4 Luglio, 2022 17:00

Seminario Matematico e Fisico di Milano

The Schrödinger equation as inspiration of beautiful mathematics

Gigliola Staffilani, Massachusetts Institute of Technology

Sala Consiglio 7 piano, Edificio La Nave

abstract

abstract

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on to the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

4 Luglio, 2022 11:00

MOX Seminar

Functional Inequalities in Broken Sobolev Spaces and Applications to Polyhedral Discretization Methods

Michele Botti, MOX, Dipartimento di Matematica, Politecnico di Milano

Aula Saleri, MOX - Dipartimento di Matematica - Politecnico di Milano

abstract

abstract

In this talk, I will present several functional analysis tools that are instrumental to establish the stability and convergence analysis of nonconforming approximation methods on polyhedral meshes for a wide class of nonlinear problems. Starting from some preliminary results concerning broken Sobolev spaces, I will develop novel arguments to prove the broken version of Poincaré, Korn, trace inequalities and Sobolev embeddings. The main ingredient is a generalization of the local continuous trace inequality allowing to establish the results without restricting to piecewise polynomials or other discrete functional spaces and without requiring the definition of an interpolator mapping the discontinuous functions to continuous ones.

Contatto: paola.antonietti@polimi.it

30 Giugno, 2022 14:00

MOX Seminar

Electro-mechanical dysfunction in genetic-based cardiomyopathies

Cecilia Ferrantini, Department of Experimental and Clinical Medicine, University of Florence

Live: Aula Saleri, MOX - Dipartimento di Matematica - Politecnico di Milano
Online: mox.polimi.it/elenco-seminari/?id_evento=2173

abstract

abstract

Natural muscles provide billions of work cycles involving tension development or movement and convert the chemical energy to mechanical energy with an exceptional high efficiency. The contractile material of the cardiomyocyte may be primarily affected by gene mutations, with detrimental mechanical and energetic consequences. Faster cross-bridge detachment and increased tension cost, for instance, are hallmarks of hypertrophic cardiomyopathy, identified in human myocardial samples and recapitulated by in vitro models. In selected contexts of HCM, cellular mechanisms leading to enhanced excitability may be as, or more, important than reentry macro-substrates in predicting clinical manifestations.

29 Giugno, 2022 17:00

Seminario Matematico e Fisico di Milano

Approach to equilibrium in translation-invariant quantum systems: some structural results

Vojkan Jaksic, McGill University, Montreal, Canada

Sala di Rappresentanza, 7 piano, Dip. di Matematica, via Bonardi 9

abstract

abstract

We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase of entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase of the specific (mean) entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.

This is a joint work with C. Tauber and C.-A. Pillet

24 Giugno, 2022 14:00

MOX Seminar

Finite Neuron Method for Numerical PDEs

Jinchao Xu, Department of Mathematics at the Pennsylvania State University, University Park

Aula Saleri, VI piano, Dip. di Matematica

abstract

abstract

In this talk, I will present several results on the finite neuron method, namely a class of methods on the application of neural networks for numerical solution of partial differential equations. Topics include approximation properties of neural network functions, error estimates for the finite neuron method, comparison and analysis of different training algorithms.
Contact: alfio.quarteroni@polimi.it

16 Giugno, 2022 16:30

Seminario Matematico e Fisico di Milano

Quadratic forms arising from geometry

Giuseppe Ancona, Università di Strasburgo

Sala di Rappresentanza del Dip. di Matematica, via C. Saldini 50

abstract

abstract

The cup product on topological manifolds or the intersection product
on algebraic varieties induce quadratic forms which turn out to be a
fine invariant of these geometric objects. We will discuss some old
theorems on the signature of these quadratic forms and some
applications both of geometric and arithmetic origins. Finally we will
study an old conjecture on Grothendieck about those signatures and
explain some new evidences.

