Speakers, titles and abstracts

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Speakers, titles and abstracts

Nathanael Berestycki (Vienna)

Can you hear the shape of LQG? We obtain a Weyl law for the eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows linearly with n, with the proportionality constant given by the Liouville area of the domain (times a certain deterministic constant depending on γ∈ (0, 2)). At the heart of the proof we obtain estimates of independent interest on the small-time behaviour of the on-diagonal heat kernel. Interestingly, we show that the scaled heat kernel displays nontrivial pointwise fluctuations, which however vanish when integrating over a domain. This is joint work in preparation with Mo-Dick Wong (Durham).

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Sergio Conti (Bonn)

Dislocations are topological singularities of the strain field, which play a crucial role in the plastic deformation of crystals. Their models offer a interesting playground in the calculus of variations. I shall give an overview of recent progress in the variational analysis of dislocation models in linear and finite elasticity, of their relation to discrete models, and in the derivation of line-tension models via Gamma convergence. The talk is based on joint work with Adriana Garroni, Stefan Müller and Michael Ortiz.
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Margherita Disertori (Bonn)

We study the Integrated Density of States (IDS) of the random Schrödinger operator appearing in the study of certain reinforced random processes in connection with a supersymmetric sigma-model. We rely on previous results on the supersymmetric sigma-model to obtain lower and upper bounds on the asymptotic behavior of the IDS near the bottom of the spectrum in all dimension. We show a phase transition for the IDS between weak and strong disorder regime in dimension larger or equal to three, that follows from a phase transition in the corresponding random process and supersymmetric sigma-model. In particular, we show that the IDS does not exhibit Lifshitz tails in the strong disorder regime, in stark contrast with other disordered systems, like the standard Anderson model. This is joint work with V. Rapenne, C. Rojas-Molina and X. Zeng.
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Hugo Duminil-Copin (IHES and Geneva)

We will present the scaling relations between the critical exponents β, γ, δ, η, ν, ζ as well as α (when α⋝0) for the random-cluster model on the square lattice. As a key input, we show and use the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing.
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Clement Erignoux (INRIA Lille Nord)

The facilitated exclusion process is a degenerate kinetically constrained lattice gas, in which particles can hop to an empty neighboring site provided that another neighboring site is also occupied. Because of the kinetic constraint, the system in dimension 1 exhibits two phases, a frozen phase at density less than 1/2, and a diffusive phase at density larger than 1/2. For this reason, standard entropic tools cannot be used to straightforwardly derive its macroscopic behavior, in particular its hydrodynamic limit. To workaround this issue, we exploit a classical mapping between exclusion and zero-range processes, and deduce from the hydrodynamic limit of the zero-range process that of the exclusion process. Despite the lack of smoothness of the various macroscopic limits, thanks to the attractiveness of the zero-range process, this allows to derive the hydrodynamic limits in both the symmetric and asymmetric case. Based on JW With O. Blondel, M. Sasada, M. Simon and L. Zhao
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Marco Falconi (Politecnico Milano)

In this talk I will discuss a new approach to global well-posedness in the classical electrodynamics of extended charges in interaction with the electromagnetic field, that stems from the proof of Bohr's correspondence principle in quantum electrodynamics. Based on a joint work with Z. Ammari and F. Hiroshima.
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Davide Fermi (Politecnico Milano)

We address the problem of characterizing the ground states of a 2D Ising model with nearest-neighbor ferromagnetic and long-range dipolar interactions. Numerical evidence and varia- tional computations support a conjecture maintaining that, for sufficiently large values of the ferro- magnetic coupling constant, the competition between the two kinds of interaction produces striped periodic ground states. We present an analytic proof of the fact that such striped configurations are indeed minimizers within the variational class of states whose domain walls consist of arbitrary collections of horizontal and vertical straight lines. The proof relies on Block Reflection Positivity and appropriate spin flips. Joint work with Alessandro Giuliani (Roma Tre University).
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Christophe Garban (Lyon 1)

I will start by introducing and motivating the (two-component) Coulomb gas on the d-dimensional lattice Z^d. I will then present some puzzling properties of the fluctuations of this Coulomb gas. The connection of this model with integer-valued fields and compact-valued spin systems will be emphasised through the talk. This is based on joint works with Avelio Sepúlveda and David Dereudre.
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Adriana Garroni (Roma Sapienza)

