2 Dicembre, 2021 17:00

Seminario Matematico e Fisico di Milano

Nonlocal Aggregation-Diffusion Equations: entropies, gradient flows, phase transitions and applications

Carrillo José Antonio, University of Oxford

polimi-it.zoom.us/j/85300866421

abstract

abstract

This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

1 Dicembre, 2021 15:00

Seminari FDS

“DE NEXU INTER MULTITUDINEM CLASSIUM, IN QUAS FORMAE BINARIAE SECUNDI GRADUS DISTRIBUUNTUR, EARUMQUE DETERMINANTEM" (GAUSS 1834)

Leonardo Colzani, Università di Milano Bicocca

online tiny.cc/fds22webex

abstract

abstract

Nonostante il titolo, il seminario sarà elementare. Si parlerà del problema di contare il numero di punti a coordinate intere in una regione nel piano o nello spazio, un problema non banale con molte applicazioni.

30 Novembre, 2021 15:15

Sezione di Analisi

Sign-indefinite logistic growth models with flux-saturated diffusion

Elisa Sovrano, Università degli Studi di Modena e Reggio Emilia

Aula Seminari III Piano

abstract

abstract

Reaction-diffusion processes can be based on Fick-Fourier's law. Changing perspective, we deal with a dispersive flux which is a nonlinear bounded function of the gradient. In a bounded domain with a regular boundary, we investigate a Dirichlet problem associated with a quasilinear reaction-diffusion equation where the mean curvature operator drives the diffusion process. As for the reaction, we consider the product of a logistic-type nonlinearity and a sign-indefinite weight function modeling spatial heterogeneities. For this problem, we present some recent results concerning the existence and the multiplicity of positive solutions. Depending on the logistic term's behavior at zero, we prove three qualitatively different bifurcation diagrams by varying the diffusivity parameter. We point out a new multiplicity phenomenon without any similarity with the case of linear-diffusion logistic-growth models. This talk is based on joint works with Pierpaolo Omari (University of Trieste).

29 Novembre, 2021 14:00 oclock

Sezione di Fisica Matematica

Correlation energy of mean-field Fermi gases

Marcello Porta, SISSA Trieste

Aula Seminario III Piano - Dipartimento di Matematica, Politecnico di Milano

abstract

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In this talk I will discuss the ground state properties of homogeneous, interacting Fermi gases, in the mean-field scaling. In this regime, Hartree-Fock theory provides a good approximation for the ground state energy of the system; this approximation is based on the replacement of the space of fermionic wave functions with the smaller set of Slater determinants, where the only correlations among the particles are those induced by the Pauli principle. I will discuss a rigorous approach that allows to go beyond the Hartree-Fock approximation, and that in particular allows to compute the leading order of the correlation energy, defined as the difference between the many-body and Hartree-Fock ground state energies. The expression we obtain reproduces the ground state energy of a non-interacting Bose gas, and agrees with the prediction of the random-phase approximation. The proof is based on a rigorous bosonization method, that allows to describe the particle-hole excitations around the Fermi surface in terms of a quasi-free Bose gas. Joint work with N. Benedikter, P. T. Nam, B. Schlein and R. Seiringer.

25 Novembre, 2021 14:00

MOX Seminar

Eigen sign-flip test for semiparametric regression

Federico Ferraccioli, Università degli Studi di Padova

Live: Aula Saleri, MOX - Dipartimento di Matematica - Politecnico di Milano
Online: mox.polimi.it/mox-seminars/?id_evento=2103

abstract

abstract

In this seminar, a nonparametric test for semiparametric penalized regression models is presented. The test is based on random sign-flipping of an appropriate transformation of the vector of residuals, that exploits a spectral decomposition of the residualizing matrix associated with the nonparametric part of the model. The formulation allows for a wide applicability of the test to a vast class of extensively used semiparametric regression models, including for instance models based on univariate or multivariate splines. The good asymptotic properties of the proposed test are discussed, together with its good control of the Type I error and power, even in challenging data scenarios.

