Dipartimento di Matematica

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During its late stage of development, the Volvox embryo, which is a sphere made of a single layer of cells, turns itself inside out and forms complex folds along the way. We want to reveal the hidden mechanism behind it using the mathematical approach together with experimental data. We develop a new morphoelastic shell model, and derive algebraic-differential equations for the growth functions incorporating the experimental data into the coefficients. We assume a special scaling of the pressure, and construct asymptotic analytical solutions for the growth functions and the residual stress distribution of the cell sheet is also obtained. We find that the cell sheet uses two different growth rates to generate the folds: one that makes the fold shapes without causing tension in the shell, and another that produces tension to balance the pressure without changing the shapes. We also discover how the folds move along the cell sheet, which is like a race with two runners going in opposite directions. We study the residual stress and find out how it signals the coordination of the movement of the two folds.

Contatto:
pasquale.ciarletta@polimi.it

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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This talk will start with a brief review of the early history of the expression of Werner Heisenberg’s famous (1927) Uncertainty Principle for position (Q) and momentum (P) in Quantum Mechanics. In his 1927 paper, Heisenberg only gave a qualitative formulation of the Uncertainty Principle, but within a few months, the physical community settled on a mathematical expression of this principle in terms of an inequality first presented by E.H. Kennard, wherein the product of the standard deviations of Q and P is greater or equal than half of the (reduced) Planck constant. This is still the most common and well-known version of an uncertainty relation today. Nevertheless, it will be shown in this talk that Kennard’s inequality, is not strong enough to provide a foundation for most of the common examples used in textbooks as illustrations of the Uncertainty Principle. I will argue, therefore, that conceptually stronger inequalities are needed for the purpose. I will review several approaches to obtain such inequalities: the Landau & Pollak inequalities (1961); the entropic uncertainty inequality (Bialynicki-Birula & Mycielski,1975) and a statistico-geometric approach to the uncertainty relation by (Hilgevoord & Uffink,1991), and discuss their strengths and weaknesses vis-a-vis the common textbook examples.

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Solving differential problems using full order models (FOMs), like the finite element method, incurs prohibitively computational costs in real-time simulations and multi-query routines. Reduced order modeling aims at replacing FOMs with reduced order models (ROMs), that exhibit significantly reduced complexity while retaining the ability to capture the essential physical characteristics of the system.
Traditional ROMs, rooted in the assumption of linear modal superimposition, such as proper orthogonal decomposition (POD), can prove inadequate when addressing nonlinear time-dependent parametrized partial differential equations (PDEs), especially for scenarios involving coherent structures evolving over time. In response, an alternative approach is proposed, based on deep learning (DL) algorithms, leveraging tools like convolutional neural networks (CNNs), to construct an efficient nonlinear surrogate. In the resulting DL-ROM both the nonlinear trial manifold and the nonlinear reduced dynamics are learned non-intrusively through DL models trained on a dataset of FOM snapshots corresponding to different parameter values. Accuracy and efficiency of the DL-ROM technique are evaluated across various applications, demonstrating the capability to perform real-time computations for new queries.
After presenting some recent achievements in this field, a quick overview of the ongoing research, sketching a list of open issues of potential interest, will be provided.

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Neurodegenerative disorders are an increasing health and societal issue worldwide. Many of these diseases (such as Alzheimer’s and Parkinson’s) are related to an impairment of the waste clearance system in the brain. Indeed, in physiological conditions, brain activity produces waste proteins that are collected and removed by the Cerebrospinal Fluid (CSF), the main component of the glymphatic system. In this talk, I present a multiphysics model coupling Stokes equations for the CSF flow in the brain ventricles and subarachnoid space with dynamic Multiple-Network Poroelastic (MPE) equations for the perfusion of the brain tissue by CSF and blood [Corti et al. 2023; Causemann et al. 2022]. The model is discretized by a discontinuous Galerkin method based on polytopal grids, which can naturally comply with the geometrical complexity of the domain. An a priori analysis of the method is provided, together with numerical tests in realistic geometries. Then, perspectives are discussed on further developments of the model, its numerical discretization, and its employment to address clinical questions.

