# One world IAMP mathematical physics seminar

This online seminar takes place on Tuesdays, starting at UTC 14.

Current organisers are Jan Dereziński (Warsaw), Marcello Porta (SISSA Trieste), Kasia Rejzner (York), and Daniel Ueltschi (Warwick).

Scientific committee: Nalini Anantharaman (Strassbourg), Mihalis Dafermos (Cambridge), Stephan De Bièvre (Lille), Krzysztof Gawedzki (ENS Lyon), Bernard Helffer (Nantes), Vojkan Jaksic (McGill), Flora Koukiou (Cergy), Antti Kupiainen (Helsinki), Mathieu Lewin (Paris Dauphine), Bruno Nachtergaele (UC Davis), Claude-Alain Pillet (Toulon), Robert Seiringer (IST Austria), Jan Philip Solovej (Copenhagen), Hal Tasaki (Gakushuin).

 May 11, 2021 David Brydges (University of British Columbia) Lace expansions and spin models I will review lace expansions starting with their relation to the other expansions in statistical mechanics and then discussing their application to spin models. Akira Sakai opened this avenue in 2007 with his proof that in sufficiently high dimension $d$ the two point function for the critical Ising model is asymptotic to $c_d |x-y|^{-(d-2)}$. In arxiv.org/abs/1905.09605 Tyler Helmuth, Mark Holmes and I have introduced a different way to derive lace expansions based on the Symanzik random walk representation. Video link: youtu.be/KAuRLtSuX9c May 4, 2021 Chris Fewster (University of York) Measurement of quantum fields in curved spacetimes A standard account of the measurement chain in quantum mechanics involves a probe (itself a quantum system) coupled temporarily to the system of interest. Once the coupling is removed, the probe is measured and the results are interpreted as the measurement of a system observable. Measurement schemes of this type have been studied extensively in Quantum Measurement Theory, but they are rarely discussed in the context of quantum fields and still less on curved spacetimes. In this talk I will describe how measurement schemes may be formulated for quantum fields on curved spacetime within the general setting of algebraic QFT. In particular, I will show how measurements of probe observables correspond to measurements of "induced" local system observables and describe the role of state update rules. The framework is local and fully covariant, allowing the consistent description of measurements made in spacelike separated regions. Furthermore, specific models can be given in which the framework may be exemplified by concrete calculations. I will also explain how this framework can shed light on an old problem due to Sorkin concerning "impossible measurements" in which measurement apparently conflicts with causality. Finally, I will discuss new results indicating that the system observables that can be induced by measurement schemes form a large subspace of the algebra of all local observables. The talk is based on work with Rainer Verch [Leipzig], (Comm. Math. Phys. 378, 851–889(2020), arXiv:1810.06512; see also arXiv:1904.06944 for a summary), with Henning Bostelmann and Maximilian Ruep [York] ( Phys. Rev. D 103 (2021) 025017; arXiv:2003.04660) and work in progress with Ian Jubb [Dublin] and Maximilian Ruep. Video link: youtu.be/sdnYBumtkeM April 27, 2021 Sabine Jansen (LMU Munich) Virial expansion for mixtures: some examples & recent results Mayer's expansion is an expansion of the pressure in powers of the activity (fugacity) in equilibrium statistical mechanics; the virial expansion is an expansion in powers of the density. Rigorous convergence results have been available for decades, nevertheless for mixtures the theory of density expansions is less advanced than the theory of activity expansions. I will review the setting, and discuss some recent results for the virial expansion for mixtures (multi-species systems) and illustrate them with three concrete models: mixtures of hard spheres of different sizes, non-overlapping rods of different lengths, and a hierarchical mixture of cubes in Z^d. The last two examples are helpful toy models for which concrete formulas are available and phase transitions can be studied. Based in part on joint works with Tobias Kuna, Stephen Tate, Dimitrios Tsagkarogiannis and Daniel Ueltschi. Video link: youtu.be/YEKH8dsrcpY April 20, 2021 Eric Carlen (Rutgers University) Kac Master Equations, Classical and Quantum This lecture will review recent progress and open problems concerning Kac Master Equations in both the classical and quantum setting. It will be based largely on recent papers written in collaboration with Maria Carvalho, Michael Loss and Bernt Wennberg. Video link: youtu.be/Q6WcqC0aOMM April 13, 2021 Wojciech De Roeck (KU Leuven) Impurities and boundaries for a class of gapped ground states In the last two decades, a lot of rigorous results have been proven for quantum ground states protected by a spectral gap in their Hamiltonian. Think for example of decay of correlations, the topological classification of states, and area laws of entanglement. For all of those results, the spectral gap is a necessary input, both on the conceptual and on the technical level. Imagine now that we consider such a gapped Hamiltonian that is perturbed