Several applications in physics, biology, and chemistry involve systems with multiple components, such as gas mixtures, competing population species, and reacting chemical substances. These problems are modeled with nonlinear reaction-diffusion equations that include cross-diffusion terms. Cross diffusion occurs when the flux of one component is driven by the gradient of another component. The main challenges in designing numerical methods for approximating nonlinear cross-diffusion systems are that the diffusion matrix may not be symmetric or positive semidefinite, and that a maximum principle may not be available. In this talk, we present numerical methods based on the boundedness-by-entropy framework introduced by A. Jüngel in 2015. Motivated by the inherent entropy structure of the PDE system, nonlinear transformations involving the entropy variable allow for the enforcement of positivity in the approximate solutions. Specifically, we focus on a Local Discontinuous Galerkin method. By appropriately introducing auxiliary variables, the problem is reformulated so that nonlinearities do not appear within differential operators or interface terms. This results in nonlinear operators that can be naturally evaluated in parallel. The method allows for arbitrary degrees of approximation in space, preserves boundedness of the physical unknowns without requiring postprocessing or slope limiters, and satisfies a discrete version of the entropy stability estimate of the continuous problem.
Contatto: paola.antonietti@polimi.it
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
Dopo una breve introduzione sui principi di progettazione, nel seminario verrà mostrato il progetto “Didattica multimediale per la matematica” che fa uso di applet geogebra per favorire l'apprendimento della matematica a diversi livelli e con diverse finalità specifiche (visualizzazione, auto-valutazione, gestione di rappresentazioni multiple, supporto al problem solving). In particolare, si approfondirà l’uso delle risorse didattiche per l’insegnamento-apprendimento dello studio di funzione.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
Durante il seminario si analizzeranno e discuteranno macro-fenomeni emersi in sede di valutazione standardizzata inerenti quesiti che mostrano un ampio divario tra la percentuale di risposte corrette di maschi e femmine. Con le lenti teoriche di alcune delle principali ricerche di didattica della matematica sul tema si discuteranno cause e motivazioni. Obiettivo principale del seminario è lo sviluppo di una maggiore consapevolezza rispetto ai fattori culturali, ma anche didattici e di natura metacognitiva, che portano alla formazione del gender gap in matematica.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
Among the Cohen-Macaulay rings, Gorenstein rings have a prominent role in many respects. Looking for other rings with similar properties, many authors have proposed several generalizations, and one of the most studied in the last years is the notion of nearly Gorenstein ring. In this talk, after an introduction to these rings, I will focus on some nearly Gorenstein semigroup rings. In particular, I will explain a characterization using the properties of the associated semigroup and I will give some results about their Cohen-Macaulay type. This is based on joint works with Alessio Moscariello, Raheleh Jafari and Santiago Zarzuela Armengou.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
In a three-dimensional bounded domain, we consider the compressible Navier-Stokes equations for a barotropic fluid with general non-linear density dependent viscosities and no-slip boundary conditions. A nonlinear drag term is added to the momentum equation. We establish two conditional Kato-type criteria for the convergence of the weak solutions to such a system towards the strong solution of the compressible Euler system when the viscosity coefficient and the drag term parameter tend to zero. Joint work with L. Bisconti and M. Caggio.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
Il metodo CHUNK rappresenta un approccio innovativo nell'ambito dell'apprendimento adattivo, basato su un sistema di raccomandazione che tiene conto degli interessi individuali e delle necessità educative. Fondato su quattro componenti principali, il metodo CHUNK consente di approfondire la rilevanza e le applicazioni pratiche dei contenuti di studio, esplorarli attraverso metodologie flessibili e verificarne la comprensione in modo personalizzato.
