Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of nonlinear Hamiltonian PDEs. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari.
In the second part of the talk, we will explain a general principle that allows us to obtain almost sure global solutions for Hamiltonian PDEs provided that one has a stationary probability measure. In this context, stationarity refers to a solution of the associated Liouville equation. This more general notion replaces the invariance from before. The second part of the talk is joint work with Zied Ammari and Shahnaz Farhat.

Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

A classical result in approximation theory due to Korovkin asserts that a sequence of positive unital maps on C([0,1]) converges pointwise to the identity if they merely converge to the identity on the functions 1,x,x^2. This result was later generalized by Saskin, who showed that convergence to the identity on a generating function system implies convergence to the identity everywhere if and only if the system has full Choquet boundary. Arveson's last open conjecture in his seminal work on non-commutative boundary theory predicts that a non-commutative analogue of Saskin's result holds. We refute Arveson's conjecture with an elementary counterexample. All notions will be explained during the talk.

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

During this talk we discuss the emergence of secular growths in the correlation functions of interacting quantum field theories when treated with perturbation methods. It is known in the literature that these effects are present if the interaction Lagrangian density changes adiabatically in a finite interval of time. If this happens, the perturbative approach cannot furnish reliable results in the evaluation of scattering amplitudes or in the evaluation of various expectation values. We show, during this talk, that these effects can be avoided for adiabatically switched-on interactions, if the spatial support of the interaction is compact and if the background state is suitably chosen. In particular, this is the case when the background state is chosen to be at equilibrium and when thermalisation occurs at late time. The same result holds also if the background state is only invariant under time translation or if the explicit time dependence is not too strong, in a precise sense which will be discussed in the talk.

Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

Digital models (DMs) are designed to be replicas of systems and processes. At the core of a digital model (DM) is a physical/mathematical model that captures the behavior of the real system across temporal and spatial scales. One of the key roles of DMs is enabling “what if” scenario testing of hypothetical simulations to understand the implications at any point throughout the life cycle of the process, to monitor the process, to calibrate parameters to match the actual process and to quantify the uncertainties. In this talk, we will present various (faster than) real-time Scientific Deep Learning (SciDL) approaches for forward, inverse, and UQ problems. Both theoretical and numerical results for various problems including transport, heat, Burgers, (transonic and supersonic) Euler, and Navier-Stokes equations will be presented.

When quantum fields are represented as operators on a Hilbert space, their two-point distributions naturally give rise to distributions of positive type. A number of basic results on such distributions, especially for translation invariant two-point distributions, have been known for a long time. E.g., the Bochner-Schwartz Theorem fully characterises translation invariant distributions of positive type. In this talk I will present two apparently new results on distributions of positive type, one pertaining to pointwise products and the other to methods for cutting and pasting. Both results were motivated by physical questions and extend the toolbox of theoretical physics. I will present the results in a general mathematical context before discussing their applications to quantum energy inequalities and to separable states for a free scalar QFT.

  We study quantum principal bundles on projective varieties using a sheaf theoretic approach. Differential calculi are introduced in this context. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. We next introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. The examples of principal (covariant) calculi on the quantum principal bundles SL_q(2,C) and GL_q(2,C) over the projective space P¹(C) are presented.

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions when blowup points are either regular points or non-quantized singular sources. In particular the uniqueness result covers the most general case extending or improving all previous works. For example, unlike previous results, we drop the assumption of singular sources being critical points of a suitably defined Kirchoff-Routh type functional. Our argument is based on refined estimates, robust and flexible enough to be applied to a wide range of problems requiring a delicate blowup analysis. In particular we come up with a major simplification of previous uniqueness proofs. This is a joint work with Daniele Bartolucci and Wen Yang.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Optimization problems subject to PDE constraints form a mathematical tool that can be applied to a wide range of scientific processes, including fluid flow control, medical imaging, option pricing, biological and chemical processes, and electromagnetic inverse problems, to name a few. These problems involve minimizing some function arising from a particular physical objective, while at the same time obeying a system of PDEs which describe the process. It is necessary to obtain accurate solutions to such problems within a reasonable CPU time, in particular for time-dependent problems, for which the “all-at-once” solution can lead to extremely large linear systems. In this talk we consider iterative methods, in particular Krylov subspace methods, to solve such systems, accelerated by fast and robust preconditioning strategies. In particular, we will survey several new developments, including block preconditioners for fluid flow control problems, a circulant preconditioning framework for solving certain optimization problems constrained by fractional differential equations, and multiple saddle-point preconditioners for block tridiagonal linear systems. We will illustrate the benefit of using these new approaches through a range of numerical experiments. This talk is based on work with Santolo Leveque (Scuola Normale Superiore, Pisa), Spyros Pougkakiotis (Yale University), Jacek Gondzio (University of Edinburgh), and Andreas Potschka (TU Clausthal). This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

  Like many discrete statistical mechanics models, stochastic PDEs can exhibit a “critical dimen-sion” beyond which their large-scale behaviour is expected to be trivial (i.e. governed by Gaussian fluc-tuations). While such sweeping heuristics allow us to formulate rather precise conjectures, there are relatively few cases where these have actually been proven. In this talk, we will mainly focus on the KPZ equation, a standard model of interface fluctuations. There has recently been substantial progress in our mathematical understanding of its large-scale behaviour in the supercritical regime.

I will report on a joint work with Amine Marrakchi. Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra that admits a state with trivial centralizer must be a type III_1 factor, but the converse remained open. I will present a solution of this problem, proving that such ergodic states form a dense G_delta set among all normal states on any III_1 factor with separable predual. Through Connes' Radon-Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which I will discuss in the second half of the talk.

Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)