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The present talk deals with numerical techniques to discretize wave propagation problems. In the first part of the talk I will focus on the interior Helmholtz equation as well as on scattering problems. Due to oscillations in the analytical solutions, accurate finite element approximations are computationally expensive and time-consuming, already for moderate frequencies. Therefore, when responses at many frequencies are of interest, their direct computation is unaffordable. I will discuss a model order reduction method aimed at reducing the computational costs by producing an approximation of (some functional of) the frequency response map.
In the second part of the talk, I will present a space-time discretization for the scalar-valued dissipative wave equation in two- and three-dimensions. It is a structured approach based on the Virtual Element Method for space discretization coupled with the Discontinuous Galerkin finite element method for time integration. I will discuss the theoretical analysis of the scheme and present some numerical examples to demonstrate its practical capabilities. Finally, I will conclude with some perspectives and future developments.

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Algebraic Multigrid (AMG) is the preconditioner of choice in several applications due to its effectiveness in solving linear systems arising from the discretization of PDEs. The key feature of AMG is its optimality that is its ability to guarantee, at least in some problems, a convergence rate that is independent of the mesh size. This feature is obtained through a good interplay between the smoother and the interpolation. Unfortunately, for difficult problems such as those arising from linear elasticity or diffusion problems with strong contrasts in the coefficients, standard smoothers and interpolation techniques are not enough to ensure fast convergence. In these cases, improving the prolongation operator by minimizing its energy norm may greatly enhance AMG convergence. In this talk, we show how energy minimization can be seen as a constrained minimization problem, where the constraint is twofold: the prolongation must be sparse, and its range must contain the operator near-kernel. To solve this problem, we propose an iterative algorithm based on the null-space method and preconditioning to speed-up the set-up time. We will show, with the aid of a few numerical experiments arising from real-world applications, that this approach is perfectly parallel and can lead to convergence rates significantly higher than those from standard interpolation techniques at a reasonable set-up cost.


Contatto: luca.formaggia@polimi.it

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La teoria dei giochi, nata verso la metà del secolo scorso, è una disciplina matematica che ambisce a studiare il comportamento degli esseri umani (e non solo) nelle situazioni di interazione, in cui sia necessario compiere delle scelte strategiche. Negli ultimi decenni diversi matematici hanno vinto il premio Nobel per l'economia grazie alle loro ricerche nell'ambito della teoria dei giochi, che si è dimostrata estremamente proficua ed efficace in svariati ambiti di applicazione: politica, medicina, informatica. Nonostante ciò, le ipotesi della teoria e alcuni dei concetti fondamentali, come quello di equilibrio di Nash, sono semplici e possono essere accessibili a tutti.
In questo seminario illustreremo cosa sia la teoria dei giochi, fornendo oltre agli elementi teorici della disciplina e alcuni esempi di applicazione anche degli spunti didattici per affrontare l'argomento con gli studenti della scuola secondaria.

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Geomstats is an open-source Python package for computations, and statistics on nonlinear manifolds. We provide object-oriented and extensively unit-tested implementations. Manifolds can be equipped with Riemannian metrics with associated exponential and logarithmic maps, geodesics, and parallel transport. Some manifolds can also be endowed with additional mathematical structures, such as Lie group, or fiber bundle structures. Statistics and learning algorithms provide methods for estimation, clustering, and dimensionality reduction on manifolds. All associated operations are supported in different backends, namely NumPy, Autograd, and PyTorch.
In this talk, we briefly introduce the main concepts in Riemannian geometry and discuss the package design. We show that Geomstats provides reliable building blocks to both foster research in differential geometry and statistics and democratize the use of Riemannian geometry in statistics and machine learning. The source code is freely available under the MIT license at github.com/geomstats/geomstats.

