Luigi Ambrosio (Pisa)

"Calculus, heat flow and curvature-dimension bounds in metric measure spaces"

I will present some recent developments at the interface between Analysis, Geometry, Optimal Transport. In particular I will focus on calculus in nonsmooth structures, heat flow and curvature/dimension bounds in metric measure spaces.

Viviane Baladi (Paris)

"Sinai billiards from a mathematician's viewpoint"

Sinai billiards (or the periodic Lorentz gas) are natural dynamical systems which have been challenging mathematicians for half a century. In the past decade, a new mathematical tool to study them has emerged : Ruelle transfer operators acting on scales of anisotropic Banach spaces.

I will survey rigorous results obtained on dispersing billiards (for both discrete and continuous time) using this approach, leading to recent work on the measure of maximal entropy.

Horacio Casini (Bariloche)

"Facets of the QFT entropy cone"

The question that motivates this investigation is whether vacuum entanglement entropy (EE) can be thought as a "statistical correlator" providing a universal description for QFT, uniquely characterising the different allowed models. In this sense it would be desirable to count with a set of "axioms" that would be sufficient to characterize EE in QFT. We collect all known properties of relativistic EE and show that they lead to unitarity bounds for specific types of fields, where the saturation of the bound gives a free field. However, the same ideas do not allow to obtain unitarity bounds for currents, for example. We also show that the EMI model for EE satisfies all the known constraints but does not come from any QFT for d>2. Both these observations point to the incompleteness of our current understanding of the QFT entropy cone.

Alberto De Sole (Rome)

"Vertex algebra and Poisson vertex algebra cohomology"

We review cohomology theories corresponding to the chiral and classical operads. The first one is the cohomology theory of vertex algebras, while the second one is the classical cohomology of Poisson vertex algebras (PVA).

Using a spectral sequence relating them, we are able to compute the cohomology of vertex algebras in many interesting cases. We then describe a unified approach to integrability through vanishing of the first cohomology, which is applicable to both classical and quantum systems of Hamiltonian PDEs.

Klaus Fredenhagen (Hamburg)

"Construction of Haag-Kastler nets: Time slice axiom, Noether theorem and anomalies"

A general method for the construction of C*-algebraic Haag-Kastler nets is developed. It depends on a classical Lagrangian and a cocycle about classical symmetries which describes possible anomalies. It is shown that these Haag-Kastler nets satisfy the time-slice axiom, fulfill a unitary version of Noether’s theorem and exhibit a renormalization group flow induced by anomalies. The talk is based on joint work with Romeo Brunetti, Michael Dütsch and Kasia Rejzner and on previous joint work with Detlev Buchholz.

Alba Grassi (CERN)

"Quantization conditions in gauge theory and topological strings"

Supersymmetric gauge theories and topological string have provided us with new analytic tools to study the spectral theory of quantum mechanical operators leading to new exact solutions. In this talk I will review some of these developments and show some applications.

Massimiliano Gubinelli (Bonn)

"Stochastic analysis of Euclidean quantum fields"

I will review recent progresses in the construction and study of Euclidean quantum fields via techniques from stochastic analysis, including various kinds of stochastic quantisation and a stochastic optimal control formulation recently introduced. All these methods hint to an underlying structure of superrenormalizable models which we tentatively identify as a kind of stochastic analysis based on a Gaussian source of randomness.

Daniel Harlow (MIT)

"Continuity of symmetry operators in quantum field theory"

In this talk I will discuss the continuity properties of internal symmetries in quantum field theory, both in their action on the Hilbert space and also on the algebra of local operators. I will motivate a requirement that the latter should be strongly continuous on any uniformly-bounded subset of the algebra of operators associated to any finite region, and I will show that this implies that internal symmetries which act non-unitarily on the fields of a quantum field theory, such as the shift symmetry of a scalar field, must be spontaneously broken.

Stefan Hollands (Leipzig)

"Black Hole Interiors"

In this talk I will discuss how quantum effects affect the nature of the singularity inside physical black holes, and in particular how such effects are relevant in the context of the famous so-called "strong cosmic censorship hypothesis".

The discussion takes into account recent developments in mathematical relativity which have greatly increased our understanding of wave propagation on black hole spacetimes. These are combined with the methods of quantum field theory on curved spacetimes.

Arthur Jaffe (Harvard)

"Quantum Fourier Analysis"

Some inequalities for quantum entropy are inspired by inequalities in classical Fourier analysis. We relate this to planar algebras and to picture language describing problems in quantum information.

