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 Hal Tasaki (Gakushuin).

Scientific committee: Nalini Anantharaman (Strassbourg), Mihalis Dafermos (Cambridge), Stephan De Bièvre (Lille), 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), Daniel Ueltschi (Warwick).

If you would like to receive seminar announcements, please send an email to with “subscribe” in the subject line; or “unsubscribe” to have your email address removed. You can also email comments or suggestions.

Other One World seminars are listed on the website of the probability seminar, which initiated the series. The website lists further mathematical-physics seminars.

Tribute to the IAMP seminar, by Hal Tasaki, Robert Seiringer, Bruno Nachtergaele, and Elliott Lieb:

Upcoming seminars

September 27, 2022 Dorothea Bahns (Georg-August-Universität Göttingen)
Title TBA

Livestream link will appear here.
October 4, 2022 Paweł Duch (Adam Mickiewicz University)
Flow equation approach to singular stochastic PDEs
Most stochastic PDEs arising in physics, such as the KPZ equation describing the motion of a growing interface or the stochastic quantization equation of the $\Phi^4$ Euclidean QFT, are ill-posed in the classical analytic sense due to irregular nature of random terms. Equations of this type are called singular. Regularization and renormalization are usually necessary to give mathematical meaning to such equations. In the talk, I will present a novel technique of solving singular stochastic PDEs. The technique is based on the renormalization group flow equation. It is applicable to a large class of parabolic or elliptic SPDEs with fractional Laplacian, additive noise and polynomial non-linearity. It covers equations in the whole super-renormalizable regime. A nice feature of the method is that it avoids the algebraic and combinatorial problems arising in different approaches. Based on arXiv:2109.11380 and arXiv:2201.05031.
Livestream link will appear here.
October 11, 2022 Christoph Kopper (Ecole Polytechnique)
Title TBA

Livestream link will appear here.
October 18, 2022 Karol Kozlowski (ENS Lyon)
Title TBA

Livestream link will appear here.
October 25, 2022 Sohei Ashida (Gakushuin University)
Title TBA

Livestream link will appear here.
November 1, 2022 Giovanni Felder (ETH Zürich)
Title TBA

Livestream link will appear here.
November 15, 2022 Anton Alekseev (University of Geneva)
Title TBA

Livestream link will appear here.
November 22, 2022 Ivan Avramidi (New Mexico Tech)
Title TBA

Livestream link will appear here.
November 29, 2022 Camillo de Lellis (IAS Princeton)
Title TBA

Livestream link will appear here.

