of category theory. Then I will review the notion of quantum entanglement in this context. As a further preparation to quantum computations I will discuss the question of complexity of quantum algorithms and quantum states. I will introduce a complexity measure for a simple class of the "torus knot states" and review some alternative recent measures and approaches from the literature.

To be announced.

*Alexander Atland*, University of Cologne.

Machine learning for Fluid Mechanics.

*Steve Brunton*, University of Washington.

## Abstract

Many tasks in fluid mechanics, such as design optimization and control, are challenging because fluids are nonlinear and exhibit a large range of scales in both space and time. This range of scales necessitates exceedingly high-dimensional measurements and computational discretization to resolve all relevant features, resulting in vast data sets and time-intensive computations. Indeed, fluid dynamics is one of the original big data fields, and many high-performance computing architectures, experimental measurement techniques, and advanced data processing and visualization algorithms were driven by decades of research in fluid mechanics. Machine learning constitutes a growing set of powerful techniques to extract patterns and build models from this data, complementing the existing theoretical, numerical, and experimental efforts in fluid mechanics. In this talk, we will explore current goals and opportunities for machine learning in fluid mechanics, and we will highlight a number of recent technical advances. Because fluid dynamics is central to transportation, health, and defense systems, we will emphasize the importance of machine learning solutions that are interpretable, explainable, generalizable, and that respect known physics.

Some problems and some solutions in shape and topology optimization of structures built by additive manufacturing.

*Grégoire Allaire*, CMAP, École Polytechnique.

## Abstract

Additive manufacturing (or 3-d printing) is a new exciting way of building structures without any restriction on their topologies. However, it comes with its own difficulties or new issues. Therefore, it is a source of many interesting new problems for optimization. I shall discuss two of them and propose solutions to these problems, but there is still a lot of room for improvement!

First, additive manufacturing technologies are able to build finely graded microstructures (called lattice materials). Their optimization is therefore an important issue but also an opportunity for the resurrection of the homogenization method ! Indeed, homogenization is the right technique to deal with microstructured materials where anisotropy plays a key role, a feature which is absent from more popular methods, like SIMP. I will describe recent work on the topology optimization of these lattice materials, based on a combination of homogenization theory and geometrical methods for the overall deformation of the lattice grid.

Second, additive manufacturing, especially in its powder bed fusion technique, is a very slow process because a laser beam must travel along a trajectory, which covers the entire structure, to melt the powder. Therefore, the optimization of the laser path is an important issue. Not only do we propose an optimization strategy for the laser path, but we couple it with the usual shape and topology optimization of the structure. Numerical results show that these two optimizations are tightly coupled.

This is a joint work with many colleagues, including two former PhD students, P. Geoffroy-Donders and M. Boissier.

To be announced.

*Marco Mazzuchelli*, École normale supérieure de Lyon.

To be announced.

*Juan Juan Cai*, Department of Econometrics and Data Science, School of Business and Economics, Vrije Universiteit Amsterdam.

To be announced.

*Juan Juan Cai*, Department of Econometrics and Data Science, School of Business and Economics, Vrije Universiteit Amsterdam.

To be announced.

*Valeria Chiadò Piat*, Politecnico di Torino.

Intrinsic non-perturbative topological strings.

*Murad Alim*, University of Hamburg.

## Abstract

We study difference equations which are obtained from the asymptotic expansion of topological string theory on the deformed and the resolved conifold geometries as well as for topological string theory on arbitrary families of Calabi-Yau manifolds near generic singularities at finite distance in the moduli space. Analytic solutions in the topological string coupling to these equations are found. The solutions are given by known special functions and can be used to extract the strong coupling expansion as well as the non-perturbative content. The strong coupling expansions show the characteristics of D-brane and NS5-brane contributions, this is illustrated for the quintic Calabi-Yau threefold. For the resolved conifold, an expression involving both the Gopakumar-Vafa resummation as well as the refined topological string in the Nekrasov-Shatashvili limit is obtained and compared to expected results in the literature. Furthermore, a precise relation between the non-perturbative partition function of topological strings and the generating function of non-commutative Donaldson-Thomas invariants is given. Moreover, the expansion of the topological string on the resolved conifold near its singular small volume locus is studied. Exact expressions for the leading singular term as well as the regular terms in this expansion are provided and proved. The constant term of this expansion turns out to be the known Gromov-Witten constant map contribution.

