Dipartimento di Matematica

Petros Koumoutsakos is Herbert S. Winokur, Jr., Professor of Engineering and Applied Sciences and Area Chair for Applied Mathematics at Harvard’s John A. Paulson School of Engineering and Applied Sciences (SEAS). He studied Naval Architecture (Diploma-NTU of Athens, M.Eng.-U. of Michigan), Aeronautics, and Applied Mathematics (PhD-Caltech) and has served as the Chair of Computational Science at ETH Zurich (1997-2020). Petros is elected Fellow of the American Society of Mechanical Engineers (ASME), the American Physical Society (APS), and the Society of Industrial and Applied Mathematics (SIAM). He is the recipient of the Advanced Investigator Award by the European Research Council and the ACM Gordon Bell prize in Supercomputing. He is elected International Member of the US National Academy of Engineering (NAE). His research interests are on the fundamentals and applications of computing and artificial intelligence to understand, predict and optimize fluid flows in engineering, nanotechnology, and medicine.

abstract

abstract

Over the last thirty years, we have experienced more than a billion-fold increase in hardware capabilities and a dizzying pace of acquiring and transmitting massive amounts of data. Artificial Intelligence (AI) has been the beneficiaries of these advances, and today it is increasingly embedded in technologies that touch every aspect of humanity. However, along with the abundance of promise, there is an ever increasing amount of hype, in particular regarding the capabilities of learning algorithms to model, predict and control complex fluid mechanics problems.
In this talk, I will offer a perspective on forming alloys of AI and simulations for the prediction and control of complex flow systems. I will present novel algorithms for learning the Effective Dynamics (LED) of complex flows and a fusion of multi-agent reinforcement learning and scientific computing (SciMARL) for modeling and control of complex flow-structure interactions. I will juxtapose successes and failures and argue that the proper fusion of fluid mechanics knowledge and AI expertise are essential to advance scientific frontiers.

Contatto: alfio.quarteroni@polimi.it

abstract

abstract

The Casimir effect is a manifestation of van der Waals polarization forces acting between atoms and molecules, in a regime where relativistic retardation effects cannot be neglected. An effective description can be given in terms of quantum fields influenced by classical objects,
such as boundaries or external potentials. This leads to the emergence of ultraviolet divergences which can be treated by means of the so-called zeta-function regularization. In this seminar I first present a rigorous implementation of the zeta-function technique, using the language of canonical quantization for a scalar field. To this purpose, complex powers of the elliptic operator describing the dynamics are used to define smeared versions of the Wightman field and renormalization of expectation values is ultimately attained by analytic continuation. Next, I address the issue of boundary anomalies, discussing some explicitly solvable models with zero-range potentials modeling semi-transparent walls.
Based on joint works with Livio Pizzocchero (University of Milano).

abstract

abstract

Environmental, social, and governance (ESG) factors have acquired great relevance for investors in recent years. This paper examines the intersection between ESG and crowdfunding, one of the most popular types of fintech. Using data on the population of 508 platforms established in the 37 OECD countries between 2007 and 2020, we document that platforms with higher levels of ESG criteria are more likely to survive over time. Results are robust to different measures of ESG and endogeneity concerns. The importance of ESG criteria depends on cultural differences, as it is more pronounced for those platforms operating in countries where the level of power distance is lower.

abstract

abstract

How to identify the geometry of a nanostructure given the desired effective thermal conductivity tensor? While several topology optimization methods have been developed for macroscale scenarios, they can’t be readily applied to nanoscale devices. In fact, when the feature size of the material becomes comparable with the particles’ mean-free path, flux becomes nondiffusive, and a momentum-resolved model is needed. On the other side, standard material interpolation methods, such as SIMP, are mostly applied to averaged quantities, such as the local thermal conductivity. In this talk, I will discuss our recent attempt at solving this issue [1], based on a novel method, termed the “Transmission Interpolation Method” (TIM). Instead of working with locally resolved variables, TIM acts on the interfacial phonon flux and links it to the material distribution. The talk will show the application of this approach to two examples: Tuning the effective thermal conductivity tensor of a nanostructure and maximizing phonon size effects. The talk will conclude with a demo and remarks for future developments.
[1] Giuseppe Romano and Steven G Johnson, Structural and Multidisciplinary Optimization 65, 297 (2022)