Contatti:
paola.antonietti@polimi.it
edie.miglio@polimi.it
marco.verani@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.

Alain Goriely is a mathematician and scientist with a broad range of interests in mathematical methods, mechanics, sciences, and engineering. He is particularly renowned for his contributions to the field of solid mechanics, both fundamental and applied, and for developing a mathematical theory of biological growth, which he explores in detail in his monograph "The Mathematics of Mechanics in Biological Growth".

Goriely earned his PhD from the University of Brussels in 1994, where he later became a lecturer. In 1996, he joined the University of Arizona and established a research group within the highly regarded Program of Applied Mathematics. In 2010, he took up a position at the University of Oxford as the inaugural Statutory Professor of Mathematical Modelling and Fellow of St. Catherine's College. He currently serves as the Director of the Oxford Centre for Industrial and Applied Mathematics.

In addition to his academic work, Goriely is also active in scientific outreach, sharing his insights and findings on a variety of topics related to his research, including tendril perversion in plants, twining plants, umbilical cord knotting, whip cracking, the shape of seashells, and brain modelling. For his contributions to mathematics and sciences, Goriely was elected as a Fellow of the Royal Society in 2022.

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Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no known cure. Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. This evolution is associated with the aggregation of key toxic proteins. In this talk, I will show how we use multiscale modelling to gain insight into this process and, doing so, gain some understanding on how the brain works. In particular, by looking at protein dynamics on the neuronal network, we can unravel some of the universal features associated with dementia that are driven by both the network topology and protein kinetics. By further coupling this approach with functional models of the brain, we will show that we can explain important aspects of cognitive loss during disease development.

Contatto: paola.antonietti@polimi.it

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E se un gioco di carte, apparentemente innocuo, nascondesse tanta di quella geometria da tenerci un seminario? A partire dal gioco Dobble, intraprenderemo un viaggio che metterà a dura prova le nostre certezze geometriche, toccando aspetti sia proiettivi che combinatori. Vedremo come alcune rappresentazioni ed alcuni concetti, tipici del modello di piano euclideo al quale solitamente pensiamo, cederanno il passo, mentre altri verranno considerati sotto una luce nuova.

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Saper risolvere problemi e, in generale, saperli gestire, è una delle competenze matematiche fondamentali. Tuttavia, la risoluzione dei problemi, e in particolare di “word problems”, è un’attività che può creare difficoltà a studenti di ogni ordine e grado, spesso legate alla fase di comprensione del testo. A tal proposito, è stata progettata un’attività didattica che prevede che gli studenti, individualmente e poi in maniera collaborativa, analizzino dapprima criticamente il testo di “word problems” per poi affrontarne la risoluzione. Tale attività è stata sperimentata con studenti del primo anno della scuola secondaria di secondo grado e i risultati sembrano mostrarne l’efficacia nel favorire l’attivazione di adeguati processi risolutivi da parte degli studenti.

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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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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.

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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There is a quantum singularity theory developed by Fan, Jarvis, and Ruan that produces a theory similar to Gromov-Witten theory, but from the input of a potential function and a group of symmetries. This can be considered as a Landau-Ginzburg A-model. There is also a Landau-Ginzburg B-model that can be constructed from similar input of a potential function and a group of symmetries. Finally, there is a construction of Berglund-Hübsch and Krawitz that links the two, and this is known as BHK mirror symmetry. However, this only works when the group of symmetries is abelian.
Recently there have been generalizations of both the A-model and the B-model to include nonabelian groups of symmetries. In this talk we will review the previously mentioned theories, and I will describe an extension to BHK mirror symmetry that is conjectured to give an isomorphism between the LG A-model and the LG B-model for nonabelian groups of symmetries. I will describe a result giving an isomorphism between the state spaces for the LG A-model and the LG B-model.

