Antonino De Martino, Politecnico di Milano, Spectral theories on the S-Spectrum, Tuesday, December 12, 2023, time 15:15, Aula Seminari - III Piano
Abstract:Abstract:
One of the deepest results in hypercomplex analysis is the Fueter extension theorem. It gives a two steps procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step, this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, was widely studied by McIntosh and collaborators. In this talk, I will discuss the main notions of the S-spectrum and some concepts of the monogenic functional calculus. Moreover, I will also give some ideas on the new research direction of the fine structures.

Fabio Zanolin, Università di Udine, An introduction to chaotic dynamics, with some applications, Thursday, November 16, 2023, time 15:45, Aula Seminari - III Piano (nell'ambito delle iniziative del Dipartimento di Eccellenza)
Abstract:Abstract:
In this lectures we will present a brief introduction to chaotic dynamics, together with some applications, according to the following outline: 1) A brief historical introduction 2) Different concepts of chaos 3) A paradigmatic example: the "Bernoulli shift" 4) The Smale horseshoe 5) Topological horseshoes 6) Fixed points and periodic points for contractive/expansive maps 7) A topological method for seeking chaotic dynamics 8) "Linked Twist Maps" 9) Some applications, topological fluid mixing, fluid stirring

Fabio Zanolin, Università di Udine, An introduction to chaotic dynamics, with some applications, Tuesday, November 14, 2023, time 15:45, Aula Seminari - III Piano (nell'ambito delle iniziative del Dipartimento di Eccellenza)
Abstract:Abstract:
In this lectures we will present a brief introduction to chaotic dynamics, together with some applications, according to the following outline: 1) A brief historical introduction 2) Different concepts of chaos 3) A paradigmatic example: the "Bernoulli shift" 4) The Smale horseshoe 5) Topological horseshoes 6) Fixed points and periodic points for contractive/expansive maps 7) A topological method for seeking chaotic dynamics 8) "Linked Twist Maps" 9) Some applications, topological fluid mixing, fluid stirring

Luca Gennaioli, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Asymptotics as s -> 0+ of the fractional perimeter on Riemannian manifolds, Thursday, October 26, 2023, time 14:15, Aula seminari MOX - VI piano
Abstract:Abstract:
In this work we study the asymptotics of the fractional Laplacian as s -> 0+ on any complete Riemannian manifold (M, g), both of finite and infinite volume. Surprisingly enough, when M is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on M. As a corollary, we can find the asymptotics of the fractional s-perimeter on (essentially) every complete manifold, generalizing both the existing results: the classical result for Rn by Dipierro-Figalli-Palatucci-Valdinoci (2012) and the recent one for the Gaussian space by Carbotti-Cito-La Manna-Pallara (2021). In doing so, from many sets E contained in M we are able to produce a bounded harmonic function associated to E, which in general can be non-constant.

Ángel Castro, Instituto de Ciencias Matemáticas (Madrid), Traveling waves near shear flows, Monday, July 03, 2023, time 15:15, Aula Seminari III piano (nell'ambito delle iniziative del Dipartimento di Eccellenza)
Abstract:Abstract:
In this talk we will consider the existence of traveling waves arbitrarily close to shear flows for the 2D incompressible Euler equations. In particular we shall present some results concerning the existence of such solutions near the Couette, Taylor-Couette and the Poiseuille flows. In the first part of the talk we will introduce the problem and review some well known results on this topic. In the second one some of the ideas behind the construction of our traveling waves will be sketched.

Luigi Berselli, Università di Pisa, Energy conservation or anomalous dissipation for incompressible fluids, Thursday, June 29, 2023, time 15:15, Aula Seminari III piano (nell'ambito delle iniziative del Dipartimento di Eccellenza)
Abstract:Abstract:
We provide an overview and propose elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Holder continuous functions. Our focus is on exploring the interplay between space and time regularity. Additionally, we delve into the potential extension of these results to the Navier-Stokes equations in the presence of a solid boundary. Specifically, we consider the case of Dirichlet boundary conditions and our approach avoids any additional assumptions on the kinematic pressure.