Past seminars

September 20, 2022 Slava Rychkov (IHES)
Tensor Renormalization Group at Low Temperatures
Tensor RG is a real-space approach to renormalization in lattice models. It shows impressive numerical results, but its rigorous theory is still in its infancy. I will start by reminding the basics of Tensor RG, and what makes it potentially more powerful than Wilson-Kadanoff approach. I will then explain how Tensor RG can be used to understand the Ising model in nonzero magnetic field at low temperatures. This gives an alternative to the Pirogov-Sinai theory and to the renormalization group methods in the contour representation from the 1980’s. Joint work with Tom Kennedy, to appear soon.
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August 30, 2022 Jian Ding (Peking University)
Repeated emergence of 4/3-exponent
In this talk, I will describe the emergence of the 4/3-exponent in two seemingly unrelated models: random distance of Liouville quantum gravity and correlation length for the two-dimensional random field Ising model. I will then explain that such 4/3-exponent, while being unexpected among respective communities even from a physics perspective, has in fact been hinted in Leighton-Shor (1989) and Talagrand (2014) where the 4/3-exponent emerges in a random matching problem. Finally, I will present the heuristic computation which leads to the emergence of the 4/3-exponent. While I will review related progress on these topics, the two papers featuring 4/3-exponent are with Subhajit Goswami and with Mateo Wirth.
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August 16, 2022 Jonathan Dimock (SUNY at Buffalo)
Ultraviolet Stability for Quantum Electrodynamics in d=3
We report on results for quantum electrodynamics on a finite volume Euclidean spacetime in dimension d=3. The theory is formulated as a functional integral on a fine toroidal lattice involving both fermion fields and abelian gauge fields. The main result is that, after renormalization, the partition function is bounded uniformly in the lattice spacing. This is a first step toward the construction of the model. The result is obtained by renormalization group analysis pioneered by Balaban. A single renormalization group transformation involves block averaging, a split into large and small field regions, and an iden- tification of effective actions in the small field regions via cluster expansions. This leads to flow equations for the parameters of the theory. Renormalization is accomplished by fine-tuning the initial conditions for these equations. Large field regions need no renormalization, but are shown to give a tiny contribution.
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August 2, 2022 Martin Fraas (UC Davis)
Projections, parallel transport and adiabatic theory
I will give an overview of adiabatic theory with the focus on a geometric point of view. The talk will cover traditionally adiabatic theory, adiabatic theory in many-body systems, and a recent work with W. De Roeck and A. Elgart on the adiabatic theory for disordered systems.
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July 26, 2022 Alexander Strohmaier (University of Leeds)
A mathematical analysis of Casimir interactions and determinant formulae
I will explain a mathematical treatment of Casimir interactions of perfect conductors in which the Casimir energy is written as the trace of an operator without the need for regularisations. I will also show some consequences of this approach, relations to microlocal analysis, and will prove a trace formula that allows to compute the Casimir energy in terms of determinants of single layer operators. Such formulae have been derived by other methods in the physics literature and we will show that all these approaches give the same well defined Casimir energy. (Based on joint work with F. Hanisch and A. Waters, as well as numerical work with T. Betcke and X. Sun).
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July 19, 2022 Tomohiro Sasamoto (Tokyo Institute of Technology)
Mapping macroscopic fluctuation theory for the symmetric simple exclusion process to a classically integrable system
The large deviation principle for symmetric simple exclusion process(SEP) had been established by Kipnis, Olla, Varadhan in 1989 [1]. A somewhat different formulation, known as the macroscopic fluctuation theory (MFT), was initiated and developed by Jona-Lasinio et al in 2000’s [2]. The basic equations of the theory, MFT equations, are coupled nonlinear partial differential equations and have resisted exact analysis except for stationary situation. In this talk we introduce a novel generalization of the Cole-Hopf transformation and show that it maps the MFT equations for SEP to the classically integrable Ablowitz-Kaup-Newell-Segur(AKNS) system. This allows us to solve the equations exactly in time dependent regime by adapting standard ideas of inverse scattering method. The talk is based on a joint work with Kirone Mallick and Hiroki Moriya [3]. References [1] C. Kipnis, S. Olla, S. R. S. Varadhan, Hydrodynamics and large deviations for simple exclusion processes, Comm. Pure Appl. Math., 42:115–137, 1989. [2] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory, Rev. Mod. Phys., 87:593–636, 2015. [3] Kirone Mallick, Hiroki Moriya, Tomohiro Sasamoto, Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process, arXiv: 2202.05213, to appear in Phys. Rev. Lett.
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July 12, 2022 Stefan Teufel (University of Tübingen)
A slightly different look at the bulk-edge correspondence in quantum Hall systems
The bulk-edge correspondence of transport coefficients in quantum Hall systems is usually shown for systems with a (mobility) gap in the bulk and at zero temperature. I will present recent results showing equality of magnetization in the bulk and edge current in microscopic models without and with interaction between electrons. This equality holds at positive temperature and without assuming a (mobility) gap in the bulk, and is robust to perturbations near the edges. I then discuss how the equality of transport coefficients can be recovered from this form of bulk-edge correspondence, at least in some cases. My talk is based on joint work with Horia Cornean, Jonas Lampart, Massimo Moscolari, and Tom Wessel.
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July 5, 2022 Michele Correggi (Politecnico di Milano)
Ground State Properties in the Quasi-Classical Regime
We study the ground state energy and ground states of systems coupling non-relativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasi-classical approximation. The latter is very useful whenever the force-carrying field has a very large number of excitations, and thus behaves in a semiclassical way, while the non-relativistic particles, on the other hand, retain their microscopic features. We prove that the ground state energy of the fully microscopic model converges to the one of a nonlinear quasi classical functional depending on both the particles' wave function and the classical configuration of the field. Equivalently, this energy can be interpreted as the lowest energy of a Pekar-like functional with an effective nonlinear interaction for the particles only. If the particles are confined, the ground state of the microscopic system converges as well, to a probability measure concentrated on the set of minimizers of the quasi classical energy.
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June 28, 2022 Owen Gwilliam (University of Massachusetts Amherst)
Observables in the Batalin-Vilkovisky formalism: from Feynman diagrams to commutative diagrams
This talk will discuss a convergence between perturbative QFT and higher algebra, in the sense of homological algebra, operads, and higher categories, although we do not expect the listener to have any expertise in those topics. A key notion here is a factorization algebra, which captures the local-to-global nature of the observables of a field theory and which is due to Beilinson and Drinfeld. With a focus on the example of Chern-Simons theory, we will discuss how the BV/factorization package offers a new view on the emergence of algebraic structures, like braided monoidal categories, from physics.
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June 21, 2022 Chiara Saffirio (University of Basel)
Mean-field evolution and semiclassical limit of many interacting fermions
We will review recent progresses in the derivation of effective evolution equations for the dynamics of many weakly interacting fermions. We will focus on the mean-field regime and couple it with a semiclassical limit to obtain, as the number of particles goes to infinity, the Hartree-Fock and the Vlasov equations. We will comment on the class of singular interactions and quantum states that we are able to treat, drawing a comparison with the PDE theory of the limiting equations. Based on joint works with J. Chong and L. Laflèche.
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June 7, 2022 Michael Aizenman (Princeton University)
A new perspective on a pair of two dimensional phenomena: delocalization in random height functions and the Berezinskii-Kosterlitz-Thouless phase of O(2) symmetric spin models
Delocalization in integer-restricted Gaussian field, and other random height functions formulated over planar doubly-periodic lattices, is shown to imply slow (power law) decay of correlations in the corresponding dual O(2) symmetric two-component spin model. The link proceeds through a lower bound on the spin-spin correlation in terms of the probability of their sites being on a common level loop of the dual random height function. Motivated by this observation, we have extended the recent proof by P. Lammers of delocalization transition in two dimensional graphs of degree three, to all doubly periodic graphs, in particular to Z^2. The extension is established through a monotonicity argument based on lattice inequalities of O. Regev and N. Stephens-Davidowitz. Taken together the results yield a new perspective on the BKT phase transition in O(2)-invariant models and complete a new proof of delocalization in two-dimensional integer-valued height functions. (Both phenomena are unique to two dimension, and have been previously proven and studied by other means). That talk is based on a joint work by M. Aizenman, M. Harel J. Shapiro and R. Peled.
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May 31, 2022 Benjamin Doyon (King's College London)
Ergodicity, large-scale correlations and hydrodynamics in many-body systems
Long-time behaviours in statistical ensembles of many-body systems are notoriously difficult to access. Hydrodynamics, which is the theory for the emergent large-scale dynamics, gives a lot of information, such as exact asymptotics of correlation functions. It turns out that at the Euler scale, the emergent theory for extended systems is largely universal. I will discuss a number of such universal results in one dimension of space. Some can be shown rigorously: a notion of ergodicity at all frequencies hold for correlation functions in stationary states of all short-range quantum spin chains. The Boltzmann-Gibbs principle, where local observables project onto hydrodynamic modes in two-point functions, and the linearised Euler equations, are also established. The complete space of hydrodynamic modes is the space of ``extensive conserved quantities”, which is defined unambiguously. I will illustrate these concepts in integrable systems, using generalised hydrodynamics. For correlations in non-stationary states, much less is established. I will describe a macroscopic fluctuation theory for the Euler scale which provides a framework for these. In particular, I will explain how a new type of long-range correlations, hitherto not observed, appear when the system is subject to fluid motion, which breaks the paradigm that separate fluid cells are not correlated.
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May 24, 2022 Kevin Costello (Perimeter Institute)
Form factors of gauge theory as correlators of a vertex algebra
Form factors are scattering amplitudes in the presence of a local operator. I will explain that, for certain gauge groups, form factors of self-dual Yang-Mills theory (with some additional fields) are the correlators of a vertex algebra. This is joint work with Natalie Paquette, and is closely related to the celestial holography program.
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May 17, 2022 Wojciech Dybalski (Adam Mickiewicz University)
Interacting massless excitations in 1+1 dimensional QFT
Massless 1+1 dimensional quantum field theories have a peculiar scattering theory. Since the motion of the excitations is dispersionless, only the distinction between left-movers and right-movers is meaningful. Thus complete particle interpretation (asymptotic completeness) can be expected already at the level of two-body scattering. This effect will be illustrated by certain wedge-local models of interacting Wigner particles. In the second part of the talk I will move to the more exotic case of infraparticles, i.e., excitations which are not of Wigner type. It is known that infraparticles abound in 1+1 massless QFT, even in the familiar case of free field theory. A natural definition of a scattering amplitude for infraparticles will be proposed and tested in this latter model. Paradoxically, infraparticles in free field theory exhibit non-trivial scattering. The talk is based on joint works with Yoh Tanimoto (CMP 305, 427-440 (2011)) and Jens Mund (arXiv:2109.02128).
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May 10, 2022 Vieri Mastropietro (University of Milan)
Anomalies and non-perturbative QFT
Adler and Bardeen in 1969 established the non-renormalization of the chiral anomaly, writing it as a perturbative expansion which is order by order vanishing; since then, this property has found uncountable applications, including the Standard Model consistence via the anomaly cancellation condition (Bouchiat, Iliopoulos, Meyer 1972). After reviewing briefly this notion, we prove the anomaly non renormalization at a non-perturbative level in the case of lattice vector boson-fermion models in d=1+1 (uniformly in the lattice) and in d=3+1 (up to a cut-off of the order of the inverse coupling). The proof relies on bounds on the large distance decay of correlations and Ward Identities, both exact and emerging. In the case of chiral models, like the effective electroweak theory with quartic Fermi interaction in d=3+1, some Ward Identites are violated and the anomaly can be proved to vanish up to subdominant corrections which are rigorously bounded, under the cancellation condition on charges and with a cut-off of the order of the inverse coupling. Open problems and conjectures will be finally discussed.
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May 3, 2022 Daniel Ueltschi (University of Warwick)
Random loop models and their universal behaviour in dimension 3+
I will discuss several loop soup models that represent classical or quantum systems of statistical physics. These systems undergo phase transitions that are characterised by the occurrence of macroscopic loops. As it turns out, the joint distribution of the loop lengths exhibits a universal behaviour: In 3+ dimension it is always given by a Poisson-Dirichlet distribution. The heuristics is based on the fact that the loops are so intertwined that they behave effectively in mean-field fashion. Hence the connection to the split-merge process, whose invariant measure is Poisson-Dirichlet. I will also discuss consequences about symmetry breaking in certain quantum spin systems. This talk is based on collaborations over the years with D. Gandolfo, J. Ruiz, C. Goldschmidt, P. Windridge, V. Betz, S. Grosskinsky, A. Lovisolo, C. Benassi, J. Björnberg, J. Fröhlich.
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April 26, 2022 Deepak Dhar (IISER Pune)
Hard rigid rods on hypercubical lattices
I consider a system of hard rigid rods of length $k$ and width $1$, on $d$-dimensional hypercubic lattices. I will show that in the limit of large $k$, the entropy per site at full packing $ s(d,k)$ for $d=2$ satisfies $$ \lim_{k \rightarrow \infty} \frac { s(d,k) k^2}{\log k} =1, $$ and give heuristic arguments that this result would be true also for all dimensions $d>2.$ If $k$ is large enough ($ k> 6 $ in $2$-d, and $k>4$ in $3$-d), this model is known to undergo two phase transitions when chemical potential is increased: from a low density isotropic phase to an intermediate density nematic phase, and on further increase to a high-density phase with no orientational order. I will present evidence to suport the conjecture that, for large $k$, the second phase transition is a first order transition with a discontinuity in density as a function of the chemical potential, in all dimensions greater than 1, and if the chemical potential per rod at the transition is $\mu^*(k,d)$, and the density of holes jumps from a value $\epsilon_1(k,d)$ to $\epsilon_2(k,d)$, we have for all $d \geq 2$ $$\lim_{ k \rightarrow \infty} \frac{ \epsilon_1(d, k) k^2}{\log k} =1,$$ $$\lim_{k \rightarrow \infty} \frac{ \exp\left[ \frac{\mu^*(d, k)}{k} \right] \log k}{k} =1,$$ $$\lim_{k \rightarrow \infty} \frac{\epsilon_2(d,k) k ^m}{\epsilon_1(d,k)} =0, ~{\rm for~ all}~m>0.$$ References: Deepak Dhar and R. Rajesh, Phys. Rev. E 103, 042130 (2021); Aagam Shah, D. Dhar and R. Rajesh, Phys. Rev. E. 105, 034103 (2022).
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April 19, 2022 Hirosi Ooguri (Caltech & Kavli IPMU)
Symmetry in QFT and Gravity
I will review aspects of symmetry in quantum field theory and combine them with the AdS/CFT correspondence to derive constraints on symmetry in quantum gravity. The quantum gravity constraints to be discussed include the no-go theorem on global symmetry, the completeness of gauge charges, and the decomposition of high energy states into gauge group representations.
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April 12, 2022 Edward Witten (IAS)
Black Hole Entropies and Algebras
I will review the idea of black hole entropy, as originally discovered by Bekenstein and Hawking roughly half a century ago, and then explain how to interpret black hole entropy in terms of an appropriate algebra of observables.
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April 5, 2022 Jacob Shapiro (Princeton University)
Continuum and strongly disordered topological insulators
A natural question to ask in the field of topological insulators is whether continuum descriptions on L^2(R^d) and effective discrete descriptions on l^2(Z^d) agree, at the level of the long time dynamics and at the level of the topological indices. This is shown to be the case at least for a class of models of the integer quantum Hall effect. The basic tool to approach the problem is homotopies of Fredholm operators. This same tool is also shown to apply to other problems in topological insulators, such as the bulk-edge correspondence for Z_2 time-reversal invariant strongly disordered discrete systems. This talk is based on joint works with Michael I. Weinstein and with Jeff Schenker and Alex Bols.
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March 29, 2022 Serena Cenatiempo (GSSI L'Aquila)
Trial states for the zero temperature dilute Bose gas
Non-relativistic interacting bosons at zero temperature, in two and three dimensions, are expected to exhibit a fascinating critical phase, famously known as condensate phase. Even though a proof of Bose-Einstein condensation in the thermodynamic limit is still beyond reach of the current available methods, the mathematical physics community has recently gained an enhanced comprehension of other aspects of the macroscopic behavior of low temperature Bose gases, in several interesting regimes. In this talk we review part of these advances, by describing trial states for the dilute three dimensional Bose gas, capturing the celebrated Lee-Huang-Yang sub-leading correction to the ground state energy. We conclude with some open questions and perspectives. Based on joint works with G. Basti, C. Boccato, C. Brennecke, C. Caraci, A. Olgiati, G. Pasqualetti and B. Schlein.