Contatto: giulia.pozzi@polimi.it

abstract

abstract

Il seminario si propone di analizzare rappresentazioni esterne spontanee o indotte messe in atto da studenti durante la risoluzione di problemi. Le rappresentazioni esterne possono fare da ponte per l’elaborazione di una strategia risolutiva. A volte si trovano schemi personali che riflettono il processo di pensiero utilizzato. Spesso esse sostituiscono anche un’esplicitazione verbale del processo risolutivo, o comunque coesistono con questa in un intreccio tra registro verbale e iconico, o grafico-simbolico. Durante il seminario saranno analizzate le risposte degli studenti con lo scopo di fornire indicazioni didattiche durante attività di problem-solving.

abstract

abstract

Il seminario prende in considerazione il ragionamento diagrammatico in geometria elementare e le sue origini nella pratica matematica greca. Si discutono alcuni limiti ed alcune potenzialità di questo approccio ala geometria. Si mostra che il ruolo delle inferenze diagrammatiche diminuì nel corso della storia della geometria moderna anche prima di arrivare alla matematica astratta del diciannovesimo secolo. Le ragioni dell'abbandono delle pratiche diagrammatiche sembra legato alla trasformazione della geometria da una scienza delle figure in una scienza dello spazio. Si mostra allora che tale trasformazione ha preceduto e reso possibile la nascita della matematica astratta e di un ideale dimostrativo non-diagrammatico.

abstract

abstract

In recent years, natural hazards involving large mass movements such as landslides, debris flows, and mud flows have been increasing their frequency and intensity as a consequence of climate change and other related factors. These phenomena often carry huge rocks and heavy materials that may, directly or indirectly, cause damage to our structures resulting in a relevant socio-economic impact.
The numerical simulation of the above events still represents a big challenge mainly for two reasons: the need to deal with large strain regimes and the intrinsic multiphysics nature of such events.
While the Finite Element Method (FEM) represents a recognized, well established and widely used technique in many engineering fields, unfortunately it shows some limitation when dealing with problems where large deformation occurs. In the last decades many possible alternatives have been proposed and developed to overcome this drawback, such as the use of the so called particle-based methods. Among these, the Material Point Method (MPM) blends the advantages of both mesh-based and mesh-less methods. MPM avoids the problems of mesh tangling while preserving the accuracy of Lagrangian FEM and it is especially suited for non linear problems in solid mechanics and fluid dynamics.
The talk will show some recent advances in MPM formulations [1], presenting both an irreducible and mixed formulation stabilized using variational multiscale techniques, as well as the partitioned strategies to couple MPM with other techniques such as FEM or DEM [2, 3]. All algorithms are implemented within the Kratos-Multiphysics open-source framework and available under the BSD license.
References
[1] Iaconeta, I., Larese, A., Rossi, R. and Guo, Z., Comparison of a Material Point Method and
a Meshfree Galerkin Method for the simulation of cohesive-frictional materials, Materials, 10 , 1150, (2017).
[2] Chandra, B., Singer, V., Teschemacher, T., Wuechner, R. and Larese, A., Nonconforming
Dirichlet boundary conditions in Implicit Material Point Method by means of penalty augmentation,
Acta Geotechnica, 16(8), 2315-2335 (2021).
[3] Singer, V., Sautter, K.B., Larese, A., Wuchner, R. and Bletzinger, K.U.,, A Partitioned
Material Point Method and Discrete Element Method Coupling Scheme , Under revision in Advanced Modeling and Simulation in Engineering Sciences (2022).
Contatto: anna.scotti@polimi.it

