Mathematical Physics Group

Seminars 2022-23

Oct 28th: Anatoly Nikitin (Institute of Mathematics of NAS of Ukrain)

Title: Superintegrable Systems with Position Dependent Mass Invariant w.r.t. Dilatation Transformations

Abstract: Classical Lie symmetries of fundamental equations of quantum mechanics are revised. higher order integrals of motion for both the Schroedinger-Pauli equations and Shroedinger equations with position dependent masses are classified. The related superintegrable quantum mechanical systems which appear to be supersymmetric  are discussed.

Jan 18th: Davide Astesiano (Iceland U.) (Slides)

Title: Extended Rotating Sources in General Relativity

Abstract: We discuss the exact class of solutions for stationary self gravitating dust in general relativity. Subsequently, we discuss the weak field approximation, arriving at the gravitoelectromagnetic limit.These solutions of Einstein's equations in the weak-field and slow-motion approximation can be studied to investigate the impact of General Relativity on galactic dynamics. We propose an analytic toy model with explicit applications to some galactic models to show that they may introduce non-negligible corrections to the Newtonian velocity profile.

Mar 16th: Alessandro Tomasiello (Milano B.)

Title: Gravity, Entropy and Optimal Transport

Abstract: The mathematical field of Optimal Transport gives a way of understanding the Einstein equations in terms of the evolution of the entropy of probability distribution of particles. This is particularly useful for gravity compactifications, where the warping function is unified with the internal Ricci tensor. Moreover, this field provides a way to think about curvature in a 'synthetic' way, that still applies in presence of singularities. As a concrete application, I will present several theorems on the masses of KK spin-two fields, which are relevant for gravity localization and for scale separation. 

Mar 23rd: Giovanni Ortenzi (UniTo)

Title: Catastrophes and Blowups for Homogeneous Euler Equations

Abstract: The homogeneous Euler equations are obtained neglecting the pressure in the classical fluid dynamics Euler equations. The interest for this simple nonlinear 3D model has been pointed out by Ze'ldovich (1970) during the study of the large scale structure of the universe. From the mathematical point of view, this PDE is a multidimensional generalization of the Riemann-Hopf equation, the prototype for the study of gradient catastrophes.In this talk I will present a complete blowup characterization for the homogeneous Euler equations.

Apr 3rd: Pierre Bieliavsky (UC Louvain)

Title: Symmetric Spaces and Quantization

Abstract: I will present some (possibly unexpected) implications of symmetric space theory in Wentzel–Kramers–Brillouin analysis and quantization.

Apr 20th: Fabrizio Nieri (UniTo)

Title: Exact Methods and Hidden Integrable/Algebraic Structures in Supersymmetric Gauge Theories

Abstract: Classical Lie symmetries of fundamental equations of quantum mechanics are revised. higher order integrals of motion for both the Schroedinger-Pauli equations and Shroedinger equations with position dependent masses are classified. The related superintegrable quantum mechanical systems which appear to be supersymmetric  are discussed.

May 4th: Rita Fioresi (UniBo)

Title: Geometric structure of Data Through Deep Learning

Abstract: A Deep Learning classifier can see a low-dimensional Riemannian manifold structure on data. Such structure comes via the local data matrix, a variation of the Fisher information matrix, where the role of the model parameters is taken by the data variables. We obtain a foliation of the data domain and we show that the dataset on which the model is trained lies on a leaf, the data leaf, whose dimension is bounded by the number of classification labels. We validate our results with some experiments with the MNIST dataset: paths on the data leaf connect valid images, while other leaves cover noisy images. At the end of the talk, we give a Cartan Geometric perspective and new geometric questions on the data manifold.

May 8th: Antonio Racioppi (National Institute of Chemical Physics and Biophysics Tallinn)

Title: Slow-roll Inflation in Palatini F(R) Gravity

Abstract: We study single field slow-roll inflation embedded in Palatini F(R) gravity. In contrast to metric F(R), when rewritten in terms of an auxiliary field and moved to the Einstein frame, Palatini F(R) does not develop a new dynamical degree of freedom. However, it is not possible to analytically solve the constraint equation of the auxiliary field for a general F(R). We propose a method that allows us to circumvent this issue and compute the inflationary observables. Moreover, we prove that Palatini F(R)'s which, for infinite curvature, diverge faster than R^2, have a universal limit described by a Palatini quadratic gravity where the Einstein-Hilbert term has the wrong sign. Such a configuration is a powerful tool in order to realize hilltop inflaton potentials.

Jun 1st: Pietro Grassi (UniPO and INFN)

Title: The Geometry of Supergravity

Abstract: I will introduce some of the tools used in supergravity and in supersymmetric theories such as superspaces and supermanifolds. Then I will discuss their geometry and I will define some new differential operators needed for supergravity. Finally, I will show how supergravity can be rewritten in a purely geometric way using the language of differential forms.

Jun 22nd: Donato Bini (CNR and INFN)

Title: Gravitational Scattering in a Two-body System: Conservative and Radiative Aspects

Abstract: The scattering angle in hyperboliclike encounters contains gauge-invariant information associated with the conservative dynamics of a binary system (like the energy, the angular momentum, the redshift, etc.). The use of these information  has led to the determination of  the 5PN (modulo 2 unknown) first and the 6PN (modulo 2+4 unknowns)  Hamiltonian for the conservative motion of the system (TuttiFrutti approach). Radiative effects in the scattering  have been the focus of intense recent investigations which will be also summarized here.