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March 22, 2022 Vojkan Jaksic (McGill University)
Approach to equilibrium in translation-invariant quantum systems: some structural results
We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase of entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase of the specific (mean) entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest. This is a joint work with C. Tauber and C.-A. Pillet
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March 15, 2022 Jerzy Lewandowski (University of Warsaw)
Horizon equations
The extremity assumptions imposes a set of non-trivial equations on the intrinsic and extrinsic geometry of the horizon. Some of them already got more attention since they are applied in the near horizon geometries (NHG), some other are less know. All of them are very important for the existence and uniqueness of extremal black hole solutions to Einstein's equations. In 4d spacetime, an integrability condition for the NHG equation is available that is satisfied by the family of non-extremal horizon geometries, namely by those that are of the Petrov type D. What is special about that equation, is that its solutions exhibit similar properties to those proved in the global black hole spacetime theory: spherical topology of horizon cross sections, rigidity, no-hair. Extension of the research to the horizons that had a Hopf bundle structure led to interesting results about the global structure of the Kerr-NUT-(A)dS spaces. Misner's construction was generalised to the case when none of the three parameters (NUT, Kerr, and the cosmological constant) vanishes. The resulting family contains also exact solutions to Einstein's equations that admit no horizon or nowhere time like Killing vector and describe a topologically spherical universe evolving from the past scry to the future scry in a non-singular manner.
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March 8, 2022 Makiko Sasada (University of Tokyo)
Topological structures and the role of symmetry in the hydrodynamic limit of nongradient models
Recently, we introduce a general framework in order to systematically investigate hydrodynamics limits of various microscopic stochastic large scale interacting systems in a unified fashion. In particular, we introduced a new cohomology theory called the uniformly cohomology to investigate the underlying topological structure of the interacting system. Our theory gives a new interpretation of the macroscopic parameters, the role played by the group action on the microscopic system, and the origin of the diffusion matrix associated to the macroscopic hydrodynamic equation. Furthermore, we rigorously formulate and prove for a relatively general class of models Varadhan’s decomposition of closed forms, which plays a key role in the proof of hydrodynamic limits of nongradient models. Our result is applicable for many important models including generalized exclusion processes, multi-species exclusion processes, exclusion processes on crystal lattices and so on. Based on joint papers with Kenichi Bannai and Yukio Kametani
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March 1, 2022 Marek Biskup (UCLA)
Extremal points of random walks on planar and tree graphs
I will review recent progress on the description of points heavily visited by paths of random walks. The focus will be on the situations where the random walk has an approximate scale-invariant structure and the associated local time process is thus logarithmically correlated in space. Two geometric settings will be considered: the simple random walk on finite subsets of the square lattice and the random walk on homogeneous trees of finite depth. In the former case, a full description will be given of the scaling limit of thick, thin and avoided points for the walk run up to the times proportional to the cover time. For the latter setting, the law of the most frequently visited leaf-vertex, and the time spent there, will be given in the limit of the tree depth tending to infinity. In both cases, the spatial laws will be determined by a corresponding multiplicative chaos measure. Based on joint papers with Y. Abe, S. Lee and O. Louidor.
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February 22, 2022 Lorenzo Taggi (Sapienza Università di Roma)
Macroscopic loops in the interacting Bose gas, Spin O(N) and related models
We consider a general system of interacting random loops which includes several models of interest, such as the spin O(N) model, the double dimer model, random lattice permutations, and is related to the loop O(N) model and to the interacting Bose gas in discrete space. We present an overview on these models, introduce some of the main open questions about the size and the geometry of the loops, and present some recent results about the occurrence of macroscopic loops in dimensions d > 2 as the inverse temperature is large enough.
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February 15, 2022 Martin Zirnbauer (University of Cologne)
Field theory of Anderson transitions reviewed: Color-Flavor Transformation
This talk is in two parts. First, the field-theoretic approach to (de-)localization in Anderson-type models for disordered electrons is reviewed, with emphasis placed on the presence of a hyperbolic target-space sector and the expected pattern of spontaneous symmetry breaking. The second part is a review of the "Color-Flavor (CF) Transformation" (MZ, 1996). Conceived as a variant of the Efetov-Wegner supersymmetry method, the CF Transformation is tailored to quantum systems with disorder distributed according to Haar measure for any compact Lie group of classical type (A, B, C, or D). It has been applied to Dyson's Circular Ensembles, random-link network models, quantum chaotic graphs, disordered Floquet dynamics, and more. It is reviewed here as a rigorous tool to derive the effective field theory for systems in the metallic regime.
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February 8, 2022 Erik Skibsted (Aarhus University)
Stationary scattering theory, the N-body long-range case
Within the class of Dereziński-Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known, we show that all entries of the N-body quantum scattering matrix have a well-defined meaning at any given non-threshold energy. As a function of the energy parameter the scattering matrix is weakly continuous. This result generalizes a similar one obtained previously by Yafaev for systems of particles interacting by short-range potentials. As for Yafaev’s works we do not make any assumption on the decay of channel eigenstates. The main part of the proof consists in establishing a number of Kato-smoothness bounds needed for justifying a new formula for the scattering matrix. Similarly we construct and show strong continuity of channel wave matrices for all non-threshold energies. The set of so-called stationary complete energies has full measure. We show that the scattering and channel wave matrices constitute a well-defined ‘scattering theory’ at such energies, in particular at any stationary complete energy the scattering matrix is unitary, strongly continuous and characterized by asymptotics of minimum generalized eigenfunctions.
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February 1, 2022 Daisuke Shiraishi (Kyoto University)
Recent progress on loop-erased random walk in three dimensions
Loop-erased random walk (LERW) is a model for a random simple path, which is created by running a simple random walk and, whenever the random walk hits its path, removing the resulting loop and continuing. LERW was originally introduced by Greg Lawler in 1980. Since then, it has been studied extensively both in mathematics and physics literature. Indeed, LERW has a strong connection with other models in statistical physics, especially the uniform spanning tree which arises in statistical physics in conjunction with the Potts model. In this talk, I will talk about some recent progress on LERW while focusing on the three-dimensional case. This is joint work with Xinyi Li.
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January 25, 2022 Michele Schiavina (ETH Zürich)
Ruelle Zeta Function from Field Theory
I will discuss a field-theoretic interpretation of Ruelle's zeta function, which "counts" prime geodesics on hyperbolic manifolds, as the partition function for a topological field theory (BF) with an unusual gauge fixing condition available on contact manifolds. This suggests a rephrasing of a conjecture due to Fried, on the equivalence between Ruelle's zeta function (at zero) and the analytic torsion, as gauge-fixing independence in the Batalin--Vilkovisky formalism.
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January 18, 2022 Marcin Lis (University of Vienna)
An elementary proof of phase transition in the planar XY model
Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer and establishes a Berezinskii-Kosterlitz-Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers on delocalisation of integer-valued height functions, and classical correlation inequalities. This is joint work with Diederik van Engelenburg.
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January 11, 2022 Claudio Dappiaggi (Università di Pavia)
Stochastic Partial Differential Equations and Renormalization à la Epstein-Glaser
We present a novel framework for the study of a large class of nonlinear stochastic partial differential equations, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use specific techniques proper of microlocal analysis. These allow us to deal with renormalization using an Epstein-Glaser perspective, hence without resorting to any specific regularization scheme. As a concrete example we shall use this method to discuss the stochastic $\Phi^3_d$ model and we shall comment on its applicability to the stochastic nonlinear Schrödinger equation -- Joint work with Nicolò Drago, Paolo Rinaldi and Lorenzo Zambotti.
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December 21, 2021 Christoph Schweigert (Universität Hamburg)
More about CFT correlators
I will explain two recent developments concerning bulk fields in two-dimensional rational conformal field theories: the importance of the relative Serre functor to study bulk fields for logarithmic conformal field theories and the use of stringnet techniques to simplify the construction of correlators for semisimple modular tensor categories.
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December 14, 2021 Nicolai Reshetikhin (University of California, Berkeley)
On the asymptotic behavior of character measures in large tensor powers of finite dimensional representations of simple Lie algebras
Let $V$ be a finite dimensional representation of a simple Lie algebra and $H$ be an positive element of its Cartan subalgebra ("magnetic field"). On the space $V^{\otimes N}$ we have a natural density matrix $N\exp(-H)$where $H$ acts diagonally: $H(x\otimes y\otimes z\dots)=Hx\otimes y\otimes z\dots +x\otimes y\otimes z\dots+x\otimes y\otimes z\dots$. The space $V^{\otimes N}$ decomposes into a direct sum of irreducible subrepresentations: \[ V^{\otimes N}\simeq \oplus_{\lambda} V_\lambda^{\oplus m(\lambda, N)} \] where $\lambda$ is the highest weight of the representation $V_\lambda$ and $m(\lambda, N)$ is its multiplicity in the tensor product. The character distribution assigns the probability \[ p_{\lambda}(N,H)=\frac{m(\lambda, N)Tr_{V_\lambda}(e^{-H)})}{(Tr_V(e^{-H}))^N} \] to each $\lambda$ in the decomposition of the tensor product. One of the natural problems for this distribution is to find its asymptotic in the limit $N\to \infty$ and $\lambda\to \infty$ in the appropriate way. When $H=0$ the character distribution becomes a uniform distribution. In this case, in such generality the asymptotic was studied by Ph. Biane, 1993 by T. Tate and S. Zelditch, 2004. For tensor powers of vector representations the asymptotic was derived by S. Kerov in 1986. When $H$ is generic, i.e. when $H$ is strictly inside of the principal Weyl chamber it was computed by O.Postnova and N.R. in 2018. This talk is based on a joint work with O. Postnova and V. Serganova (to appear on the arxiv).
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December 7, 2021 Nina Holden (ETH Zurich)
Conformal welding in Liouville quantum gravity: recent results and applications
Liouville quantum gravity (LQG) is a natural model for a random fractal surface with origins in conformal field theory and string theory. A powerful tool in the study of LQG is conformal welding, where multiple LQG surfaces are combined into a single LQG surface. The interfaces between the original LQG surfaces are typically described by variants of the random fractal curves known as Schramm-Loewner evolutions (SLE). We will present a few recent conformal welding results for LQG surfaces and applications to the integrability of SLE and LQG partition functions. Based on joint work with Ang and Sun and with Lehmkuehler.
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November 30, 2021 Niklas Beisert (ETH Zurich)
Planar Integrability and Yangian Symmetry in N=4 Supersymmetric Yang–Mills
The discovery and utilisation of integrability in the planar limit of certain gauge and string theory models has made available many exciting non-perturbative results and insights for the AdS/CFT correspondence. In this talk I will introduce the prototypical AdS/CFT duality between string theory on AdS₅×S⁵ and N=4 supersymmetric Yang–Mills theory, and highlight the usefulness of planar integrability through some achievements. I will then present a formal notion of integrability as a hidden Yangian symmetry of the action. Importantly, this symmetry is realised on field theory objects in a non-standard fashion for which a rigorous framework to deal with its implications is lacking. I will demonstrate that this symmetry indeed leads to non-trivial relations for quantum correlations functions of this model. In order to make them work out, I also show in what sense the symmetry is compatible with BRST gauge fixing.
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November 23, 2021 Julian Sonner (University of Geneva)
Random matrices and black holes: new connections from holographic duality
Quantum chaotic systems are often defined via the assertion that their spectral statistics coincides with, or is well approximated by, random matrix theory (RMT). In turn, the late-time ergodic behaviour of chaotic quantum systems is expected to fall into a small number of universality classes of RMT dynamics. Recent developments in holographic duality have made it possible to characterise the ergodic late-time behaviour of black holes in asymptotically anti de Sitter space in similar terms. In this talk I will start by introducing the basic ideas and notions of holographic duality (also known as AdS/CFT), before moving on to these more recent developments, mostly focussing on low-dimensional examples.
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November 16, 2021 Paweł Nurowski (Polish Academy of Sciences, Warsaw)
Split real form of Lie group G2: simple, exceptional and naturally appearing in physics
A simple exceptional Lie group G2 was predicted to exist by Wilhelm Killing in 1887. Its geometric realization appeared later, in 1893; it is due to Friedrich Engel and Elie Cartan. They realized this group as a symmetry group of two quite different geometric structures on five manifolds. In my talk I will try to explain how the idea of these G2 symmetric geometric structures could came to minds of Cartan and Engel. My explanation will be based on properties of the root diagram of G2. I will keep my lecture self contained, so I will define the G2 root diagram and will explain how to use it. After nailing down the two 5-dimensional G2 -symmetric geometries of Cartan and Engel, I will show how they appear naturally in constrained classical mechanics.
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November 9, 2021 Gaultier Lambert (University of Zurich)
Universality for free fermions point processes
I report on recent results obtained with A. Deleporte on universality of local statistics for the ground state of a free Fermi gas confined in a generic potential on ℝ^n. This model was introduced by Macchi in 1975 and it forms a determinantal point process whose kernel is the spectral projector associated to a Schrödinger operator −ℏ^2Δ+V. I will explain how to obtain the asymptotics of the kernel of this projector as ℏ→0 using methods from semiclassical analysis and discuss some probabilistic consequences. In particular, this implies universality of microscopic fluctuations, both in the bulk and around regular boundary points.
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November 2, 2021 Lea Bossmann (IST Austria)
Asymptotic expansion of low-energy excitations for weakly interacting bosons
We consider a system of N bosons in the mean-field scaling regime in an external trapping potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/N. We show that the structure of the ground state and of the non-degenerate low-energy eigenstates is preserved by the dynamics if the external trap is switched off. This talk is based on joint works with Sören Petrat, Peter Pickl, Robert Seiringer and Avy Soffer.
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October 26, 2021 Tomaž Prosen (University of Ljubljana)
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 - specifically, the so-called dual unitary circuits - where the spectral form factor is proven to match with circular ensembles of random matrix theory. The key ideas of novel methodology needed in the proofs will be discussed which are very different than the standard periodic-orbit based methods in quantum chaos of few body semiclassical systems.
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October 19, 2021 Rupert L. Frank (Caltech and LMU Munich)
Eigenvalue bounds for Schrödinger operators with complex potentials
We discuss open problems and recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We seek bounds for individual eigenvalues or sums of them which depend on the potential only through some L^p norm. While the analogues of these questions are (almost) completely understood for real potentials, the complex case leads to completely new phenomena, which are related to interesting questions in harmonic and complex analysis.
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October 12, 2021 Christophe Garban (Université Lyon 1)
Continuous symmetry breaking along the Nishimori line
I will discuss a new way to prove continuous symmetry breaking for (classical) spin systems on Z^d, d\geq 3 which does not rely on "reflection positivity". Our method applies to models whose spins take values in S^1, SU(n) or SO(n) in the presence of a certain quenched disorder called the Nishimori line. The proof of continuous symmetry breaking is based on two ingredients
1) the notion of "group synchronization" in Bayesian statistics. In particular a recent result by Abbe, Massoulié, Montanari, Sly and Srivastava (2018) which proves group synchronization when d\geq 3.
2) a gauge transformation on both the disorder and the spin configurations due to Nishimori (1981).
I will end the talk with an application of these techniques to a deconfining transition for U(1) lattice gauge theory on the Nishimori line. This is a joint work with Tom Spencer (