abstract

abstract

This talk presents a framework for forward uncertainty quantification problems in spatially-extended neurobiological networks. We will consider networks in which the cortex is represented as a continuum domain, and local neuronal activity evolves according to an integro-differential equation, collecting inputs nonlocally, from the whole cortex. These models are sometimes referred to as neural field equations.
Large-scale brain simulations of such models are currently performed heuristically, and the numerical analysis of these problems is largely unexplored. In the first part of the talk I will summarise recent developments for the rigorous numerical analysis of projection schemes [1] for deterministic neural fields, which sets the foundation for developing Finite-Element and Spectral schemes for large-scale problems.
The second part of the talk will discuss the case of systems in the presence of uncertainties modelled with random data, in particular: random synaptic connections, external stimuli, neuronal firing rates, and initial conditions (and any combination thereof). Such problems give rise to random solutions, whose mean, variance, or other quantities of interest have to be estimated using numerical simulations. This so-called forward uncertainty quantification problem is challenging because it couples spatially nonlocal, nonlinear problems to large-dimensional random data.
I will present a family of schemes that couple a spatial projector for the spatial discretisation, to stochastic collocation for the random data. We will analyse the time-dependent problem with random data and the schemes from a functional analytic viewpoint, and show that the proposed methods can achieve spectral accuracy, provided the random data is sufficiently regular.
Acknowledgements. This talk presents joint work with Francesca Cavallini (VU Amsterdam), Svetlana Dubinkina (VU Amsterdam), and Gabriel Lord (Radboud University).
References
[1] Avitabile D, Projection methods for Neural Field Equations arXiv e-prints,
arXiv:2112.03244 doi.org/10.48550/arXiv.2112.03244

Contatto: simona.perotto@polimi.it

abstract

abstract

L’idea di “democrazia” che è ormai ben radicata dentro ciascuno di noi, spesso ci porta a considerare la questione dei sistemi elettorali (ovvero metodi per ottenere, a partire dalle preferenze degli individui di una certa popolazione, una lista di preferenze unica) con una certa leggerezza. Invece, appena si approfondisce la teoria dei sistemi di voto, ci si imbatte immediatamente in situazioni paradossali e problemi imprevisti, quasi sempre di difficile soluzione. Inizieremo questa conferenza discutendo alcuni dei principali problemi e paradossi che si presentano nella progettazione di un sistema elettorale. Nella parte finale ci occuperemo di un problema di diversa natura, ma di notevole importanza, ovvero quello della definizione “imparziale” dei collegi elettorali.

abstract

abstract

Tra il XVI e il XVIII secolo è avvenuta, in Europa, una rivoluzione cosmologica. Si tratta di molto più di un semplice cambiamento “geometrico” (il Sole al centro invece della Terra al centro); si tratta della sostituzione di tutto un insieme di conoscenze e di concezioni. Le nuove concezioni sono state sviluppate in modo non sincrono, talvolta secondo le osservazioni empiriche ma talvolta anche contro le osservazioni empiriche, e hanno preso il sopravvento quando sono arrivate a costituire un insieme sufficientemente completo e coerente.

Vengono percorsi i passaggi principali, attraverso i protagonisti chiave (Copernico, Tycho Brahe, Keplero, Galileo, Newton), sia astronomi osservatori sia puramente speculativi, le loro scoperte e i loro tentativi di interpretazione, cercando di mettere in luce alcuni snodi che sono particolarmente interessanti dal punto di vista epistemologico.

abstract

abstract

“A cosa serve la matematica?” Qual è quel docente che non si è mai sentito rivolgere questa domanda? Spesso è una domanda provocatoria, posta dallo studente più "simpatico" della classe. Ma a volte è una domanda sincera, posta da uno sconosciuto in treno. Solitamente siamo troppo stanchi o demoralizzati per rispondere a questa domanda. Ovviamente la matematica è utilissima, nel mondo di oggi, pieno di dati e in cui ogni aspetto della vita è intrecciato ad applicazioni, banali o profonde della matematica. Forse però vale la pena di soffermarsi un attimo in più su tale domanda e cercare di sviscerarla meglio...
Cosa vuol dire "a cosa serve la matematica?"?
La matematica pura è utile?
Perché i politici devono sapere la matematica?
A cosa serve a me la matematica?
L'utilità della matematica è interessante per chi la studia?
Nel seminario più che fornire le mie personali risposte proverò a dare un po' di materiale per pensarci su. Spero possa essere utile per avere qualche idea in più la prossima volta che sentirete queste o altre domande...