The talk will focus on the role played by material defects in determination of the surface tension carried by grain boundaries in polycrystals. Defects induce elastic distortion in the bulk, therefore a variational model accounting for both, elastic energy and presence of defects, as the one proposed by Lauteri and Luckhaus, is the starting point for our analysis.
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Rafael Greenblatt (Roma Tor Vergata)

I will discuss the evolution of many-body fermionic lattice models with finite-range interactions starting in thermal equilibrium and subject to a weak time-dependent perturbation varying on a specified time scale. Under suitable assumptions on the form of the perturbation, expectation values of local observables can be approximated by a convergent series expressed in terms of truncated Euclidean correlation functions of the initial equilibrium state; this makes it possible to bound the difference from a Gibbs state of the instantaneous (perturbed) Hamiltonian. Assuming that these correlation functions decay in a way that can be verified for many weakly-interacting models with a spectral gap, the convergence of the series is uniform in the system size and temperature, and the range of allowed time scales grows as the temperature is taken to zero, leading to a many-body adiabatic theorem in the zero temperature limit. Based on arXiv:2211.16836; joint work with Markus Lange, Giovanna Marcelli, and Marcello Porta.
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Antti Kupiainen (Helsinki)

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas.
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Vieri Mastropietro (Università di Milano)

The anomaly cancellation is a basic property of the Standard Model, crucial for its consistence. We present a rigorous proof that the anomaly cancels in a finite lattice regularization of the U(1) Electroweak sector with non-compact massive gauge fields, up to subleading corrections and under the same condition as in the continuum. The analysis uses a method which explores the analogy with universality in quantum materials and is based on expansions convergent up to a cut-off of the order of the inverse coupling, such that non-perturbative effects are excluded.

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Domenico Monaco (Roma Sapienza)

Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost stationary state, we give a new proof that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime. In particular, we prove that, in quantum Hall systems and Chern insulators, the transverse response current is quantized up to any order in the strength of the inducing electric field. The latter is introduced as a perturbation to a periodic, spectrally gapped equilibrium Hamiltonian by means of a linear potential; existing proofs of the exactness of Kubo formula rely instead on a time-dependent magnetic potential. The result applies to both continuum and discrete crystalline systems modelling the quantum (anomalous) Hall effect. This is joint work with Giovanna Marcelli [Lett. Math. Phys 112 (2022) / arXiv:2112.03071].
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Cyrill Muratov (Pisa)

We present a variational treatment of confined magnetic skyrmions in a minimal micromagnetic model of ultrathin ferromagnetic films with interfacial Dzylashinksii-Moriya interaction (DMI) in competition with the exchange energy. Under Dirichlet boundary conditions, we prove existence of topologically non-trivial energy minimizers that concentrate on points in the domain as the DMI strength parameter tends to zero. Furthermore, we derive the leading order non-trivial term in the Γ-expansion of the energy in the limit of vanishing DMI strength that allows us to completely characterize the limiting magnetization profiles and interpret them as particle-like states whose radius and position are determined by minimizing a renormalized energy functional. In particular, we show that in our setting the skyrmions are strongly repelled from the domain boundaries, which imparts them with stability that is highly desirable for applications.
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Bruno Nachtergaele (UC Davis)

A salient feature of topological insulators and symmetry protected topological phases of quantum lattice systems is the spectral gap above the ground state. I will review recent results of two types. The first are proofs of lower bounds for the spectral gap for a variety of specific models. The second type of results is concerned with the stability of the spectral gap and other fundamental characteristics of the system under uniformly small perturbations of the interactions in the model Hamiltonian. I will discuss the basic strategies used to approach both types of problems.
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Sébastien Ott (Fribourg)

