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Direttore Vicario: Prof. Gabriele Grillo
Responsabile Gestionale: Dr.ssa Franca Di Censo

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  • Preconditioning of multiphysics problems with applications to the biomechanics of the brain
    Kent - Andre Mardal, University of Oslo and Simula Research Laboratory
    giovedì 11 luglio 2019 alle ore 14:00, Aula consiglio VII piano, Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO
    ABSTRACT
    In this talk we will discuss preconditioning algorithms for monolithic schemes of coupled problems involving the coupling of porous and viscous flow as well as fluid-structure interaction and dimension reduction problems.
    We explore how fractional Laplacian solvers may be utilized to obtain parameter robust schemes. The schemes are discussed in the context of biomechanical modelling of the waste clearance processes in the brain that is believed to fail in various forms of dementia such as Alzheimer's and Parkinson's diseases.

    Contact: paolo.zunino@polimi.it

  • Implementing zonal harmonics with the Fueter principle
    Amedeo Altavilla, Università Politecnica delle Marche
    lunedì 8 luglio 2019 alle ore 14:00, Aula Seminari del 3 piano, Edificio La Nave, Via Ponzio 32-34
    ABSTRACT
    The Fueter theorem is a well studied result in hypercomplex analysis which gives a procedure to construct hyperholomorphic functions (in some sense), starting from complex holomorphic ones.
    In this talk I will show how to adapt such result in order to obtain formulas to represent real zonal harmonics in every dimension.
  • Shift invariant subspaces of the quaternionic space of slice $L^2$ functions
    Alessandro Monguzzi, Università degli Studi di Milano Bicocca
    lunedì 8 luglio 2019 alle ore 15:00, Aula Seminari del 3 piano, Edificio La Nave, Via Ponzio 32-34
    ABSTRACT
    I will give a characterization of the closed shift invariant subspaces of the quaternionic space of slice $L^2$ functions. As a consequence, the inner-outer factorization theorem for the quaternionic Hardy space $H^2$ on the unit ball is obtained. Therefore, I will present some properties of inner and outer functions in the quaternionic setting, providing in particular sufficient conditions as well as necessary ones for functions to be inner or outer.

    This talk is based on joint works with Giulia Sarfatti and Daniel Seco.
  • A scalable computational framework for large-scale simulation of fluid-driven fracture propagation
    Bianca Giovanardi, MIT, Cambridge, MA, USA
    venerdì 5 luglio 2019 alle ore 14:00, aula Saleri VI piano
    ABSTRACT
    Fluid-driven crack propagation concerns several areas of engineering, including structural, geotechnical, and petroleum engineering. The development of simulation tools for pressurized cracks propagating in realistic scenarios needs to tackle the complexity arising from the non-linear hydro-mechanical coupling of the fluid and the cracked solid in a suitable framework for large-scale applications.
    In this talk, I will introduce the governing equations of hydraulic fracture and discuss well-known propagation regimes where the prob- lem can be treated analytically. I will then present a computational ap- proach to model the hydro-mechanical coupling of the fracturing solid and the fluid flow inside the propagating cracks. Benchmarks in 2D and 3D demonstrate the capability of the computational framework to successfully deal with a priori unknown and arbitrarily intricate crack paths, and with the need for large-scale simulations.
    Time permitting, I will describe other recent efforts in developing advanced computational algorithms for large-scale simulation of complex material response, including elastic instabilities and material failure.


    Contatto: edie.miglio@polimi.it
  • A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis
    Colette De Coster, Univ. Valenciennes
    martedì 2 luglio 2019 alle ore 15:15, Aula seminari 3° piano
    ABSTRACT
    In this talk, we survey some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation
    \begin{equation*}
    {\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},
    \end{equation*}
    in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$, with $a,b>0$ parameters.
    This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids.

    In this talk, we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.


    This seminar is organized within the PRIN 2015 Research project «Partial Differential Equations and related Analytic-Geometric Inequalities» Grant Registration number 2015HY8JCC _003, funded by MIUR – Project coordinator Prof. Filippo Gazzola
  • The mathematics of spreading droplets
    Lorenzo Giacomelli, Università La Sapienza, Roma
    lunedì 24 giugno 2019 alle ore 16:30 precise, Sala Consiglio, 7 piano, Edificio La Nave, Via Ponzio 31-33
    ABSTRACT
    Wetting phenomena at small scales are an area where chemistry,
    physics, mathematics, and engineering intersect. In recent years,
    driven also by molecular dynamics, new concepts have been introduced
    to describe the statics and dynamics of wetting, allowing new
    insights into the old problems of surface forces. Among these
    problems, two prominent ones are an appropriate mathematical modeling
    of the moving contact line where liquid, solid, and surrounding vapor
    meet, and how such models influence the macroscopic properties of the
    flow. After a general framing -- the classical setting of droplets'
    statics and dynamics, diffuse and sharp interface models, classical
    and new descriptions of the contact line -- I shall review the PDE
    theory for one of such models -- the so-called thin-film equation --, mainly focusing on the two aforementioned problems and on some of the most interesting current challenges.
  • Pure Traction Problems between Linear and Finite Elasticity
    Franco Tomarelli, Politecnico di Milano
    mercoledì 19 giugno 2019 alle ore 15:15, Aula seminari 3° piano
    ABSTRACT
    A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energies of a hyperelastic material body subject to an equilibrated force field.
    We show that the strains of minimizing sequences associated to re-scaled nonlinear energies weakly converge, up to subsequences, to the strains of minimizers of a limit energy, provided an additional compatibility condition is fulfilled by the force field. The limit energy functional exhibits a gap that makes it different from the classical linear elasticity functional; nevertheless the compatibility condition entails the coincidence of related minima and minimizers. A strong violation of this condition provides a limit energy which is unbounded
    from below, while a mild violation may produce unboundedness of strains and a limit energy which has infinitely many extra minimizers which are not minimizers of standard linear elastic energy. A consequence of this analysis is that a rigorous validation of linear elasticity fails for compressive force fields that infringe such compatibility condition.
  • A new approach to quantum mechanics
    Yakir Aharonov, Chapman University
    martedì 18 giugno 2019 alle ore 16:00 precise, Sala Consiglio, 7 piano, Edificio La Nave, Via Ponzio 31-33
    ABSTRACT
    In my talk, I will discuss a reformulation of quantum mechanics in which each quantum system at any time is described by two Hilbert space vectors rather than one. One of the vectors propagates from past boundary condition towards the present and the other propagates back to the present from a future boundary condition. I will show that this reformulation uncovers a host of fascinating new physical phenomena and new mathematics. Examples of the former, i.e. new physical phenomena, are Weak Measurements and Weak Values. Examples of the latter, i.e. new mathematics, are superoscillations. Both examples will be described in detail within this talk.