Nick Trefethen is Professor of Numerical Analysis and head of the Numerical Analysis Group at Oxford University. He was educated at Harvard and Stanford and held positions at NYU, MIT, and Cornell before moving to Oxford in 1997. He is a Fellow of the Royal Society and a member of the US National Academy of Engineering, and served during 2011-2012 as President of SIAM. He has won many prizes including the Gold Medal of the Institute for Mathematics and its Applications, the Naylor Prize of the London Mathematical Society, and the Polya and von Neumann Prizes from SIAM. He holds honorary doctorates from the University of Fribourg and Stellenbosch University.
As an author Trefethen is known for his books including Numerical Linear Algebra (1997), Spectral Methods in MATLAB (2000), Spectra and Pseudospectra (2005), Approximation Theory and Approximation Practice (2013/2019), Exploring ODEs (2018), and An Applied Mathematician's Apology (2022). He organized the SIAM 100-Dollar, 100-Digit Challenge in 2002 and is the inventor of Chebfun.

abstract

abstract

For the first time, a method has recently become available for fast computation of near-best rational approximations on arbitrary sets in the real line or complex plane: the AAA algorithm (Nakatsukasa-Sete-T. 2018). We will present the algorithm and then demonstrate a number of applications, including
* detection of singularities
* model order reduction
* analytic continuation
* functions of matrices
* nonlinear eigenvalue problems
* interpolation of equispaced data
* smooth extension of multivariate real functions
* extrapolation of ODE and PDE solutions into the complex plane
* solution of Laplace problems
* conformal mapping
* Wiener-Hopf factorization

(joint work with Stefano Costa and others)

Contatto: paola.antonietti@polimi.it

abstract

abstract

By the early 1970's two important conjectures, the Wigner-Yanasee-Dyson conjecture and the Strong Subadditivity of Quantum Entropy conjecture, had attracted the attention on many mathematicians and physicists. These conjectures were both proved in 1973 by Leib and by Lieb and Ruskai respectively. The methods introduced for their solution were powerful, and immediately found other applications, for example in Lindblad's 1975 proof of the Data Processing Inequality, now one of the cornerstones of quantum information theory. In this talk I will briefly explain the history before turning to modern developments and a modern perspective that has led to new inequalities and stronger versions of known inequalities. In this latter part, I will focus on recent work of myself, and myself in collaboration with Alexander Mueller-Hermes and with Haonan Zhang. I will also briefly discuss some recent applications.

abstract

abstract

We will describe recent progress on describing derived categories of coherent sheaves for certain singular projective varieties in terms of derived categories of finite dimensional algebras, which are typically noncommutative.
This is based on ongoing joint works with Yujiro Kawamata & Nebojsa Pavic and with Carlo Klapproth, Nebojsa Pavic & Evgeny Shinder.

abstract

abstract

The starting point of the seminar is the well-known generalized Lane-Emden equation
\begin{equation}
\Delta_p u + \vert u \vert^{q-1} u =0 \quad \text{ in $\mathbb{R}^n$ } \, , \quad \quad \text{(LE)}
\end{equation}
where $\Delta_p$ is the usual $p-$Laplace operator with $ 1 < p < n $ and $ q > 1 $. I will discuss several non-existence and classification results for positive solutions of (LE) in the subcritical ($q < p^\ast - 1 $) and in the critical case ($q = p^\ast - 1 $). In the critical case, it has been recently shown, exploiting the moving planes method, that positive solutions to the critical $ p-$ Laplace equation (i.e. (LE) with $ q = p^\ast -1 $) and with finite energy, i.e. such that $ u \in L^{p^\ast}(\mathbb{R}^n) $ and $ \nabla u \in L^p(\mathbb{R}^n) $, can be completely classified. In this talk, I will present some recent classification results for positive solutions to the critical $p-$Laplace equation with (possibly) infinite energy satisfying suitable conditions at infinity. Moreover, if time permits I will discuss analogous results in the anisotropic, conical and Riemannian settings.