In questo seminario, presenterò i fondamenti teorici e l’applicazione del metodo CHUNK, illustrando come sia stato adottato presso la Naval Postgraduate School di Monterey, CA, per rispondere alla diversità delle classi. Condividerò alcune riflessioni su come questo approccio potrebbe essere adattato al contesto scolastico, con l’obiettivo di discutere l'integrazione del metodo CHUNK in un ambiente educativo che promuova un apprendimento non solo individuale, ma anche collaborativo e partecipativo.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
Pathwise uniqueness plays a crucial role in the investigation of the existence of strong solutions to Stochastic Differential Equations (SDEs) since the seminal result by Yamada and Watanabe in 1971, where they proved that weak existence and pathwise uniqueness imply strong existence. A few years later, Zvonkin introduced the so-called Zvonkin transformation, which allows to remove a drift term by means of a suitable change of coordinates using the Ito formula, and then applies the result of Yamada and Watanabe to construct strong solutions to a class of SDEs with rough drift coefficient. In recent years, these techniques have been extended to Stochastic Partial Differential Equations (SPDEs) . One of the main tools for proving pathwise uniqueness in infinite dimensions is the so-called Itô-Tanaka trick, which involves replacing the "bad" drift term with the solution to a suitable Kolmogorov equation. In this talk, we will examine these topics for a class of parabolic and hyperbolic SPDEs.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
This talk will be an introduction to the \(K(\pi,1)\) conjecture, a 50-year-old open problem on the topology of configuration spaces associated with Coxeter and Artin groups. Then I will give a high-level overview of combinatorial approaches to this conjecture which led to its solution in the spherical and affine cases.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
The celebrated Euclidean isoperimetric inequality provides a sharp estimate of the measure of the boundary of subsets of the Euclidean space in terms of their volume. This inequality is also rigid, as equality is achieved only by Euclidean balls. The study of sharp and rigid isoperimetric inequalities on Riemannian manifolds has been an active area of research over the past few decades and is closely connected to curvature bounds.
In this introductory talk, we will review some classical and recent isoperimetric inequalities on classes of Riemannian manifolds with curvature bounded below. A possible unified approach to the proofs of such inequalities arises from the sharp concavity properties of the isoperimetric profile function. This latter result was first obtained on compact manifolds by C. Bavard--P. Pansu and S. Gallot in the '80s, and it has recently been extended to the noncompact setting in a joint work with G. Antonelli, E. Pasqualetto, and D. Semola.
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).
In recent years, machine learning (ML) and artificial intelligence (AI) methods have begun to play a more and more enabling role in the sciences and in industry. In particular, the advent of large and/or complex data corpora has given rise to new technological challenges and possibilities. In his talk, Müller will touch upon the topic of ML applications in the sciences, in particular in chemistry and physics. He will also discuss possibilities for extracting information from machine learning models to further our understanding by explaining nonlinear ML models. Finally, Müller will briefly discuss perspectives and limitations.
Klaus-Robert Müller has been a professor of computer science at Technische Universität Berlin since 2006; at the same time he is directing rsp. co-directing the Berlin Machine Learning Center and the Berlin Big Data Center and most recently BIFOLD . He studied physics in Karlsruhe from 1984 to 1989 and obtained his Ph.D. degree in computer science at Technische Universität Karlsruhe in 1992. After completing a postdoctoral position at GMD FIRST in Berlin, he was a research fellow at the University of Tokyo from 1994 to 1995. In 1995, he founded the Intelligent Data Analysis group at GMD-FIRST (later Fraunhofer FIRST) and directed it until 2008. From 1999 to 2006, he was a professor at the University of Potsdam. From 2012 he has been Distinguished Professor at Korea University in Seoul. In 2020/2021 he spent his sabbatical at Google Brain as a Principal Scientist. Among others, he was awarded the Olympus Prize for Pattern Recognition (1999), the SEL Alcatel Communication Award (2006), the Science Prize of Berlin by the Governing Mayor of Berlin (2014), the Vodafone Innovations Award (2017), Hector Science Award (2024), Pattern Recognition Best Paper award (2020), Digital Signal Processing Best Paper award (2022). In 2012, he was elected member of the German National Academy of Sciences-Leopoldina, in 2017 of the Berlin Brandenburg Academy of Sciences, in 2021 of the German National Academy of Science and Engineering and also in 2017 external scientific member of the Max Planck Society. From 2019 on he became an ISI Highly Cited researcher in the cross-disciplinary area. His research interests are intelligent data analysis and Machine Learning in the sciences (Neuroscience (specifically Brain-Computer Interfaces, Physics, Chemistry) and in industry.
Seminari Matematici al
Politecnico di Milano
- Analisi
- Cultura Matematica
- Seminari FDS
- Geometria e Algebra
- Probabilità e Statistica Matematica
- Probabilità Quantistica