Dipartimento di Matematica

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We reflect on the MSCA-IF project MiDiROM, presenting its successes and challenges over the past two years, and discussing the remaining open questions. The project's main objective was to develop efficient computational methods for modeling processes in fractured porous media, by combining mixed-dimensional modeling with deep learning approaches.

This talk focuses on two prominent research themes within MiDiROM. Firstly, we developed multipoint mixed finite element methods for rotation-based formulations of Stokes flow and Biot poroelasticity. By using exact discrete complexes and a carefully selected quadrature rule, we obtained efficient and stable discretization methods using the lowest-order Raviart-Thomas finite element pair.
Secondly, we introduced reduced order modeling techniques for Darcy flow systems capable of generating locally mass conservative flux fields. These methods combine concepts from reduced basis methods, exact discrete complexes, and trained neural networks. Mass conservation is then guaranteed through a "spanning tree solve" and appropriate projection operators.

To conduct numerical simulations, we simultaneously developed the Python package PyGeoN, in which discrete complexes are explicitly available. The conclusion of the talk will emphasize various capabilities of this open-source software, such as mixed-dimensional differential operators and a polygonal mesh generator using aperiodic monotiles.

Contatto:
alessio.fumagalli@polimi.it

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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.

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La ricerca operativa (RO) è una branca della matematica applicata solitamente insegnata a livello universitario. Tuttavia, negli ultimi anni sono state sviluppate diverse iniziative per introdurre la RO agli studenti delle scuole secondarie. A queste iniziative si aggiunge ROAR (Ricerca Operativa Applicazioni Reali), un corso triennale composto da tre unità didattiche, pensato per gli studenti del triennio delle scuole secondarie. In questo seminario ci si concentrerà sulla prima unità didattica, mettendo in luce sia gli aspetti contenutistici sia le metodologie utilizzate per la sua implementazione con gli studenti. In particolare, si delineeranno gli aspetti teorici che hanno guidato la progettazione e poi si mostreranno esempi di problemi di ottimizzazione proposti agli studenti, che mirano a sviluppare le loro capacità di modellizzazione e problem-solving.

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The classification of minimal surfaces of general type is a classical and long-standing research topic. In this context fixing invariants turns out to be fundamental.
In this talk we will review some recent results on the case pg=q=2, which is still widely open in spite of several contributions by many authors over the last two decades. More specifically, given a minimal surface S of general type with pg=q=2, it turns out that the self-intersection K^2 of the canonical divisor K is 4

Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics.

His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE).

In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).

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In this talk we consider the finite element approximation of some mixed formulations of linear elasticity, namely, the displacement-pressure, the displacement-stress and the displacement-pressure-stress approaches. As for any mixed formulation, involving unknowns belonging to different functional spaces, the global inf-sup stability that ensures well posedness of the problem is a consequence of “little” inf-sup conditions that need to be satisfied between the interpolating spaces of the different unknowns. Approximations satisfying these conditions are sometimes difficult to implement, and even very rare, as in the case of the displacement-pressure-stress approach. The alternative is to modify the discrete problem by adding stabilisation terms to the Galerkin equations that yield stable approximations for any choice of the interpolating spaces. We discuss this possibility and show that one may prove stability and convergence in the same norms as for the continuous problem.

Contatto: simona.perotto@polimi.it

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Verranno osservati e discussi alcuni fenomeni di interazione fluido-struttura della vita quotidiana. I comportamenti sia dei fluidi che delle strutture sembrano mettere in discussione i principi elementari del calcolo vettoriale.

David Keyes directs the Extreme Computing Research Center at the King Abdullah University of Science and Technology (KAUST), where he was a founding Dean in 2009 and currently serves in the Office of the President as Senior Associate. He is a professor in the programs of Applied Mathematics, Computer Science, and Mechanical Engineering. He is also an Adjunct Professor of Applied Mathematics and Applied Physics at Columbia University, where he formerly held the Fu Foundation Chair. He works at the interface between parallel computing and PDEs and statistics, with a focus on scalable algorithms that exploit data sparsity. Before joining KAUST, Keyes led multi-institutional scalable solver software projects in the SciDAC and ASCI programs of the US Department of Energy (DoE), ran university collaboration programs at US DoE and NASA institutes, and taught at Columbia, Old Dominion, and Yale Universities. He is a Fellow of SIAM, the AMS, and the AAAS. He has been awarded the Gordon Bell Prize from the ACM, the Sidney Fernbach Award from the IEEE Computer Society, and the SIAM Prize for Distinguished Service to the Profession. He earned a B.S.E. in Aerospace and Mechanical Sciences from Princeton in 1978 and a Ph.D. in Applied Mathematics from Harvard in 1984.