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October 5, 2021 Michael Loss (Georgia Institute of Technology)
Which magnetic fields support a zero mode?
I present some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. The critical quantity, is the $3/2$ norm of the magnetic field $B$. The point is that the spinor structure enters the analysis in a crucial way. This is joint work with Rupert Frank at LMU Munich.
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September 28, 2021 Rostyslav Hryniv (Ukrainian Catholic University)
Generalized soliton solutions of the Korteweg-de Vries equation
The Korteweg--de Vries (KdV) equation is a non-linear dispersive equation describing shallow-water waves and possessing many intriguing properties. One of them is existence of the so-called soliton solutions representing solitary waves travelling with constant speed and shape, as well as a special way in which several such solitons interact. Another interesting fact is that solutions of the KdV equation can be obtained as solutions of the inverse scattering problem for the family of associated Schroedinger operators, as discovered by S.Gardner, J.Green, M.Kruskal and R.Miura in 1967, and the classical soliton solutions of the KdV correspond precisely to the so-called reflectionless potentials (I.Kay and H.Moses, 1956).
The aim of this talk is two-fold. Firstly, we characterise the family of all Schroedinger operators with integrable reflectionless potentials and give an explicit formula producing all such potentials. Secondly, we use the inverse scattering transform approach to describe all solutions of the KdV equation whose initial (t=0) profile is an integrable reflectionless potential. Such solutions will stay integrable and reflectionless for all positive times and can be called generalized soliton solutions of KdV.
This research extends and specifies in several ways the previous work on reflectionless potentials by V.Marchenko, C.Remling et al. and generalized soliton solutions of the KdV equation introduced by V.Marchenko in 1991 and F.Gesztesy, W.Karwowski, and Z.Zhao in 1992. The talk is based on a joint project with B.Melnyk and Ya.Mykytyuk (Lviv Franko National University).