15 Giugno, 2022 17:00

Sezione di Geometria, Algebra e loro applicazioni

Fueter mapping theorem and generalized Appell polynomials in the poly-analytic setting

Kamal Diki, Chapman University

On line

abstract

abstract

The Fueter mapping theorem is a fundamental result in quaternionic analysis relating slice hyperholomorphic functions and Fueter regular ones. The action of the Fueter map on quaternionic monomials leads to an interesting class of functions forming an Appell system with respect to the hypercomplex derivative. In this talk I will present two extensions of the Fueter map in the case of polyanalytic functions of a quaternionic variable. The first map is built upon a suitable global operator with non-constant coefficients allowing to construct Fueter regular functions starting from poly-slice hyperholomorphic ones. The second map allows to construct polyanalytic Fueter regular functions. Based on this second construction we introduce and study the main properties of a new family of Generalized-Appell polynomials which are poly-Fueter regular. I will discuss also how the polyanalytic Fueter maps act on a poly slice hyperholomorphic Bargmann transform. This gives rise
to two integral transforms in the Fueter regular and polyanalytic Fueter regular setting.

14 Giugno, 2022 12:00

Sezione di Finanza Quantitativa

Cryptocurrencies and Stablecoins: market relationships & market efficiency

Emilio Barucci, Politecnico di Milano

Aula Seminari Terzo Piano

The Fueter mapping theorem is a fundamental result in quaternionic analysis relating slice hyperholomorphic functions and Fueter regular ones. The action of the Fueter map on quaternionic monomials leads to an interesting class of functions forming an Appell system with respect to the hypercomplex derivative. In this talk I will present two extensions of the Fueter map in the case of polyanalytic functions of a quaternionic variable. The first map is built upon a suitable global operator with non-constant coefficients allowing to construct Fueter regular functions starting from poly-slice hyperholomorphic ones. The second map allows to construct polyanalytic Fueter regular functions. Based on this second construction we introduce and study the main properties of a new family of Generalized-Appell polynomials which are poly-Fueter regular. I will discuss also how the polyanalytic Fueter maps act on a poly slice hyperholomorphic Bargmann transform. This gives rise
to two integral transforms in the Fueter regular and polyanalytic Fueter regular setting.

7 Giugno, 2022 15:30

MOX Seminar

Physics-Informed, Data-Driven and Hybrid Approaches to Space-Time Systems

Stefano Castruccio, University of Notre Dame

Live: Aula Saleri, VI piano Dip. di Matematica
Online: mox.polimi.it/mox-seminars/?id_evento=2169

abstract

abstract

In this talk I will discuss two different approaches to characterize space-time systems. This first one is model-driven and loosely inspired by physics, assumes that the system is locally diffusive through a stochastic partial differential equation, and can be efficiently approximated with a Gaussian Markov random field. This approximation will be used to produce a stochastic generator of simulated multi-decadal global temperature, thereby offering a fast alternative to the generation of large climate model ensembles.

The second approach is instead data-driven, and relies on (deep) neural networks in time. Instead of traditional machine learning methods aimed at inferring an extremely large parameter space, we instead rely on an alternative fast, sparse and computationally efficient echo state network dynamics on an appropriately dimensionally reduced spatial field. The additional computational time is then used to produce an ensemble and probabilistically calibrate the forecast. The approach will be used to produce air pollution forecasts from a citizen science network in San Francisco and forecasting wind energy in Saudi Arabia.

Towards the end of the presentation, I will discuss how these two broad frameworks could be used in synergy to allow for improved predictability and understanding of space-time systems whose physical understanding is currently limited and/or largely influenced by parametrizations.

7 Giugno, 2022 14:15

Sezione di Analisi

Noncommutative Geometry by example

Fabio Cipriani, Politecnico di Milano

Aula Seminari III Piano e online: polimi-it.zoom.us/j/95645334076?pwd=M1R1WmUvNno5bTduZkRZVk5RRSt0dz09

abstract

abstract

We will illustrate various examples of spaces which appear to be singular when analyzed by the classical tools of topology, measure theory, analysis and geometry, but which can be naturally approached by instruments of Noncommutative Geometry. Among them we will find instances of duals of non abelian groups, Clifford algebras of Riemannian manifolds, leafs space of foliations, orbit spaces of dynamical systems, orbifolds, spaces of observables and states in Quantum Mechanics, quasi-crystals, quantum information channels and quantum groups.