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Numerical software is being reinvented to provide opportunities to tune dynamically the accuracy of computation to the requirements of the application, resulting in savings of memory, time, and energy. Floating point computation in science and engineering has a history of “oversolving” relative to expectations for many models. So often are real datatypes defaulted to double precision that GPUs did not gain wide acceptance until they provided in hardware operations not required in their original domain of graphics. However, computational science is now reverting to employ lower precision arithmetic where possible. Many matrix operations considered at a blockwise level allow for lower precision and many blocks can be approximated with low rank near equivalents. This leads to smaller memory footprint, which implies higher residency on memory hierarchies, leading in turn to less time and energy spent on data copying, which may even dwarf the savings from fewer and cheaper flops. We provide examples from several application domains, including a look at entries that earned Gordon Bell Prize finalist status in 2022 and 2023.

Contatti:
alfio.quarteroni@polimi.it
paola.antonietti@polimi.it

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Il tema dell'argomentazione, nella sua declinazione nei diversi ambiti disciplinari, è chiaramente espresso nelle indicazioni Nazionali, ma spesso disatteso nella pratica scolastica.
Educare all'argomentazione presenta problemi didattici specifici che, nell'ambito matematico, si articolano con i problemi didattici relativi sia alla costruzione dei concetti che all'introduzione ad una prospettiva teorica, ed in particolare alla pratica della dimostrazione.
Da qualche decennio la ricerca in didattica della matematica affronta i problemi che emergono in ambito scolastico in relazione all'argomentare, al dimostrare e al loro mutuo rapporto.
Il seminario ha l'obiettivo di discutere alcuni di questi problemi didattici proponendo, alla luce dalla ricerca in questo ambito, qualche esempio di buone pratiche da sviluppare in classe.

Sauro Succi serves as Senior Research Executive and Principal Investigator at the Center for Life Nano-Neuro Sciences at la Sapienza of the Italian Institute of Technology and Research Affiliate of the Physics Department of Harvard University. He holds a degree in nuclear engineering from the University of Bologna and a PhD in physics from EPFL. His research activity covers a broad range of topics related to the mathematical modeling and simulation of complex states of flowing matter, such as thermonuclear plasmas, fluid turbulence, soft matter as well as quantum and subnuclear fluids such as quark gluon plasmas. He is best known for his contributions to the early inception, development and application of the Lattice Boltzmann method, for which he has received a number of international awards, including the Alexander von Humboldt Award in Physics (2002), the Raman Chair of the Indian Academy of Sciences (2012), the APS Aneesur Rahman Prize in Computational Physics (2017), and the CECAM Berni Alder Prize (2019) for exceptional contributions to the microscopic simulation of matter. He is also a ERC Advanced Grant and PoC awardee (2017,2022), an elected member of Academia Europaea (2015) and a Honorary Professor at University College London (2022).

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We present a pedagogical introduction to the current state of quantum computing algorithms for the simulation of classical fluids. Different strategies, such as quantum nonlinear ODE solvers, Carleman-Koopman linearization, functional Liouville formulations and variational quantum eigensolvers, will be briefly discussed and commented on. We shall then focus on two distinct quantum computing implementations of the Carleman-Koopman procedure, one based on quantum linear algebra solvers and the other on a second-quantized Dirac-Majorana formulation of the Lattice Boltzmann equation. Finally, we shall offer a brief mid-term perspective of quantum versus exascale computing for fluid flows.

Contatto: alfio.quarteroni@polimi.it

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Quale forma assume una catena omogenea appesa ai suoi estremi quando è sottoposta alla sola forza peso? Nel XVII secolo questa domanda ha impegnato le menti di grandi scienziati, che hanno individuato nella cosiddetta catenaria (una curva esprimibile in termini di funzioni iperboliche) la risposta. Per via di questa sua particolarità, essa è stata utilizzata in architettura, soprattutto da Antoni Gaudí. In questo seminario vedremo qualche esempio di questo utilizzo e mostreremo che la catenaria svolge un importante ruolo anche nella descrizione delle forme assunte dalle pellicole di acqua saponata.

