### Seminari

### Prossimi Seminari

- Nonintrusive reduced order models using physics informed neural networks

Jan S. Hesthaven, Chair of Computational Mathematics and Simulation Science, EPFL, Lausanne, CH

giovedì 29 ottobre 2020 alle ore 14:00 precise, Online seminar: https://mox.polimi.it/elenco-seminari/?id_evento=1979&t=763724

### Seminari Passati

- The LLV decomposition for hyper-Kaehler cohomology

Radu Laza, Stony Brook

venerdì 12 luglio 2019 alle ore 14:30, Sala di Rappresentanza, Via C. Saldini 50ABSTRACTLooijenga—Lunts and Verbitsky (LLV) have shown that the cohomology of

a compact hyper-Kaehler manifold admits the action of a big Lie

algebra g, generalizing the usual sl(2) Hard Lefschetz action. We

compute the LLV decomposition of the cohomology for the known classes

of hyper-Kaehler manifolds (i.e. K3^n, Kim_n, OG6, and OG10). As an

application, we easily recover the Hodge numbers of the exceptional

example OG10. In a different direction, we establish the so-called

Nagai’s conjecture (on the nilpotency index for higher degree

monodromy operators) for the known cases. More interestingly, based

on the known examples, we conjecture a new restriction on the

cohomology of compact hyper-Kaehler manifolds, which in particular

implies the vanishing of the odd cohomology as soon as the second

Betti number is large enough relative to the dimension.

This is joint work with M. Green, Y. Kim, and C. Robles.

- 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 MILANOABSTRACTIn 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 32-34ABSTRACTThe 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 32-34ABSTRACTI 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 pianoABSTRACTFluid-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° pianoABSTRACTIn 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 31-33ABSTRACTWetting 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° pianoABSTRACTA 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.