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September 21, 2021 Klaus Fredenhagen (Universität Hamburg)
Towards an algebraic construction of Quantum Field Theory
Algebraic quantum field theory provides a conceptual and mathematically precise framework for the analysis of structural features of quantum field theory. During the last decades, it has in addition provided a generalization of quantum field theory to generic Lorentzian spacetimes, and, moreover, an elegant reformulation of renormalized perturbative quantum field theory. The principles used there actually determine also a C*-algebraic formulation which is surprisingly rich and leads to some nonperturbative results. In particular, the time slice axiom can be derived, a version of Noether's theorem is obtained, and the renormalization group flow caused by anomalies becomes visible in the algebraic structure. The talk is based on joint work with Detlev Buchholz (2020) and on joint work with Romeo Brunetti, Michael Dütsch and Kasia Rejzner (2021).
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September 14, 2021 Yvan Velenik (Université de Genève)
Fluctuations of a layer of unstable phase in the planar Ising model
I'll first review some of the known results about phase separation, fluctuations of interfaces and (equilibrium) aspects of metastability in the planar Ising model. This will naturally lead us to consider a setup in which an interface separates a layer of unstable phase along the boundary of the system from the stable phase occupying the bulk. I'll then describe a recent work, in which we prove that, after a 1/3:2/3 scaling, the distribution of this interface weakly converges to the distribution of the stationary trajectories of an explicit Ferrari-Spohn diffusion. The proof relies on a rigorous reduction to an effective interface model, which I'll briefly sketch.
This is based on a joint work with Dima Ioffe, Sébastien Ott and Senya Shlosman; arXiv:2011.11997 .