locally by a term that is not small, such that we certainly cannot hope to preserve the global gap. Common sense dictates that, far away from the perturbation, the ground state should not be seriously affected by this perturbation. While we cannot prove this in the generality in which it should be true, I will describe some results that go in this direction. Video link: youtu.be/szMl4Kv1-ek April 6, 2021 Bernard Helffer (Paris-Sud / Nantes) Semiclassical methods and tunneling effects: old and new In 1982-1983 the so-called symmetric double well problem was rigorously analyzed in any dimension in the semi-classical context by B. Simon and Helffer-Sjöstrand. This involves semi-classical Agmon estimates, Harmonic approximation, WKB constructions and a very fine analysis of the so-called tunneling effect in order to establish the splitting between the lowest eigenvalues. The strategy followed by Helffer-Sjöstrand appears to be quite efficient in many other contexts. After recalling how it works on the initial double well problem, we will discuss in a rather impressionist way the main successes of the approach along the years. Finally we will focus on a quite recent result by Bonnaillie-H\'erau-Raymond devoted to the measure of a magnetic tunneling in Surface Superconductivity and discuss open problems. The presented results correspond to contributions by various subsets of the following set of authors : Bonnaillie, Fournais, Helffer, Hérau, Kachmar, Morame, Nier, Raymond, Simon, Sjöstrand and many others... Video link: youtu.be/szMl4Kv1-ek March 30, 2021 Jan Wehr (University of Arizona) Aggregation and deaggregation of interacting micro swimmers A swarm of light-sensitive robots is moving in a planar region, changing the direction of motion randomly. Each robot emits light; in turn, it adapts the speed of its motion to the total intensity of the light shed on it by the other robots. The adaptation takes time---the sensorial delay. Using a natural approximation of the equations of motion, the robots may be programmed to make the sensorial delay negative. In this case they are, in some sense, predicting the future. In a series of experiments by the Giovanni Volpe group and the University of Gothenburg (Sweden) it was shown that at a certain negative value of the delay, the collective behavior of the robots changes qualitatively from aggregation to deaggregation. I am going to explain this phenomenon by an asymptotic analysis of the system of stochastic differential equations, describing the motion of a single robot in an inhomogeneous landscape. The results were obtained in a joint work with Giovanni Volpe, Mite Mijalkov and Austin McDaniel. They display an interesting feature: while the original system is driven by a single noise source, the limiting one contains two independent ones. I will explain this by showing how one can generate two independent Wiener processes from a single one. Using this fact to rigorously justify the asymptotic results is a work in progress with Jeremiah Birrell. Video link: youtu.be/-7TrciUd5w4 March 23, 2021 Sylvia Serfaty (Courant Institute) Ginzburg-Landau vortices, old and new We present a review of results on vortices in the 2D Ginzburg-Landau model of superconductivity (also relevant for superfluidity and Bose-Einstein condensates), their onset at critical fields, interaction and patterns. We also report on recent joint work with Carlos Roman and Etienne Sandier in which we study the onset of vortex lines in the 3D model and derive an interaction energy for them. Video link: youtu.be/ayfWbNW16kA March 16, 2021 Gianluca Panati (University of Rome Sapienza) The Localization Dichotomy in Solid State Physics Solid state systems exhibit a subtle intertwining between the topology of the "manifold" of occupied states and the localization of electrons, if the latter is appropriately defined. This correspondence has been first noticed and proved in the case of periodic gapped systems: for non interacting electrons, localization is measured via the localization of the Composite Wannier Bases (CWBs) spanning the range of the Fermi projector $P_F$, while the topological information is encoded in the (first) Chern class of the Fermi projector. It has been proved, for dimension $d \leq 3$ that only two regimes are possible: i) either there exists an exponentially localized CWB, and correspondingly the Chern class of $P_F$ vanishes (ordinary insulator); ii) or any possible choice of a CWB is delocalized, in the sense that it yields an infinite expectation value for the square of the position operator, and the Chern class is non zero (Chern insulator). More recently, the previous dichotomy has been reformulated in such a way that it applies also to non-periodic systems, and a corresponding "Localization Dichotomy conjecture" for non-periodic systems has been stated. In my talk, I will gently introduce the subject, starting from the periodic case, and summarize the recent attempts, by several groups, to prove the Localization Dichotomy for non-periodic systems. The talk is based on work in collaboration with D. Monaco, M. Marcelli, M. Moscolari, A. Pisante, and