To be announced.

*David J. Luitz*, Max Planck Institute for the Physics of Complex Systems, Dresden.

Global Slodowy slices for moduli spaces of λ-connections.

*Brian Collier*, University of California Riverside.

To be announced.

*Nicola Fusco*, Università di Napoli "Federico II"

Nonlocality features of the area functional and the Plateau problem.

*Riccardo Scala*, Università degli Studi di Siena.

## Abstract

We briefly discuss the definition of relaxation of the area functional. The relaxed area functional, denoted by $A$, extends the classical area functional, which, for any "regular" map $v:U\subset \mathbb{R}^n\rightarrow \mathbb{R}^N$ evaluates the $n$-dimensional area of its graph over $U$. The problem of determining the domain and the expression of $A$ is open in codimension greater than 1. Specifically, this relaxed functional turns out to be nonlocal and cannot be expressed by an integral formula. We discuss how it is related to classical and nonclassical versions of the Plateau problem. As a main example, we try to understand what is the relaxed graph of the function $x/|x|$, a question that surprisingly remained open for decades.

To be announced.

*Juan Luis Vázquez*, Universidad Autónoma de Madrid.

Spacetime as a quantum circuit.

*Michal P. Heller*, Albert Einstein Institute.

## Abstract

We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the "complexity equals volume" conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $T\bar{T}$, we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action. Based on arXiv:2101.01185.

To be announced.

*Xiao-Gang Wen*, Massachusetts Institute of Technology.

To be announced.

*Marta Lopes*, Universidade Nova, Centro de Matemática e Aplicações.

Geometry, Light Response and Quantum Transport in Topological States of Matter.

*Karyn Le Hur*, Centre de Physique Theorique, École Polytechnique, CNRS.

An invitation to the principal series.

*Tarek Anous*, University of Amsterdam.

## Abstract

Scalar unitary representations of the isometry group of d-dimensional de Sitter space $SO(1,d)$ are labeled by their conformal weights $\Delta$. A salient feature of de Sitter space is that scalar fields with sufficiently large mass compared to the de Sitter scale $1/l$ have complex conformal weights, and physical modes of these fields fall into the unitary continuous principal series representation of $SO(1,d)$. Our goal is to study these representations in $d=2$, where the relevant group is $SL(2,R)$. We show that the generators of the isometry group of $dS_2$ acting on a massive scalar field reproduce the quantum mechanical model introduced by de Alfaro, Fubini and Furlan (DFF) in the early/late time limit. Motivated by the ambient $dS_2$ construction, we review in detail how the DFF model must be altered in order to accommodate the principal series representation. We point out a difficulty in writing down a classical Lagrangian for this model, whereas the canonical Hamiltonian formulation avoids any problem. We speculate on the meaning of the various de Sitter invariant vacua from the point of view of this toy model and discuss some potential generalizations.

To be announced.

*Clara Cordeiro*, Departamento de Matemática, Universidade do Algarve.

To be announced.

*Clara Cordeiro*, Departamento de Matemática, Universidade do Algarve.

To be announced.

*David Tong*, University of Cambridge.

To be announced.

*Igor Lesanovsky*, Universität Tübingen.

To be announced.

*Lígia Henriques‐Rodrigues*, Departamento de Matemática, Universidade de Évora.

To be announced.

*Lígia Henriques‐Rodrigues*, Departamento de Matemática, Universidade de Évora.

To be announced.

*Rubem Mondaini*, Beijing Computational Science Research Center.

Instituto Superior Técnico
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