12.01.26
Registration 9.30 - 10.00
Cyril Labbé
This talk will focus on the random Schrodinger operator on R obtained by perturbing the Laplacian with a white noise. First, I will present an Anderson localization result for this operator: the spectrum is pure point and the eigenfunctions exponentially localized. Second, I will present various results on the eigenvalues and eigenfunctions of the finite-volume approximation of this operator in the infinite volume limit: it turns out that many different behaviors can be observed according to the energy regime one focuses on. This is based on a series of joint works with Laure Dumaz.
11.00 - 11.30 Coffee Break
Ilya Chevyrev
In this talk, I will describe a family of observables for 3D quantum Yang-Mills theory based on regularising connections with the YM heat flow. I will describe how these observables can be used to show that there is a unique renormalisation of the stochastic quantisation equation of YM in 3D that preserves gauge symmetries. This complements a recent result on the existence of such renormalisations. Our analysis is based on short time expansions of SPDEs and of regularised Wilson loops, and requires a careful balance between the running time of the dynamic and the regularisation parameter coming from the YM heat flow. Based on joint work with Hao Shen.
12.30 - 14.00 Lunch Break
Fabio Toninelli
The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijren, Kutner and Spohn as a mesoscopic model for driven diffusive systems with one conserved quantity. In the subcritical dimension d=1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to an anisotropic Stochastic Heat equation. At the critical dimension d=2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections. This talk is based on the work https://arxiv.org/abs/2501.00344 joint with Giuseppe Cannizzaro and Quentin Moulard where we pin down the logarithmic superdiffusivity by identifying exactly the large-time asymptotic behaviour of the so-called diffusion matrix and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem. This is the first superdiffisive scaling limit result for a critical SPDE, beyond the weak coupling regime.
Filippo Nava
In this talk, I will present an ongoing work with D. Janssen and K. Rejzner on the gluing problem in quantum electromagnetism. Using a relative differential cohomology approach, we construct a space of field configurations for free electromagnetism that is compatible with gluing, and we discuss their dynamics. We construct an algebra of observables, roughly corresponding to smeared Wilson lines; in this setting, we prove that given a (nice enough) partition of a spacetime manifold, the algebra of observables on the full spacetime can be reconstructed by the algebras on the partitions. Eventually, we discuss its quantisation and the consequences on the superselection sectors.
Lorenzo Pettinari
Condensed Bose gases can be effectively described in terms of quasi-particles, commonly referred to as phonons. Their dynamics is captured by a c-number condensate Hamiltonian consisting of a quadratic term supplemented by third- and fourth-order perturbative corrections. These additional interaction terms render the phonons unstable, giving rise to two distinct decay processes known as Beliaev and Landau damping. From a mathematical perspective, such decay mechanisms should manifest as a broadening of the Bogoliubov dispersion relation in the thermodynamic limit.
To validate this picture, I will present two different approaches to deriving the phonon decay rates. The first is inspired by the W ^*-algebraic framework of Jakšić–Pillet, employing Standard Representations and perturbative expansions of a suitably chosen vector state. The second method is based on the analysis of two-body correlation functions. Both approaches yield the same imaginary correction to the Bogoliubov dispersion relation, which in turn determines the expected broadening. Furthermore, our approaches offer a new perspective on the decay of phonons in terms of the left and right components of these quasi-particles.
The talk is based on joint work with Jan Dereziński and may be viewed as a modern elaboration of the classical contributions of Beliaev, Hohenberg–Martin, and others.
Stefano Galanda
In this talk we present the construction of equilibrium states at positive temperature, with and without Bose-Einstein condensation, for a non-relativistic Bosonic QFT (gas of Bose particles) on $\mathbb{R}^3$, interacting through a localised two body interaction. We use methods of quantum field theory in the algebraic formulation and of quantum statistical mechanics in the operator algebraic setting to obtain this result. Moreover, in order to prove convergence of the correlation functions, we rearrange the perturbative series introducing an auxiliary stochastic Gaussian field which mediates the interaction of the Bosonic field. Limits where the localisation of the two-body interaction is removed are eventually discussed in combination with other regimes. This talk is based on a collaboration with Nicola Pinamonti.