S. Teufel, and on discussions with several colleagues. Video link: youtu.be/KbuVtavr0Qs March 9, 2021 Simone Warzel (TU Munich) An invitation to quantum spin glasses Quantum spin glass models of mean-field type are prototypes of quantum systems exhibiting phase transitions related to the spead of the eigenstates in configuration space. Originally motivated by spin glass physics, they are discussed in relation to many-body localisation phenomena, quantum adiabatic algorithms as well as in the context of models in mathematical biology. Quantum effects transform the rich phase diagram of classical spin glasses as described by Parisi theory into an even more colourful landscape of phases which range from purely quantum paramagnetic to intermediate behavior in which eigenstates occupy only a fraction of configuration space. In this talk, I will give an introduction to the multifaceted motivations and challenges behind the study of these quantum glasses and explain existing and expected results. Video link: youtu.be/2vv9ryssLR4 March 2, 2021 Margherita Disertori (University of Bonn) Supersymmetric transfer operators Transfer matrix approach is a powerful tool to study one dimensional or quasi 1d statistical mechanical models. Transfer operator kernels arising in the context of quantum diffusion and the supersymmetric approach display bosonic and fermionic components. For such kernels, the presence of fermion-boson symmetries allows to drastically simplify the problem. I will review the method and give some results for the case of random band matrices. This is joint work with Sasha Sodin and Martin Lohmann. Video link: youtu.be/lRtSNMix1Rs February 23, 2021 Tyler Helmuth (University of Durham) Efficient algorithms for low-temperature spin systems Two fundamental algorithmic tasks associated to discrete statistical mechanics models are approximate counting and approximate sampling. At high temperatures Markov chains give efficient algorithms, but at low temperatures mixing times can become impractically large, and Markov chain methods may fail to be efficient. Recently, expansion methods (cluster expansions, Pirogov--Sinai theory) have been put to use to develop provably efficient low-temperature algorithms for some discrete statistical mechanics models. I’ll introduce these algorithmic tasks, outline how expansion algorithms work, and indicate some open directions. Video link: youtu.be/btFWDlne7kU February 16, 2021 Gerald Dunne (University of Connecticut) Resurgent Asymptotics and Non-perturbative Physics There are several important conceptual and computational questions concerning path integrals, which have recently been approached from new perspectives motivated by Ecalle's theory of resurgent functions, a mathematical formalism that unifies perturbative and non-perturbative physics. I will review the basic ideas behind the connections between resurgent asymptotics and physics, with examples from quantum mechanics and matrix models. Video link: youtu.be/i1iNUMurUrM February 9, 2021 Herbert Spohn (TU Munich) Hydrodynamic equations for the classical Toda lattice For arbitrary size, the Toda lattice is an integrable system with an extensive number of local conservation laws. To then build the respective hydrodynamic equations, one has to appropriately generalize the notion of local equilibrium and to obtain average fields together with their currents in such states. In my presentation I will explain how these goals can be achieved. The unexpected relation to the repulsive log gas in one dimension is discussed. Video link: youtu.be/1n8RuSod9-g February 2, 2021 Wei Wu (NYU Shanghai) Massless phases for the Villain model in d>=3 The XY and the Villain models are mathematical idealization of real world models of liquid crystal, liquid helium, and superconductors. Their phase transition has important applications in condensed matter physics and led to the Nobel Prize in Physics in 2016. However we are still far from a complete mathematical understanding of the transition. The spin wave conjecture, originally proposed by Dyson and by Mermin and Wagner, predicts that at low temperature, large scale behaviors of these models are closely related to Gaussian free fields. I will review the historical background and discuss some recent progress on this conjecture in d>=3. Based on the joint work with Paul Dario (Tel Aviv). Video link: youtu.be/XsYk9LV-gxk January 26, 2021 Jürg Fröhlich (ETH Zürich) The Evolution of States as the Fourth Pillar of Quantum Mechanics I present ideas about how to extend the standard formalism of Quantum Mechanics in such a way that the theory actually makes sense. My tentative extension is called "ETH-Approach to Quantum Mechanics" (for "Events, Trees and Histories"). This approach supplies the last of four pillars QM can be constructed upon, which are: (i) Physical quantities characteristic of a physical system are represented by self-adjoint operators; (ii) the time evolution of these operators is given by the Heisenberg equation; (iii) meaningful notions of states and of "potential" and "actual events" have to be introduced; and (iv) a general law for the time evolution of states (superseding Schroedinger evolution, which is inadequate) must be formulated. Besides briefly sketching the ETH-Approach in the setting of non-relativistic QM, I present a family of very simple models of a very heavy atom coupled to the radiation field in a limit where the speed of light tends to infinity, An analysis of these models, which illustrate the ETH Approach, is the main subject of the talk. Video link: youtu.be/I83mvV8FOZo January 19, 2021 Robert Seiringer (IST Austria) Quantum fluctuations and dynamics of a strongly coupled polaron We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Landau--Pekar equations for the dynamics in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass. Video link: youtu.be/eR9b9hmssv0 January 12, 2021 Fabio Toninelli (Technical University Vienna) The anisotropic KPZ equation and logarithmic super-diffusivity The AKPZ equation is an anisotropic variant of the celebrated (two-dimensional) KPZ stochastic PDE, which is expected to describe the large-scale behavior of (2+1)-dimensional growth models whose average speed of growth is a non-convex function of the average slope (AKPZ universality class). Several interacting particle systems belonging to the AKPZ class are known, notably a class of two-dimensional interlaced particle systems introduced by A. Borodin and P. Ferrari. In this talk, I will show that the non-linearity of the AKPZ equation is marginally relevant in the Renormalization Group sense: in fact, while the 2d-SHE is invariant under diffusive rescaling, for AKPZ the diffusion coefficient diverges (logarithmically) for large times, implying marginal super-diffusivity. [Based on joint work with G. Cannizzaro and D. Erhard] Video link: youtu.be/d6-mN_x8ZO8 January 5, 2021 Søren Fournais (Aarhus University) The ground state energy of dilute Bose gases The rigorous calculation of the ground state energy of dilute Bose gases has been a challenging problem since the 1950s. In particular, it is of interest to understand the extent to which the Bogoliubov pairing theory correctly describes the ground state of this physical system. In this talk I will report on a recent proof of the second term in the energy expansion for dilute gases, the so-called Lee-Huang-Yang term, and its relation to Bogoliubov theory. Video link: youtu.be/JCa_n9nUK2U December 15, 2020 Yoshiko Ogata (University of Tokyo) Classification of symmetry protected topological phases in quantum spin systems A Hamiltonian is in a SPT phase with a given symmetry if it cannot be smoothly deformed into a trivial Hamiltonian without a phase transition, if the deformation preserves the symmetry, while it can be smoothly deformed into a trivial Hamiltonian without a phase transition, if the symmetry is broken during the deformation. We consider this problem for one- and two-dimensional quantum spin systems with on-site finite group symmetries. Video link: youtu.be/cXk6Fk5wD_4 December 8, 2020 Katrin Wendland (Albert-Ludwigs-Universität Freiburg) On invariants shared by geometry and conformal field theory I will recall how some conformal field theories can be given geometric interpretations. This can be useful from a practical point of view, when geometric methods are transferred from geometry to conformal field theory. I will in particular focus on certain invariants that are shared by geometry and conformal field theory, including the complex elliptic genus. As we shall see, this invariant is also useful from a theoretical viewpoint, since it captures information about generic properties of certain theories. Video link: youtu.be/zJ-zF1KBYM0 December 1, 2020 Alessandro Pizzo (University of Rome Tor Vergata) Stability of gapped quantum chains under small perturbations We consider a family of quantum chains that has attracted much interest amongst people studying topological phases of matter. Their Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian consisting of on-site terms and with a strictly positive energy gap above its ground-state energy. We prove stability of the spectral gap, uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain. We can treat fermions and bosons on the same footing, and our technique does not face a large field problem, even though bosons are involved. Furthermore the method can be extended to higher spatial dimensions and to complex Hamiltonians obtained by considering complex values of the coupling constant. (Joint work with S. Del Vecchio, J. Fröhlich, and S. Rossi.) Video link: youtu.be/S5lRLi3W6-E November 24, 2020 Roland Bauerschmidt (University of Cambridge) The Coleman correspondence at the free fermion point Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization. I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. I will then present a result that extends this correspondence to the non-Gaussian massless' sine-Gordon field on $\R^2$ at $\beta=4\pi$ and massive Dirac fermions. This is an instance of Coleman's prediction that the massless' sine-Gordon model and the massive Thirring model are equivalent. We use this correspondence to show that correlations of