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L’introduzione del calcolo letterale nell’ultimo anno della scuola secondaria di primo grado spesso si rivela come il momento in cui si manifesta la distanza tra l’essere in grado di svolgere una procedura e l’averla compresa, e che in genere segna una svolta irreversibile verso l’abbandono di interesse e impegno da parte degli studenti. Una delle cause dell’insorgere di questa distanza è la relativa facilità d’uso del linguaggio algebrico e delle sue regole sintattiche, che rischia di oscurarne la profondità semantica. Nel seminario si intende proporre alcuni spunti di riflessione affinché l’introduzione della “lettera” sia un passaggio di maturazione nell’acquisizione del linguaggio della Matematica nella pienezza della sua potenzialità.

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This talk is concerned with the notion of eigenstates of an operator in an abstract C*-algebra. After reviewing some basic and structural results, I will explore the possibility of reinterpreting certain typical concepts of quantum mechanics (dynamical equilibrium states, ground states, gapped states, Fermi surfaces) in terms of (algebraic) eigenstates.


This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027". This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

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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.

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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.

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In this talk, we will review some recent results on the modelling, analysis and numerical analysis of Koiter’s model for linearly elastic shells in the case where the shell under consideration is constrained to remain confined in a prescribed half-space.

To begin with, we will see how to recover the constrained Koiter’s model starting from the classical three-dimensional equations of linearized elasticity in the case where the shell under consideration is an elliptic membrane or a flexural shell.

Contatti: paola.antonietti@polimi.it e vittorino.pata@polimi.it

Giuseppe Savaré started his scientific career as a researcher at the CNR Institute for Numerical Analysis in Pavia and has been Professor of Mathematical Analysis since 2000, first at the University of Pavia and then, since 2020, at Bocconi University in Milan.
During 2019-2023 he was Hans-Fischer Senior Fellow at the Institute for Advanced Study of the Technical University of Munich. In 2011 he received the Ennio De Giorgi prize awarded by the Italian Mathematical Union.
His research activity concerns variational methods for evolution problems, in particular gradient flows and rate-independent models, and their interplay with optimal transport, measure theoretic tools and analysis in metric spaces.
In collaboration with Luigi Ambrosio and Nicola Gigli, he has written a monograph on Gradient Flows in Metric Spaces and in the Space of Probability Measures.

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Optimal transport theory studies the optimal strategies for transferring a given initial distribution of resources (mathematically represented by a measure) to a final configuration in a way that minimizes the cost of transport.
First addressed mathematically by G. Monge in 1781, its first modern formulation (1942) earned L.V. Kantorovich the Nobel Prize in Economics in 1975.
Later, thanks to many outstanding theoretical, applied and numerical contributions, the theory has been developed in many directions including, in addition to economics, linear and stochastic programming, statistics and probability, mean field models, functional analysis, PDEs, Riemannian geometry, machine learning.
The seminar aims to give a quick overview of some of the classical results and more recent contributions, with applications to evolution problems.

Contatto: marco.verani@polimi.it

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I will present some results about the asymptotic behavior of a class of stochastic reaction-diffusion-advection equations in the plane. I will show that as the divergence-free advection term becomes larger the solutions of such equations converge to the solution of a suitable stochastic PDE defined on the graph associated with the Hamiltonian. I will deal with the case when the stochastic perturbation is given by a singular spatially homogeneous Wiener process taking values in the space of Schwartz distributions. As in previous works, I will assume that the derivative of the period of the motion on the level sets of the Hamiltonian does not vanish. Time permitting, without assuming this condition on the derivative of the period, I will study a weaker type of convergence for the solutions of a suitable class of linear SPDEs.

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We consider the incompressible Euler equations in two or three dimensions and we show that the addition of a suitable multiplicative It\^o noise with superlinear growth prevents a smooth solution from blowing up in finite time. The result is valid for a more general hyperbolic-type SPDE. The proof is based on the Lyapunov function method.
Joint work with Marco Bagnara and Fanhui Xu.

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In this talk we discuss the Grothendieck-Lidskii formula in quaternionic Hilbert spaces, along with the particularities of the problem, which include, among others, the definition of an appropriate trace that differs from the usual one.
We will also briefly discuss r-nuclear operators in quaternionic Banach (or, more generally, locally convex) spaces X and the associated Grothendieck-Lidskii's formula for the case r greater or equal 2/3. In order to establish these results, the use of the aforementioned traces (and their invariance with respect to the basis choices) is essential.
To conclude, we briefly discuss how the (seemingly ad hoc) introduced trace actually arise as the canonical form in tensor products of quaternionic vector spaces. This is a joint work with P. Cerejeiras, F. Colombo, U. Kähler, and I. Sabadini.