16.30 - 17.00 Coffee Break
Harprit Singh
First I shall motivate the study of SPDEs in heterogeneous environments using the examples of the $\Phi^4$ and the parabolic Anderson model. Then, after recalling the solution theory of the dynamical $\Phi^4$ and g-PAM equation, I shall discuss the periodic homogenisation problem for these singular SPDEs. This is, in part, joint work with M. Hairer.
20.00 Social Dinner
13.01.26
Nicolò Drago
In this talk, we present new sufficient conditions for subcriticality in classical and quantum spin lattice systems, expressed through the uniqueness of Kubo-Martin-Schwinger (KMS) states. Our approach relies on a non-commutative analogue of the Kirkwood-Salzburg equations combined with a novel decomposition of local observables. These techniques significantly broaden the range of interactions to which the criteria apply and yield improved lower bounds on the subcritical inverse temperature. The framework is robust enough to handle models without any assumptions on the single-site potentials. This is joint work with L.Pettinari and C. J. F. van de Ven.
11.00 - 11.30 Coffee Break
Michele Schiavina
I will discuss the Hamiltonian formulation of local field theories on manifolds with corners (eg. a Cauchy lens) in the particular, yet common, case in which they admit an equivariant momentum map. In the presence of corners, the momentum map naturally splits into a part encoding “Cauchy data” (aka constraints), and a part encoding the “flux” across the corner (aka Noether charges).
This decomposition plays an important role in the construction of the reduced phase space, which is conveniently handled via symplectic reduction by stages, adapted for the occasion to local group actions.
The output of this analysis are natural Poisson structures associated to corners, leading to the concept of (classical) flux superselection sectors as their symplectic leaves, and providing a roadmap to "bottom-up" quantum superselection. If time permits, I will exemplify how this picture explains the existence of Strominger's "soft" symmetries and memory effects in electrodynamics.
12.30 - 14.00 Lunch Break
Lorenzo Zambotti
In this talk I want to explain as simply as I can the reason why some stochastic partial differential equations, like e.g. KPZ, need a procedure called renormalisation. This consists in a modification of the equation by adding new non-linearities that typically contain diverging constants. This is strongly related to a similar phenomenon in quantum field theory and its explanation in the context of SPDEs is probably more intuitive. Unfortunately a complete description of the procedure requires an algebraic machinery that most probabilists prefer to avoid. However the beauty of this technique resides also in the deep interplay between analysis, algebra and probability.
Francesco Caravenna
We investigate the Stochastic Heat Equation (SHE) in the critical space dimension two, where it is ill-defined. A non-trivial solution, known as the critical 2D Stochastic Heat Flow (SHF), can be constructed through regularisation and renormalisation. We investigate the SHF in the strong-disorder regime, showing that it vanishes locally and identifying the spatial scale governing the transition from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, which shed light into the SHE regularised via space-time discretisation: when the disorder strength is kept fixed, the solution exhibits fluctuations on a superdiffusive scale. Our proof refines classical change of measure and coarse-graining techniques, introducing new ideas of independent interest.
Based on joint works with Quentin Berger and Nicola Turchi.
Alessio Ranallo
Euclidean field theories have been extensively studied in the mathematical literature since the sixties, motivated by high-energy physics and statistical mechanics. Formally, such a theory is given by a Gibbs measure associated with a Euclidean action functional over a space of distributions. It has been shown that certain Euclidean field theories arise as high-density limits of interacting Bose gases at positive temperature, providing a rigorous derivation of them starting from a realistic microscopic model of statistical mechanics. Focusing on quartic local interaction in two dimensions, I will discuss their emergence of such a field theory as a limit of an inhomogeneous interacting Bose gas. Based on a joint work with Cristina Caraci, Antti Knowles, and Pedro Torres Giesteira.
Alberto Bonicelli
In this talk, after justifying the expansion of the semigroup of a one-dimensional Itô diffusion as a power series in time, I will build on previous results on expansions labelled by exotic rooted trees to derive an explicit expression for the combinatorial factors involved. A key step is the extension of the notion of tree factorial and Connes-Moscovici weights to this richer family of rooted trees. As a result, we obtain an exotic Butcher series representation of the semigroup, suitable for a comparison with the perturbative path integral construction of the statistics of the diffusion, known in the literature as Martin-Siggia-Rose formalism. Computations in the latter framework are based on the erroneous assumption that the measure of the path integral can be seen as a perturbation of a Gaussian measure. Resorting to multi-indices to represent pre-Feynman diagrams, I will shed some light on why, even if starting from such an assumption, the results are correct.