Giovanni Landi (Trieste)

"Hopf algebroids and noncommutative gauge transformations"

A (commutative) Hopf algebroid is somehow the dual of a groupoid (like Hopf algebras vs groups). In general the ground field gets replaced by a (noncommutative) algebra thus leading to a Hopf algebra over a (noncommutative) base algebra. In particular, we study natural bialgebroids of noncommutative principal bundles as a quantization of the gauge groupoid of a classical principal bundle; the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid. Examples include: Galois objects of Taft algebras and monopole and instanton bundles over noncommutative spheres. For this class of examples there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

Gregory W. Moore (Rutgers)

"2d Categorical Wall-Crossing With Twisted Masses, And An Application To Knot Invariants"

We review how supersymmetric quantum mechanics naturally leads to several standard constructions in homological algebra. We apply these ideas to 2d Landau-Ginzburg models with (2,2) supersymmetry to discuss wall-crossing. Some aspects of the web formalism are reviewed and applied to the categorification of the Cecotti-Vafa wall-crossing formula for BPS invariants. We then sketch the generalization to include twisted masses. In the final part of the talk we sketch how some of these ideas give a natural framework for understanding a recent conjecture of Garoufalidis, Gu, and Marino and lead to potentially new knot invariants. The talk is based on work done with Ahsan Khan and the final part is the result of discussions with Ahsan Khan, Davide Gaiotto, and Fei Yan.

Sergey Neshveyev (Oslo)

"A Kohno-Drinfeld type theorem for symmetric spaces"

The famous Kohno-Drinfeld theorem establishes an equivalence of representations of braid groups defined by monodromy of Knizhnik-Zamolodchikov (KZ) equations and by R-matrices of quantized enveloping algebras. In my talk I'll discuss an analogue of this theorem for type B braid groups, where the Yang-Baxter equation is complemented with the reflection equation, KZ-equations with 2-cyclotomic KZ-equations and R-matrices with K-matrices of quantized coideals. (Joint work with Kenny De Commer, Lars Tuset and Makoto Yamashita.)

Yoshiko Ogata (Tokyo)

"Classification of gapped ground state phases in quantum spin systems"

Recently, classification problems of gapped ground state phases attract a lot of attention in quantum statistical mechanics. We explain the operator algebraic approach to these problems.

Tomaž Prosen (Lubiana)

"Random Matrix Spectral Fluctuations in Quantum Lattice Systems"

I will discuss the problem of unreasonable effectiveness of random matrix theory for description of spectral fluctuations in extended quantum lattice systems. A class of interacting spin systems has been recently identified where the spectral form factor is proven to match with gaussian or circular ensembles of random matrix theory. The key ideas of novel methodology needed in the proofs will be discussed which are very different from the standard periodic-orbit based methods in quantum chaos of few body semiclassical systems.

Dan-Virgil Voiculescu (Berkeley)

"A Hydrodynamic Exercise in Free Probability: Free Euler Equations"

The Euler equations for a flow which preserves the Gaussian measure on Euclidean space can be translated in terms of Gaussian random variables, which raises the question about an analogue in free probability. We derive these "free" Euler equations by applying the approach of Arnold for Euler equations to a Lie algebra of infinitesimal automorphisms of the von Neumann algebra of a free group. We then extend the equations to non - commutative vector fields satisfying certain weaker non - commutative smoothness conditions. We also introduce a cyclic vorticity and show that it satisfies appropriate vorticity equations and that it gives rise to a family of conserved quantities.

Riccardo Zecchina (Milan)

"The geometric landscape of deep learning models"

Among the most surprising aspects of deep learning models are their highly overparametrized and non-convex nature. Both of these aspects are a common trait of all the deep learning models and have led to unexpected results for classical learning theory and non-convex optimization. Current deep neural networks (DNN) are composed of millions (or even billions) of connection weights and the learning process seeks to minimize a non-convex loss function that measures the number of classification errors made by the DNN. The empirical evidence shows that these highly expressive neural network models can fit the training data via simple variants of algorithms originally designed for convex optimization. Moreover, even if the learning processes are run with little control over their statistical complexity (e.g. regularisation, number of parameters, …), these models achieve unparalleled levels of prediction accuracy, contrary to what would be expected from the uniform convergence framework of classical statistical inference.

In this talk, we will discuss the geometrical structure of the space of solutions (zero error configurations) in overparametrized non-convex neural networks when trained to classify patterns taken from some natural distribution. Building on statistical physics techniques for the study of disordered systems, we analyze the geometric structure of the different minima and critical points of the error loss function as the number of parameters increases and we relate this to learning performance. Of particular interest is the role of rare flat minima which are both accessible to algorithms and have good generalisation properties, on the contrary to dominating minima which are almost impossible to sample. We will show that the appearance of rare flat minima defines a phase boundary at which algorithms start to find solutions efficiently.