This is based on a recent joint work with G. Catino and D. Monticelli.

abstract

abstract

Il seminario riporta e analizza un’esperienza didattica attuata in una classe quarta di liceo scientifico e progettata con lo scopo di sviluppare le competenze di modellistica matematica. Il percorso didattico trae origine da un problema autentico che ha consentito agli studenti di vestire in prima persona i panni del ricercatore matematico. La specifica attività didattica permetterà di introdurre il ciclo della modellistica come strumento per la progettazione e l’analisi dell’efficacia di un approccio didattico per competenze. Durante il seminario saranno presentati ulteriori percorsi didattici di modellistica matematica disponibili nel MOOC “Matematica per scenari” (accessibile sulla piattaforma POK www.pok.polimi.it), accompagnati da dettagliate indicazioni metodologico-didattiche.

abstract

abstract

In this seminar, we focus on the numerical analysis of a polygonal discontinuous Galerkin (PolyDG) scheme for the simulation of multiphysics wave propagation in heterogeneous media. We address wave phenomena in elastic, poroelastic, and acoustic materials coupled by means of physically consistent transmission conditions. The coupling terms also depend on the pores configuration at the interface between the subdomains. For all models configuration, we introduce the PolydG semi-discrete formulation, which is then coupled with implicit time marching schemes. The proposed PolyDG approximation offers a high-level of flexibility and precision thanks to its arbitrary-order accuracy and capability to support general meshes. We present a complete stability analysis and derive a-priori hp-error estimates. A wide set of verification tests and examples of physical interest are investigated in order to assess the performances of the method.
This research work has been funded by the European Commission through the H2020-MSCA-IF-EF project PDGeoFF (Grant no. 896616) and by the Istituto Nazionale di Alta Matematica through the INdAM-GNCS project no. CUP-E55F22000270001.

Contatto: anna.scotti@polimi.it

abstract

abstract

In this talk I will present a classification result for critical Dirac equations on the Euclidean space, appearing naturally in conformal spin geometry and in variational problems related to Dirac equations on spin manifolds. Exploiting the conformal invariance, ground state solutions can be classified. This is the spinorial counterpart of the well-known result for the Yamabe equation.

Time permitting, I will also present similar results obtained for conformal Dirac-Einstein equations, consisting of a Dirac equation coupled with a Laplace- type equation.

Joint work with Ali Maalaoui, Andrea Malchiodi and Ruijun Wu.

abstract

abstract

A Bayesian network is a statistical model which can be presented graphically by a directed acyclic graph. The nodes in the graph are discrete random variables, and the edges encode dependencies between the variables. Bayesian nets can also be described algebraically as varieties of homogeneous prime ideals. In this talk we will discuss connections between algebraic properties of such ideals and combinatorial properties of the graphs. In particular, we would like to understand when the variety is toric and when the ideal is quadratic.

abstract

abstract

I matematici sono piuttosto concordi nell'affermare che la matematica è bellezza, poesia e fantasia. Possiamo dire che gli oggetti matematici non solo hanno a che fare con l'arte, ma sono essi stessi opere d'arte. Proporremo un percorso didattico interdisciplinare tra matematica, arte, poesia e altri settori del sapere. I primi concetti topologici di punto interno e di frontiera saranno il pretesto per considerazioni di educazione civica. Il grafico dell'arcotangete ci condurrà agli omeomorfismi e ad incontri d'amore mentre il romanzo "la storia infinita" sarà lo spunto per parlare di compattificazione e proiezione stereografica.