the massless' sine-Gordon model decay exponentially for $\beta=4\pi$. This is joint work with C. Webb (arXiv:2010.07096). Video link: youtu.be/uckzDglTbcM November 17, 2020 Stefan Hollands (University of Leipzig) (In)determinism Inside Black Holes In classical General Relativity, the values of fields on spacetime are uniquely determined by their initial values within its "domain of dependence". However, it may occur that the spacetime under consideration extends beyond this domain, and fields, therefore, are not entirely determined by their initial data. In fact, such a naive failure of determinism occurs inside all physically relevant black holes. The boundary of the region determined by the initial data is called the "Cauchy horizon". Penrose has proposed ("strong cosmic censorship conjecture") that the Cauchy horizon is actually unstable in the sense that the slightest perturbation caused by remnant fields convert it to a final singularity. Whether or not this is the case -- and thus whether there is a real problem with determinism -- has been put into question recently. In this colloquium I ask wheter quantum effects will come to the rescue of strong cosmic censorship. Video link: youtu.be/BmZxDseHnT0 November 10, 2020 Peter Hintz (MIT) Linear stability of slowly rotating Kerr spacetimes I will describe joint work with Dietrich Häfner and András Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild or slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. The proof uses a detailed description of the resolvent of an associated wave equation on symmetric 2-tensors near zero energy. Video link: youtu.be/iTBpM_DlGaI November 3, 2020 Nalini Anantharaman (University of Strasbourg) The bottom of the spectrum of a random hyperbolic surface I will report on work in progress with Laura Monk, where we study the bottom of the spectrum of the laplacian, on a compact hyperbolic surface chosen at random, in the limit of growing genus. We pick combinatorial ideas from the study of random regular graphs, to propose a strategy to prove that, with high probability, there are no eigenvalues below $1/4-\epsilon$. Video link: youtu.be/MThzUIIZx2Y October 27, 2020 Bruno Nachtergaele (UC Davis) Gapped ground state phases of lattice systems - Stability of the bulk gap I will give an overview of stability results for the spectral gap above the ground states of quantum lattice systems obtained in the past decade and explain the overall strategy introduced by Bravyi, Hastings, and Michalakis. A new result I will present proves the stability of the bulk gap for infinite systems with an approach that bypasses possible gapless boundary modes (joint work with Bob Sims and Amanda Young). Video link: youtu.be/rKzw0mr9xg4 October 20, 2020 Jeremy Quastel (University of Toronto) Towards KPZ Universality The 1-d KPZ universality class contains random interface growth models as well as random polymer free energies and driven diffusive systems. The KPZ fixed point has now been determined, through the exact solution of a special model in the class, TASEP, and is expected to describe the asymptotic fluctuations for all models in the class. It is an integrable Markov process, with transition probabilities described by a system of integrable PDE's. Very recently, new techniques have become available to prove the convergence of the KPZ equation itself, as well as some non-integrable extensions of TASEP, to the KPZ fixed point. This talk will be a gentle introduction to these developments with no prior knowledge assumed. The results are, variously, joint works with Daniel Remenik, Konstantin Matetski, and Sourav Sarkar. Video link: youtu.be/ecnoyc5Wf-E October 13, 2020 Svitlana Mayboroda (University of Minnesota) The landscape law for the integrated density of states Complexity of the geometry, randomness of the potential, and many other irregularities of the system can cause powerful, albeit quite different, manifestations of localization, a phenomenon of sudden confinement of waves, or eigenfunctions, to a small portion of the original domain. In the present talk we show that behind a possibly disordered system there exists a structure, referred to as a landscape function, which can predict the location and shape of the localized eigenfunctions, a pattern of their exponential decay, and deliver accurate bounds for the corresponding eigenvalues. In particular, we establish the "landscape law", the first non-asymptotic estimates from above and below on the integrated density of states of the Schroedinger operator using a counting function for the minima of the localization landscape. The results are deterministic, and rely on a new uncertainty principle. Narrowing down to the context of disordered potentials, we derive the best currently available bounds on the integrated density of states for the Anderson model. Video link: youtu.be/0f4vaNuLbK8 October 6, 2020 Clotilde Fermanian Kammerer (Université Paris Est - Créteil Val de Marne) Nonlinear quantum adiabatic approximation We will discuss generalization of the quantum adiabatic theorem to a nonlinear setting. We will consider evolution equations