Schedule

Convergence of the renormalised model for the generalised KPZ equation

Yvain Bruned 

In this talk, we will present the convergence of the renormalised model for the generalised KPZ equation via local transformations that are governed by preparation maps. The main idea is an extension of a result on the convergence of a class Feynman diagrams given by Hairer and Quastel. With this extension, one is able to perform local transformations that make appear the renormalisation given for a model defined recursively via preparation maps in the context of Regularity Structures. This approach works both in the discrete and continuous settings  and could lead to a general convergence theorem. 

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Feynman path integrals for the Schrödinger equation with magnetic field

Sonia Mazzucchi

Since their introduction in the early 40s, Feynman path integrals have always been a powerful tool for theoretical physics on the one hand and a mathematical challenge on the other. Despite decades of effort, a definitive mathematical theory of Feynman path integration is still missing and while some steps have been taken in this direction, there are fundamental issues that still deserve further investigation. Remarkably, even rather simple quantum systems, such as a non-relativistic particle moving in an external magnetic field, lead to non trivial problems appearing in the Feynman path-integral construction of quantum dynamics. In this talk I shall give an overview of this topic with a historical perspective, highlighting recent developments and some open problems.

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Spectral theory of high-contrast random media

Matteo Capoferri

The talk is concerned with the rigorous mathematical description of propagation and localisation of waves in a particular class of composite materials with random microscopic geometry, called micro-resonant (or high-contrast) random media: small inclusions of a “soft” material are randomly dispersed in a “stiff” matrix. The highly contrasting physical properties of the two constituents, combined with a particular scaling of the inclusions, result in microscopic resonances, which manifest macroscopically by allowing propagation of waves in the material only within certain ranges of frequencies (band-gap spectrum).

High-contrast media with periodically distributed inclusions have been ex-tensively studied and numerous results are available in the literature. However, their stochastic counterparts, which model more realistic scenarios and may exhibit localisation, are far from being well understood from a mathematical viewpoint. In my talk I will give an overview of existing results through the prism of stochastic homogenisation and spectral theory, and discuss recent advances and ongoing work.

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Enhanced Schauder estimates for families of distributions

Francesco Caravenna

We investigate the regularising properties of singular kernels at the level of germs, i.e. families of distributions indexed by points in the Euclidean space. We first construct a suitable integration map which acts on germs that are so-called coherent. Then we focus on germs that can be decomposed along a basis (corresponding to the so-called modelled distributions in Regularity Structures) and we prove a version of Hairer's multilevel Schauder estimates in this setting, with minimal assumptions.

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Microlocal Analysis and Quantum Field Theory 

Nicola Pinamonti

During this talk we will recall how the use of micorlocal analysis permits to understand in a mathematically rigorous way the perturbative construction of interacting quantum (scalar) field theories on flat and curved backgrounds. In particular, we will characterise the wave front set of the propagators and of the two-point functions of free quantum field theories.

Hence, using the Hormander criterion of multiplication of distributions we will define an algebra generated by all possible interaction Lagrangians. The properties of the product in this algebra encodes the results of Wick theorem. Furthermore, micorlocal analysis, permits to prove that the ambiguities present in the construction of time ordered products among various interaction lagrangians are local.

Thanks to this observation, the procedure of renormalization, needed in the construction of time ordered products, can be understood as a problem of extending distribution to certain submanifolds of non vanishing  codimension. In the physical literature the ambiguities present in this procedure are called renormalization freedom. This freedom can be classified in a similar way as in the problem of extending homogeneous distributions. 

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Stochastic Differential Equations and the Martin-Siggia-Rose formalism: an algebraic correspondence 

NIcolò Drago

The Martin-Siggia-Rose (MSR) formalism is an efficient way to compute expectation values and correlation functions of a given stochastic differential equation (SDE). This method is based on a formal path integral approach and it is typically used in the physics community as a useful alternative to standard perturbative computations. However, there appears to be no mathematically rigorous proof of the correspondence between the results computed in the MSR approach and those computed in the standard perturbative setting. In this talk we will address this issue using techniques proper of the algebraic approach to quantum field theories: The latter provide a valuable framework to discuss rigorously the path integral formulation of field theories as well as the solution theory of SDE. In particular, working in this framework, we establish a correspondence, at the level of perturbation theory, between correlation functions and expectation values computed either in the SDE or in the MSR formalism.

Joint work with A. Bonicelli and C. Dappiaggi.

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On the multi-indices approach to rough paths and regularity structures 

Lorenzo Zambotti

I will present some results on the approach to rough paths and regularity structures recently developed by Linares, Otto and Tempelmayr and based on multi-indices rather than on trees. While they insisted on using a pre-Lie approach, we argue that a post-Lie structure is much more natural, as noted also by Bruned and Katsetsiadis.

Joint work with Jean-David Jacques.

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A stochastic analysis approach to the quantization of subcritical fermionic fields 

Francesco De Vecchi

We propose the construction of subcritical fermionic quantum fields through backward-forward noncommutative stochastic partial differential equations. This construction is an anticommutative generalization of the analogous variational method formulated for bosonic fields, and it is inspired by the so-called continuous renormalisation group. 

The talk is based on a joint work with Luca Fresta and Massimiliano Gubinelli. 

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Parabolic stochastic quantisation of the fractional Φ^4_3 model in the full subcritical regime 

Paolo Rinaldi

We construct the fractional Φ^4 Euclidean quantum field theory on ℝ^3 in the full subcritical regime via parabolic stochastic quantisation. Our approach is based on the use of a truncated flow equation for the effective description of the model at sufficiently small scales and on coercive estimates for the non-linear stochastic partial differential equation describing the interacting field.

Based on a joint work with P. Duch and M. Gubinelli.

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