In the study of spin O(N) models, a successful heuristic argument (vastly used in physics) is that the model is locally behaving like a Gaussian field at low enough temperature. This idea is usually referred to as the "spin wave picture". A way to make sense of that picture is to imagine that the spin O(N) model is (at least formally) a Gaussian Free Field at 0 temperature, and that positive temperatures are a "perturbation" of that Gaussian field. A useful conclusion of that idea is that one can (still formally) expand quantities of interest (like the free energy or correlation functions) as an asymptotic power series in T (the temperature) with the 0th coefficient being the Gaussian one. In this talk, I will first introduce the problem in more detail and present an approach based on reflection positivity to (rigorously) perform such expansions, therefore supporting the spin wave picture. The approach is initially due to Bricmont, Fontaine, Lebowitz, Lieb, and Spencer in the context of the XY model and was recently extended to general Ns (in dimension at least 3) in a joint work with A. Giuliani.
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Felix Otto (MPI Leipzig)

The ripple in a thin-film ferromagnet is the magnetization's anisotropic response to poly-crystallinity; it is experimentally well-documented. The randomly oriented grains come with a favorite axis for the magnetization that acts as a random field, a source of quenched noise. On a mesoscopic level, the ripple is modeled by a 2-d anisotropic and non-local variational problem with a random field term converging to white noise. White noise itself is too rough for the Burgers' type nonlinearity to have a classical sense, so that a renormalization is called for. We follow the strategy of rough path/regularity structures in the sense that we enhance the noise description. The main result is that of universality: The mesoscopic ensemble of the ripple is independent on how the limit of white noise is approached, it is thus independent of the statistics of the grain arrangement. In view of the non-convexity of the variational problem, this statement is formulated in terms of a suitable Γ-topology. On the technical side, we appeal to the spectral gap inequality to avoid the assumption of Gaussianity of the approximation. We also leverage a fluid-type identity for the Burgers equation to obtain coercivity of the renormalized variational problem. This is joint work with Radu Ignat, Tobias Ried, and Pavlos Tsatsoulis.
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Marcello Porta (SISSA Trieste)

We consider the edge transport properties of a class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model, a QFT for interacting chiral fermions in 1+1 dimensions with linear dispersion and arbitrary velocities. Furthermore, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the vanishing of the beta function for the emergent multi-channel Luttinger liquid, combined with lattice Ward identities for the microscopic 2d theory. Joint work with Vieri Mastropietro.
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Robin Reuvers (Roma Tre)

In 1963, Lieb and Liniger formulated an exactly solvable model for interacting bosons in 1D. Thanks to its exact Bethe ansatz solution, the model and its generalizations soon became popular objects of study. Later, when new techniques allowed for the creation of (quasi-)1D systems in the lab, Lieb and Liniger's model found experimental use and became even better known. In the meantime, mathematical physicists had been studying interacting bosons in 2 and 3D. Without the availability of exact solutions, rigorous results were much more difficult to obtain, and a popular goal was the derivation of the ground state energy of Bose gases in various settings in 2 and 3D. Many of these efforts focused on the dilute limit, in which the density of the gas is very low. Somehow, Bose gases in 1D were not studied in this way, most likely because the Lieb-Liniger model provides an exactly solvable set-up. Nevertheless, we can use insights from the 2 and 3D approaches to prove results about the ground state energy of dilute Bose and Fermi gases in 1D. In the talk, I will review the developments above and explain our results.
Slava Rychkov (IHES)

Rigorous results, algorithms, and numerics will be presented. (Joint work with Tom Kennedy.)
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Florian Theil (Warwick)

An interesting question is how much crystalline order is destroyed by the presence of dislocations. Analysing the Ariza-Ortiz model we can demonstrate that it accounts for the presence of temperature and grain boundaries. For small angle grain boundaries, the Read-Shockley law emerges as an upper bound. We conjecture that the corresponding lower bound holds as well.
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Daniel Ueltschi (Warwick)

The set of Gibbs states is known to be a Choquet simplex, that is, a convex set where any element can be written as convex combination of extremal points. The difficulty is to identify the extremal Gibbs states for given models. I will review the general theory in the context of quantum spin systems and I will discuss a kind of characteristic function. Calculations are possible for mean-field systems. There are many open questions.
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Simone Warzel (TUM Munich)

I will explain a classification and formulae for the spectral gap of mean-field spin Hamiltonians. The tool for their derivation is a semiclassical analysis based on spin-coherent states. For gapped systems a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian will be given hence justifying fluctuation theory at zero temperature for this case. I will also explain a shift caused by the spherical geometry in semiclassical spin space in these second-order approximations. (Based on joint work with C. Manai).
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