where the Hamiltonian operator depends on the time variable and on a finite number of parameters that are fixed on some coordinates of the unknown, making the equation non-linear. Under natural spectral hypotheses, there exist special functions that we call « Instantaneous nonlinear eigenvectors » such that, in the adiabatic limit, the solutions of the nonlinear equations with those initial data remain close to them, up to a rapidly oscillating phase. We will explain why this question arises, discuss the proof of this result and show how it can fail (works in collaboration with Alain Joye and Rémi Carles). Video link: youtu.be/kY-IrHnlw54 September 29, 2020 Alessandro Giuliani (University Roma Tre) Non-renormalization of the chiral anomaly' in interacting lattice Weyl semimetals Weyl semimetals are 3D condensed matter systems characterized by a degenerate Fermi surface, consisting of a pair of `Weyl nodes'. Correspondingly, in the infrared limit, these systems behave effectively as Weyl fermions in 3+1 dimensions. We consider a class of interacting 3D lattice models for Weyl semimetals and prove that the quadratic response of the quasi-particle flow between the Weyl nodes, which is the condensed matter analogue of the chiral anomaly in QED4, is universal, that is, independent of the interaction strength and form. Universality, which is the counterpart of the Adler-Bardeen non-renormalization property of the chiral anomaly for the infrared emergent description, is proved to hold at a non-perturbative level, notwithstanding the presence of a lattice (in contrast with the original Adler-Bardeen theorem, which is perturbative and requires relativistic invariance to hold). The proof relies on constructive bounds for the Euclidean ground state correlations combined with lattice Ward Identities, and it is valid arbitrarily close to the critical point where the Weyl points merge and the relativistic description breaks down. Joint work with V. Mastropietro and M. Porta. Video link: youtu.be/zUhM1k5nPcc September 22, 2020 Ian Jauslin (Princeton University) An effective equation to study Bose gasses at both low and high densities I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas. The interactions induce many-body correlations in the system, which makes it very difficult to study, be it analytically or numerically. A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to a one-particle problem, which incorporates the many-body correlations. The effective equation I will discuss is arrived at by making a very different set of approximations, and, like Bogolubov theory, ultimately reduces to a one-particle problem. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the many-body Bose gas at both low and at high densities. I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. That is, the analytical and numerical evidence suggest that this effective equation can capture many-body correlations in a one-particle picture beyond what Bogolubov can accomplish. Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas (about which there still are many important open questions). In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations. Video link: youtu.be/HyRG-PzvpyY September 15, 2020 Victor Ivrii (University of Toronto) Scott and Thomas-Fermi approximations to electronic density In heavy atoms and molecules, on the distances $a \ll Z^{-1/2}$ from one of the nuclei (with a charge $Z_m$), we prove that the ground state electronic density $\rho_\Psi (x)$ is approximated in $\sL^p$-norm by the ground state electronic density for a single atom in the model with no interactions between electrons. Further, on the distances $a \gg Z^{-1}$ from all of the nuclei (with a charge $Z_1,\ldots, Z_m$) we prove that $\rho_\Psi (x)$ is approximated in $\sL^p$-norm, by the Thomas-Fermi density. We cover both non-relativistic and relativistic cases. Video link: youtu.be/O25BT_-XNNE September 8, 2020 Antti Kupiainen (University of Helsinki) Integrability of Liouville Conformal Field Theory A. Polyakov introduced Liouville Conformal Field theory (LCFT) in 1981 as a way to put a natural measure on the set of Riemannian metrics over a two dimensional manifold. Ever since, the work of Polyakov has echoed in various branches of physics and mathematics, ranging from string theory to probability theory and geometry. In the context of 2D quantum gravity models, Polyakov’s approach is conjecturally equivalent to the scaling limit of Random Planar Maps and through the Alday-Gaiotto- Tachikava correspondence LCFT is conjecturally related to certain 4D Yang-Mills theories. Through the work of Dorn,Otto, Zamolodchikov and Zamolodchikov and Teschner LCFT is believed to be to a certain extent integrable. I will review a probabilistic construction of LCFT developed together with David, Rhodes and Vargas and recent proofs concerning the integrability of LCFT: -The proof in a joint work with Rhodes and Vargas of the DOZZ formula (Annals of Mathematics, 81-166,191 (2020) -The proof in a joint work with Guillarmou, Rhodes and Vargas of the bootstrap conjecture for LCFT (arXiv:2005.11530). Video link: youtu.be/0ms4gEUT2Nw July 28, 2020 Nicolas Rougerie (University of Grenoble Alpes) Two modes approximation for bosons in a double well potential We study the mean-field limit for the ground state of bosonic particles in a double-well potential, jointly with the limit of large inter-well separation/large potential energy barrier. Two one-body wave-functions are then macroscopially occupied, one for each well. The physics in this two-modes subspace is usually described by a Bose-Hubbard Hamiltonian, yielding in particular the transition from an uncorrelated "superfluid" state (each particle lives in both potential wells) to a correlated "insulating" state (half of the particles live in each potential well). Through precise energy expansions we prove that the variance of the number of particles within each well is suppressed (violation of the central limit theorem), a signature of a correlated ground state. Quantum fluctuations around the two-modes description are particularly relevant, for they give energy contributions of the same order as the energy difference due to suppressed variances in the two-modes subspace. We describe them in terms of two independent Bogoliubov Hamiltonians, one for each potential well. Joint work with Alessandro Olgiati and Dominique Spehner Video link: youtu.be/ylb6BWewlpI July 21, 2020 Hugo Duminil-Copin (IHES / University of Geneva) Marginal triviality of the scaling limits of critical 4D Ising and φ_4^4 models In this talk, we will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the λφ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis. Video link: youtu.be/DtLKEQran_Y July 14, 2020 Hal Tasaki (Gakushuin University) 'Topological' index and general Lieb-Schultz-Mattis theorems for quantum spin chains A Lieb-Schultz-Mattis (LSM) type theorem states that a quantum many-body system with certain symmetry cannot have a unique ground state accompanied by a nonzero energy gap. While the original theorem treats models with continuous U(1) symmetry, new LSM-type statements that only assume discrete symmetry have been proposed recently in close connection with topological condensed matter physics. Here we shall prove such general LSM-type theorems by using the "topological" index intensively studied in the context of symmetry protected topological phase. Operator algebraic formulation of quantum spin chains plays an essential role in our approach. Here I do not assume any advanced knowledge in quantum spin systems or operator algebra, and illustrate the ideas of the proof (which I believe to be interesting). The talk is based on a joint work with Yoshiko Ogata and Yuji Tachikawa in arXiv:2004.06458. Video link: youtu.be/q0k1sch56Dk July 7, 2020 Bruno Després (Sorbonne University) Spectral-scattering theory and fusion plasmas Motivated by fusion plasmas and Tokamaks (ITER project), I will describe recent efforts on adapting the mathematical theory of linear unbounded self-adjoint operators (Kato, Lax, Reed-Simon, ....) to problems governed by kinetic equations coupled with Maxwell equations. Firstly it will be shown that Vlasov-Poisson-Ampere equations, linearized around non homogeneous Maxwellians, can be written in the framework of abstract scattering theory (linear Landau damping is a consequence). Secondly the absorption principle applied to the hybrid resonance will be discussed. All results come from long term discussions and collaborations with many colleagues (Campos-Pinto, Charles, Colas, Heuraux, Imbert-Gérard, Lafitte, Nicolopoulos, Rege, Weder, and many others). Video link: youtu.be/lmnm1D3NFp8 June 30, 2020 Laure Saint-Raymond (ENS Lyon) Fluctuation theory in the Boltzmann-Grad limit In this talk, I will discuss a long term project with T. Bodineau, I. Gallagher and S. Simonella on hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behavior are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics. Video link: youtu.be/fLDFA7ZCagA June 23, 2020 Nilanjana Datta (University of Cambridge) Discriminating between unitary quantum processes Discriminating between unknown objects in a given set is a fundamental task in experimental science. Suppose you are given a quantum system which is in one of two given states with equal probability. Determining the actual state of the system amounts to doing a measurement on it which would allow you to discriminate between the two possible states. It is known that unless the two states are mutually orthogonal, perfect discrimination is possible only if you are given arbitrarily many identical copies of the state. In this talk we consider the task of discriminating between quantum processes, instead of quantum states. In particular, we discriminate between a pair of unitary operators acting on a quantum system whose underlying Hilbert space is possibly infinite-dimensional. We prove that in contrast to state discrimination, one needs only a finite number of copies to discriminate perfectly between the two unitaries. Furthermore, no entanglement is needed in the discrimination task. The measure of discrimination is given in terms of the energy-constrained diamond norm and one of the key