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September 7, 2021 Pierre Germain (New York University)
Derivation of the kinetic wave equation
The kinetic wave equation (KWE) is to nonlinear waves what the Boltzmann equation is to classical particles. The KWE is also very closely related to quantum versions of the Boltzmann equation. Finally, the KWE provides an entrypoint into turbulent phenomena, since it is aimed at describing weakly turbulent flows. For all these reasons, the KWE is an interesting object, and the question of its validity a fundamental one. I will review recent progress on this question.
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August 31, 2021 Antti Knowles(University of Geneva)
The Euclidean phi^4_2 theory as a limit of an interacting Bose gas
Local Euclidean field theories over d-dimensional space have been extensively studied in the mathematical literature since the sixties, motivated by high-energy physics and statistical mechanics. For d=2, I explain how the complex scalar field theory with quartic interaction arises as a limit of an interacting Bose gas at positive temperature, when the density of the gas becomes large and the range of the interaction becomes small. The proof is based on a quantitative analysis of a new functional integral representation of the interacting Bose gas combined with a Nelson-type argument for a general nonlocal field theory. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.
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August 24, 2021 Ron Peled (Tel Aviv University)
Quantitative estimates for the effect of disorder on low-dimensional lattice systems
The addition of an arbitrarily weak random field to low-dimensional classical statistical physics models leads to the "rounding" of first-order phase transitions at all temperatures, as predicted in 1975 by Imry and Ma and proved rigorously in 1989 by Aizenman and Wehr. This phenomenon was recently quantified for the two-dimensional random-field Ising model (RFIM), proving that it exhibits exponential decay of correlations at all temperatures and all positive random-field strengths. The talk will present new results on the quantitative aspects of the phenomenon for general systems with discrete and continuous symmetries, including Potts, spin O(n) and spin glass models. The discussion is further extended to real- and integer-valued height function models driven by a random field, for which we study the fluctuations of the gradient and height variables. Among the challenges presented by the latter setup is a conjectured roughening transition in the random-field strength for the three-dimensional integer-valued random-field Gaussian free field. Joint work with Paul Dario and Matan Harel.
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August 17, 2021 Sasha Sodin (Queen Mary University London)
Lower bounds on the eigenfunctions of random Schroedinger operators in a strip
It is known that the eigenfunctions of a random Schroedinger operator in a strip decay exponentially. In some regimes, the same is true in higher dimensions. It is however not clear whether the eigenfunctions have an exact rate of exponential decay. In the strip, it is natural to expect that the rate should be given by the slowest Lyapunov exponent, however, only the upper bound has been previously established.
We shall discuss some recent progress on this problem, and its connection to a question, perhaps interesting in its own right, in the theory of random matrix products. Based on joint work with Ilya Goldsheid.