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What is the optimal way to deform a projective hypersurface into another one?
We will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic projective hypersurfaces.
First, a natural topological embedding of the space of hypersurfaces of a given degree into the space of measures on the projective space is constructed.
Then, the optimal transport problem between hypersurfaces is defined through a constrained dynamical formulation, minimizing the energy of absolutely continuous curves which lie on the image of this embedding. In this way an inner Wasserstein distance on the projective space of homogeneous polynomials is introduced.
We will show the main properties of this distance and discuss applications on the regularity of the zeroes of a family of multivariate polynomials and on the condition number of polynomial systems solving.

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We investigate the existence of non trivial doubly periodic solutions of the Benjamin-Ono equation, depending on the space period, the time period, the space average, and the space regularity of the solution. In particular, for doubly periodic solutions with space average 0 and with time period and space period equal to 2\pi, we prove that L^2 regularity implies analyticity, while there exists more such solutions which belong to the Sobolev space H^s for every s<0. The crucial tool is the Birkhoff map for the Benjamin-Ono equation recently obtained in collaboration with T. Kappeler and P. Topalov.

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This talk is motivated by the strong case study of monitoring gait for early detection of gait impairment in patients diagnosed with gait-affecting disorders. To this effect, we record the rotation of the hip over time. This leads to data with a great complexity that makes any kind of statistical analysis non-trivial. In effect, we collect functional data that evaluates on the Lie group of three-dimensional rotations which is a non-Euclidean space. In this work, we develop sound statistical methods to enable joint clustering and alignment of such functional data by borrowing ideas from the existing k-means alignment approach. In the process, we will describe in great length the R package fdacluster, which is an extension of the R package fdakma that provides a fast C++ based implementation of the k-mean alignment algorithm from Sangalli et al. (2010) as well as hierarchical clustering for functional data and tools that help choosing the number of clusters.

Il link per seguire il seminario online sarà reso disponibile pochi minuti prima dell’avvio del seminario

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BigData users as well as BigData research community are fast growing bigger and bigger while statisticians at large seem to be growing divided between enthusiastic and worried, when not downright hostile. Is BigData also a big step ahead to extract information and actual knowledge from data? Is BigData underplaying statistical inference as we know it? Supplanting stats methodology as a low-cost futuristic option? As a (mainly) sampling statistician, in this webinar I shall try to decipher the multifaceted relationship connecting BigData to statistical and to sampling methodology, starting with the still blurry definition of BigData and discussing some very personal considerations and views.

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In his investigations on the spectral theory of Schrödinger operators with singular potentials, T. Kato introduced many celebrated tools and a lot of work has been done to transplant them to Riemannian manifolds. In this talk I shall mainly focus on the so called “L^p Positivity Preservation” and discuss how it is (un)related to the geometry of the underlying space.
The talk is based on a joint work with Daniele Valtorta and Giona Veronelli

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Despite the great progress that climate science and our understanding of the climate system have made in the last three decades, significant uncertainties in our knowledge of the climate system and future climate change remain. In this talk I will focus on estimates for one climate variable, the equilibrium climate sensitivity (ECS), and examine how uncertainties in its estimates are treated in the most recent IPCC report AR6. AR6 supplements a framework of probability intervals (adopted from earlier reports) by in addition considering low-likelihood but high-impact “storylines”. I will argue in this paper that including such storylines represents an important improvement compared to prior reports. More specifically, I will argue that (i) in light of the deep uncertainties plaguing estimates of the values of central climate variables seemingly probabilistic assessments of these quantities ought to be read in a coarse-grained way and interpreted within a possibilistic fra!
mework as statements of ranked possibilities; (ii) given the high stakes involved, an appropriate decision framework for climate policy decisions ought to include broadly precautionary considerations that are sensitive to high-impact and catastrophic climate futures; (iii) storylines are a useful tool for probing possible high-impact climate futures.