ingredients of the proof is a generalization of the Toeplitz-Hausdorff Theorem in convex analysis. Moreover, we employ our results to study a novel type of quantum speed limits which apply to pairs of quantum evolutions.This work was done jointly with Simon Becker (Cambridge), Ludovico Lami (Ulm) and Cambyse Rouze (Munich) Video link: youtu.be/gHEjszXSjMQ June 16, 2020 Nicola Pinamonti (University of Genova) Equilibrium states for interacting quantum field theories and their relative entropy During this talk we will review the construction of equilibrium states for interacting scalar quantum field theories, treated with perturbation theory, recently proposed by Fredenhagen and Lindner. We shall in particular see that this construction is a generalization of known results valid in the case of C*-dynamical systems. We shall furthermore discuss some properties of these states and we compare them with known results in the physical literature. In the last part of the talk, we shall show that notions like relative entropy or entropy production can be given for states which are of the form discussed in the first part of talk. We shall thus provide an extension to quantum field theory of similar concepts available in the case of C*-dynamical systems. Video link: youtu.be/excgcO7loj0 June 9, 2020 Andreas Winter (Universitat Autònoma de Barcelona) Energy-constrained diamond norms and the continuity of channel capacities and of open-system dynamics The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm). However, it is well-known that in infinite dimension, the norm topology is often "too strong" for reasonable applications. Here, we explore a recently introduced energy-constrained diamond norm on superoperators (subject to an energy bound on the input states). Our main motivation is the continuity of capacities and other entropic quantities of quantum channels, but we also present an application to the continuity of one-parameter unitary groups and certain one-parameter semigroups of quantum channels. Video link: youtu.be/05ZQPFB0aAc June 2, 2020 Mihalis Dafermos (Cambridge University) The nonlinear stability of the Schwarzschild metric without symmetry I will discuss an upcoming result proving the full finite-codimension non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region. No symmetry is assumed. The work is based on our previous understanding of linear stability of Schwarzschild in double null gauge. Joint work with G. Holzegel, I. Rodnianski and M. Taylor. Video link: youtu.be/6Vh62H0rPiA May 26, 2020 Sven Bachmann (University of British Columbia) Adiabatic quantum transport In the presence of a spectral gap above the ground state energy, slowly driven condensed matter systems may exhibit quantized transport of charge. One of the earliest instances of this fact is the Laughlin argument explaining the integrality of the Hall conductance. In this talk, I will discuss transport by adiabatic processes in the presence of interactions between the charge carriers. I will explain the central role played by the locality of the quantum dynamics in two instances: the adiabatic theorem and an index theorem for quantized charge transport. I will also relate fractional transport to the anyonic nature of elementary excitations. Video link: youtu.be/ErgMuxMR_1A May 19, 2020 Pierre Clavier (University of Potsdam) Borel-Ecalle resummation for a Quantum Field Theory Borel-Ecalle resummation of resurgent functions is a vast generalisation of the well-known Borel-Laplace resummation method. It can be decomposed into three steps: Borel transform, averaging and Laplace transform. I will start by a pedagogical introduction of each of these steps. To illustrate the feasability of the Borel-Ecalle resummation method I then use it to resum the solution of a (truncated) Schwinger-Dyson equation of a Wess-Zumino model. This will be done using known results about this Wess-Zumino model as well as Sauzin's analytical bounds on convolution of resurgent functions. Video link: youtu.be/EzRoLEZhono May 12, 2020 Jan Philip Solovej (University of Copenhagen) Universality in the structure of Atoms and Molecules Abstract: The simplest approximate model of atoms and molecules is the celebrated Thomas-Fermi model. It is known to give a good approximation to the ground state energy of heavy atoms. The understanding of this approximation relies on a beautiful and very accurate application of semi-classical analysis. Although the energy approximation is good, it is, unfortunately, far from being accurate enough to predict quantities relevant to chemistry. Thomas-Fermi theory may nevertheless tell us something surprisingly accurate about the structure of atoms and molecules. I will discuss how a certain universality in the Thomas-Fermi model, indeed, holds approximately in much more complicated models, such as the Hartree-Fock model. I will also show numerical and experimental evidence that the approximate universality may hold even for real atoms and molecules. Video link: youtu.be/FCxkP7CqtQQ May 5, 2020 Martin Hairer (Imperial College London) The Brownian Castle Video link: youtu.be/Ve_EFZDbXTU