### Seminari

### Prossimi Seminari

- A mathematical-physics approach to machine learning

Pierluigi Contucci, Dipartimento di Matematica Università di Bologna

giovedì 30 gennaio 2020 alle ore 14:00, Aula Saleri VI piano - Modeling and simulation of thermo-poroelastic processes in fractured geothermal reservoirs

Eirik Keilegavlen, Department of Mathematics, University of Bergen, Norway

giovedì 20 febbraio 2020 alle ore 11:30, Aula Saleri - VI piano

### Seminari Passati

- 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. - 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. - 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 31-33ABSTRACTIn 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. - ON A MATHEMATICAL THEORY OF REPEATED QUANTUM MEASUREMENTS

Vojkan Jaksic, McGill University

giovedì 13 giugno 2019 alle ore 14:30 precise, Aula Seminari III pianoABSTRACTThe statistics of the (finite alphabet) outcomes of repeated quantum measurements is studied by methods of thermodynamic formalism. Viewed as one-dimensional spin systems with long range interactions, repeated quantum measurements exhibit very rich (and sometimes very singular) thermodynamic behaviour. We will describe general thermodynamical formalism of these systems and illustrate its unexpected features on a number of examples. - Multiphase flow in heterogeneous geologic media

Stephan K. Matthai, Chair of Reservoir Engineering, Department of Infrastructure Engineering, The University of Melbourn

giovedì 13 giugno 2019 alle ore 14:00, Aula consiglio VII piano, Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANOABSTRACTHeterogeneous permeability, porosity, capillary pressure and mechanical properties are key characteristics of sedimentary and volcanic rocks with sufficient interconnected pores to permit fluid flow. In many cases, heterogeneities are nested, for instance, where mm-scale laminations define the internal structure of bed-forms that combine into facies associations and facies distributions on the km scale. Where such rocks have experienced an overprint by deformation or reactive fluid flow, initial heterogeneities might have been amplified even further. Thus, fracturing can impart long-range spatial correlations on the permeability structure, strongly amplifying permeability anisotropy, even without a noticeable increase in the porosity.

This presentation examines the impact of such nested heterogeneities on flow, (reactive) transport, and multiphase fluid displacement in natural heterogeneous porous media. The emphasis will be on intermediate scale features (~0.1 - 500-m in size) that are difficult to investigate in the laboratory or in the subsurface because flow patterns cannot be resolved by geophysical imaging techniques. The investigative method used here are field-data based hybrid FEM-FVM numeric simulations where an operator splitting technique is employed, solving elliptic-parabolic equations (fluid pressure, temperature, chemical diffusion) with the FEM and the hyperbolic ones (multiphase flow and tracer transport) with the FVM.

The presentation will illustrate the behaviour of the heterogeneous systems and the numeric methods used to model them, proceeding from tracer transport, to gas injection and water flooding of naturally fractured reservoirs to maximise hydrocarbon recovery. The presentation will conclude with a general discussion of the impact of heterogeneity, addressing questions such as: given the nested nature of geologic heterogeneity, is the spreading of gas plumes ergodic? – or – in view of the impact of heterogeneity on sweep, how can gas storage or hydrocarbon recovery be maximised?

Contatto: paolo.zunino@polimi.it - 'Strange' rational maps and spectra of graphs and groups

Rostislav Grigorchuk, Texas A&M University

giovedì 13 giugno 2019 alle ore 16:30 precise, 'Aula U5-3014 (Edificio 5, terzo piano) del Dipartimento di Matematica e Applicazioni dell'Università di Milano-Bicocca, Via Cozzi 55ABSTRACTI will present a few results concerning spectral theory of

graphs and groups that are based on the use of multidimentional rational

maps and self-similar groups. It will be explained how these maps

appear, how they help to compute the spectrum, and why they are

``strange'' (or better to say, non typical). As part of the story there

will be a short survey on spectra of infinite graphs and groups with

some old and new results belonging to the speaker and his collaborators

(L.Bartholdi, A.Dudko, D.Lenz, T.Nagnibeda, B.Simanek, A.Zuk). The

relation to the random Schrodinger operator will be mentioned at the end

of the talk if time permits. - Symmetry preservation for fourth order eigenvalue optimization problems

Francesca Colasuonno, Università degli Studi di Torino

mercoledì 12 giugno 2019 alle ore 15:15, Aula seminari 3° pianoABSTRACTIn this talk, we will discuss some recent results on two eigenvalue optimization problems governed by the biharmonic operator, under Dirichlet or Navier boundary conditions. From a physical point of view, in two dimensions our problem corresponds to building a plate, of prescribed shape and mass, out of materials with different densities --varying in a certain range of values-- in such a way to minimize the lowest frequency of the body. This problem is also referred to as composite plate problem. Both for the clamped plate (i.e., Dirichlet b.c.) and for the hinged plate (i.e., Navier b.c.), we will prove the existence of an optimal configuration and give an explicit representation of the minimizing densities in terms of sublevel sets of their corresponding first eigenfunctions. Finally, we will discuss symmetry preservation properties of the optimal configurations, in the presence of some symmetry and convexity of the domain. The tools used differ depending on the boundary conditions: while the hinged plate problem inherits the maximum principles for second order elliptic systems, allowing us to exploit the moving plane method to get symmetry preservation in more general domains, the situation is more complicated for the clamped plate problem, where we will use the polarization technique and the properties of the Green's function to deal with radial symmetry preservation in a ball.

This talk is based on two joint papers with Eugenio Vecchi (Trento).