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August 10, 2021 Daniela Cadamuro (University of Leipzig)
Operator-algebraic construction of quantum integrable models with bound states
The rigorous construction of quantum field theories with self-interaction is one of the longstanding problems of Mathematical Physics. Progress in this respect has been made in integrable field theories in 1+1 spacetime dimensions. These are characterized by a factorizing scattering matrix, where two-particle interaction determines scattering completely. Specifically, some of these theories (so-called scalar models without bound states) have been successfully treated in the operator-algebraic approach, which is based on quantum fields localized in infinitely extended wedge regions. The existence of strictly localized observables is then obtained by abstract W*-algebraic arguments. This avoids dealing with the functional analytic properties of pointlike interacting fields, which are difficult to control due to the convergence problem of the infinite series of their form factors. In extension of these results, we consider S-matrices with poles in the physical strip (corresponding to the notion of `bound states’ in the quantum mechanical sense). We exhibit wedge-local fields in these models, which arise as a deformation of those in the non-boundstate models by an additive term, the so called ``bound state operator''. This technique applies to a variety of theories, e.g., the Bullough-Dodd model, the Z(N)-Ising model, the affine Toda field theories and the Sine-Gordon model. The link between these wedge-local fields and strictly local operator algebras is subject to ongoing research.
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August 3, 2021 No seminars (ICMP 2021 in Geneva)
July 27, 2021 Eveliina Peltola (University of Bonn)
On large deviations of SLEs, real rational functions, and zeta-regularized determinants of Laplacians
When studying large deviations (LDP) of Schramm-Loewner evolution (SLE) curves, we recently introduced a ''Loewner potential'' that describes the rate function for the LDP. This object turned out to have several intrinsic, and perhaps surprising, connections to various fields. For instance, it has a simple expression in terms of zeta-regularized determinants of Laplace-Beltrami operators. On the other hand, minima of the Loewner potential solve a nonlinear first order PDE that arises in a semiclassical limit of certain correlation functions in conformal field theory, arguably also related to isomonodromic systems. Finally, and perhaps most interestingly, the Loewner potential minimizers classify rational functions with real critical points, thereby providing a novel proof for a version of the now well-known Shapiro-Shapiro conjecture in real enumerative geometry.
This talk is based on joint work with Yilin Wang (MIT).

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July 20, 2021 Phan Thành Nam (LMU Munich)
Bogoliubov excitation spectrum of dilute trapped Bose gases
In 1947, Bogoliubov proposed an approximation for the low-lying eigenvalues of weakly interacting Bose gases and used that to explain Landau’s criterion for superfluidity. We will discuss a rigorous derivation of Bogoliubov’s approximation for a general trapped Bose gas in the Gross-Pitaevskii regime, where the two-body scattering process of particles leads to an interesting nonlinear effect. The talk is based on joint work with Arnaud Triay.
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July 13, 2021 Mark Malamud (Peoples Friendship University of Russia, Moscow)
On the spectral theory of Schrodinger and Dirac operators with point interactions and quantum graphs
Modern concepts of extension theory of symmetric operators, including concepts of boundary triples, corresponding Weyl functions, and boundary operators will be discussed. Applications to Schrodinger and Dirac operators with point interactions, as well as to quantum graphs, will be demonstrated. It turns out that certain spectral properties of each of these operators (deficiency indices, selfadjointness, discreteness, lower semiboundedness, etc) strictly correlate to that of a special difference operator. In the first two cases the corresponding difference operator is generated by a special Jacobi matrix. This matrix appears as a boundary operator of the corresponding Schrodinger and Dirac realization in an appropriate boundary triple. In the case of quantum graphs a similar role is played by a certain weighted discrete Laplacian on the underlying discrete graph, which also appears as a boundary operator.
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July 6, 2021 Nikolaos Zygouras (University of Warwick)
On the 2d-KPZ
The Kardar-Parisi-Zhang (KPZ) equation in dimension 1 is by now fairly well understood, both in terms of its solution theory and its phenomenology. But in the critical dimension 2, the first steps of progress have only recently been made, which shows signs of a rich structure. In particular, a phase transition is observed when the noise is suitably scaled. Below the critical scaling, the 2d KPZ coincides with the Edwards-Wilkinson universality, while at the critical scaling and beyond, its behaviour is still mysterious. We will review joint works with F. Caravenna and R. Sun as well as contributions by other groups, and expose some of the tools used, including (discrete) stochastic analysis, renewal theory, and mathematical physics ideas from the study of disorder relevance and Schrodinger operators with point interactions.
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June 29, 2021 Mathieu Lewin (Paris Dauphine University)
Density Functional Theory: some recent results
The quantum electrons in an atom or a molecule are in principle described by the Schrödinger multi-particle linear equation. Unfortunately, in most cases it is essentially impossible to find a sufficiently precise numerical approximation of the solution, due to the very high dimensionality of the problem. It is absolutely necessary to resort to approximate models, most of which being nonlinear. Density Functional Theory (DFT) is probably the most successful and widely used method in Chemistry and Physics. In this talk I will explain what DFT is and outline its mathematical formulation. I will then describe some recent results obtained in collaboration with Elliott H. Lieb (Princeton) and Robert Seiringer (IST Austria).
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June 22, 2021 Clément Tauber (University of Strasbourg)
Topological indices for shallow-water waves.
I will apply tools from topological insulators to a fluid dynamics problem: the rotating shallow-water wave model with odd viscosity. The bulk-edge correspondence explains the presence of remarkably stable waves propagating towards the east along the equator and observed in some Earth oceanic layers. The odd viscous term is a small-scale regularization that provides a well defined Chern number for this continuous model where momentum space is unbounded. Equatorial waves then appear as interface modes between two hemispheres with a different topology. However, in presence of a sharp boundary there is a surprising mismatch in the bulk-edge correspondence: the number of edge modes depends on the boundary condition. I will explain the origin of such a mismatch using scattering theory and Levinson’s theorem. This talk is based on a series of joint works with Pierre Delplace, Antoine Venaille, Gian Michele Graf and Hansueli Jud.
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June 15, 2021 Gordon Slade (UBC Vancouver)
Mean-field tricritical polymers
We provide a full description of a tricritical phase diagram in the setting of a mean-field random walk model of a polymer density transition. The model involves a continuous-time random walk on the complete graph in the limit as the number of vertices N in the graph grows to infinity. The walk has a repulsive self-interaction, as well as a competing attractive self-interaction whose strength is controlled by a parameter g. A chemical potential ν controls the walk length. We determine the phase diagram in the (g,ν) plane, as a model of a density transition for a single linear polymer chain. The proof uses a supersymmetric representation for the random walk model, followed by a single block-spin renormalisation group step to reduce the problem to a 1-dimensional integral, followed by application of the Laplace method for an integral with a large parameter. The talk is based on joint work with Roland Bauerschmidt, Probability and Mathematical Physics 1, 167-204 (2020); arXiv:1911.00395 .
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June 8, 2021 Laszlo Erdös (IST Austria)
Eigenstate thermalisation hypothesis and Gaussian fluctuations for Wigner matrices
We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with an optimal error inversely proportional to the square root of the dimension. This verifies a strong form of Quantum Unique Ergodicity with an optimal convergence rate and we also prove Gaussian fluctuations around this convergence. The key technical tool is a new multi-resolvent local law for Wigner ensemble and the Dyson Brownian motion for eigenvector overlaps.
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June 1, 2021 Benjamin Schlein (University of Zürich)
Correlation energy of weakly interacting Fermi gases
We consider Fermi gases in a mean-field regime. To leading order, the ground state energy is given by Hartree-Fock theory. The correction to the Hartree-Fock energy, produced by the many-body interaction, is known as correlation energy. In this talk, we obtain a formula for the correlation energy, based on the observation that, energetically, the main excitations of the Fermi sea are particle-hole pairs that behave, approximately, as bosons and can therefore be dealt with through (bosonic) Bogoliubov theory. The talk is based on joint works with N. Benedikter, P.T. Nam, M. Porta and R. Seiringer.
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May 25, 2021 Constanza Rojas-Molina (Cergy Paris University)
Fractional random Schrödinger operators, integrated density of states and localization
We will review some recent results on the fractional Anderson model, a random Schrödinger operator driven by a fractional laplacian. The interest of the latter lies in its association to stable Levy processes, random walks with long jumps and anomalous diffusion. We discuss the interplay between the non-locality of the fractional laplacian and the localization properties of the random potential in the fractional Anderson model, in both the continuous and discrete settings. In the discrete setting we study the integrated density of states and show a fractional version of Lifshitz tails. This coincides with results obtained in the continuous setting by the probability community. This talk is based on joint work with M. Gebert (LMU Munich).
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May 18, 2021 Marcel Griesemer (University of Stuttgart)
From Short-Range to Contact Interactions in Many-Body Quantum Systems
Many-body quantum systems with short-ranged two-body interactions, such as ultracold quantum gases are often described by simplified models with contact interactions (sometimes called delta-potentials). The merit of such models is their simplicity, solvability (to some extent) and lack of irrelevant or unknown detail. They are also toy models of renormalization. On the other hand, in dimensions $d\geq 2$ contact interactions are not small perturbations of the free energy, which makes their self adjoint implementation in the case of $N>2$ particles a subtle business. This talk gives an introduction to the mathematics of many-particle quantum systems with two-body contact interactions and their approximation by Schrödinger operators.
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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 Tyler Helmuth, Mark Holmes and I have introduced a different way to derive lace expansions based on the Symanzik random walk representation.
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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.

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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.
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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.
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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.
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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...

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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.

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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.
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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.

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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.
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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.

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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.
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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.
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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.
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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).
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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.

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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.
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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]
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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.
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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.
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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.
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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.)

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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).
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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.

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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.
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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 $1/4-\epsilon$.
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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).
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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.

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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.
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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).
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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.
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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.

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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.

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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).

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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

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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.
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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.

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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).
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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.
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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)

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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.
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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.
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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.
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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.
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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.
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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.
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May 5, 2020 Martin Hairer (Imperial College London)
The Brownian Castle
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