24 Novembre, 2021 15:15

Sezione di Analisi

A relaxation approach to optimal transport with applications to the unbalanced case

Giacomo E. Sodini, TUM

Aula Seminari III Piano

abstract

abstract

In this talk, after briefly presenting the classical optimal transport problem, we will discuss a new interpretation of the optimal transport cost as the largest lower semicontinuous convex functional extending the cost between pairs of delta measures.
With this in mind, we will introduce a notion of optimal transport cost for (nonnegative) measures with possibly different masses and discuss its metric and topological properties.
This talk is based on a joint work with Giuseppe Savaré (Bocconi University).

22 Novembre, 2021 14:00

MOX Colloquia

Energy based modeling, simulation, and control. A path to real world digital twins

Volker Mehrmann, Technische Universität Berlin, Germany

Online seminar: mox.polimi.it/mox-seminars/?id_evento=2100

abstract

abstract

In the future we will have digital twins for every technological product or process. To create and operate those twins, a paradigm shift is needed for modeling simulation and optimization in all areas of science and engineering. A good candidate for this new paradigm is the concept of energy based modeling via port-Hamiltonian systems. These systems have wonderful mathematical properties, concerning their analytic, geometric and algebraic properties, but also with respect to their use in numerical algorithms for space-time discretization, model reduction and control. We will introduce the model class and their mathematical properties and we illustrate their usefulness with several real world applications.

Contatto: christian.vergara@polimi.it

Volker Mehrmann received his Diploma in mathematics in 1979, his Ph.D. in 1982, and his habilitation in 1987 from the University of Bielefeld, Germany. He spent research years at Kent State University in 1979-1980, at the University of Wisconsin in 1984-1985, and at IBM Research Center in Heidelberg in 1988--1989. After spending the years 1990-1992 as a visiting full professor at the RWTH Aachen, he was a full professor at TU Chemnitz from 1993 to 2000. Since then he is a full professor for Mathematics at TU Berlin.

He is a member of acatech (the German academy of engineering), academia europaea and the European Academy of Scienes, he was president of GAMM the (International association of Applied Mathematics and Mechanics), vice president of the European Mathematical Society (EMS), chair of MATHEON, the Research Center `Mathematics for key technologies´ and chair of the Einstein Center ECMath in Berlin. Since January 2019 he is president of the EMS.

He is SIAM Fellow, has held an ERC Advanced Grant and also was member of the ERC Mathematics Panel. He is editor of several journals in numerical analysis and editor-in-chief of Linear Algebra and its Applications. His research interests are in the areas of numerical mathematics/scientific computing, applied and numerical linear algebra, control theory, the theory and numerical solution of differential-algebraic equations, and recently also in energy based mathematical modeling.

22 Novembre, 2021 17:30 oclock

Sezione di Probabilità e Statistica Matematica

The spectral gap of random compact hyperbolic surfaces

Laura Monk, Max-Planck-Institut für Mathematik, Bonn

abstract

abstract

The focus of this talk will be the first non-zero eigenvalue of the Laplacian on a compact hyperbolic surface, also called its spectral gap. Similarly to expanders for graphs, surfaces with a large spectral gap are well-connected, and have good mixing properties. In spite of significant effort in the last 35 years, we have failed to construct examples of large surfaces with an optimal spectral gap.
In this talk, based on joint work with Nalini Anantharaman, I will explain how a new probabilistic approach has led to great progress in this question in recent years. Rather than trying to exhibit examples, we now aim at showing that random surfaces have a large spectral gap with probability close to one. This requires us to have a good model to sample random surfaces, and I will briefly introduce two such models.

Volker Mehrmann received his Diploma in mathematics in 1979, his Ph.D. in 1982, and his habilitation in 1987 from the University of Bielefeld, Germany. He spent research years at Kent State University in 1979-1980, at the University of Wisconsin in 1984-1985, and at IBM Research Center in Heidelberg in 1988--1989. After spending the years 1990-1992 as a visiting full professor at the RWTH Aachen, he was a full professor at TU Chemnitz from 1993 to 2000. Since then he is a full professor for Mathematics at TU Berlin.

He is a member of acatech (the German academy of engineering), academia europaea and the European Academy of Scienes, he was president of GAMM the (International association of Applied Mathematics and Mechanics), vice president of the European Mathematical Society (EMS), chair of MATHEON, the Research Center `Mathematics for key technologies´ and chair of the Einstein Center ECMath in Berlin. Since January 2019 he is president of the EMS.

He is SIAM Fellow, has held an ERC Advanced Grant and also was member of the ERC Mathematics Panel. He is editor of several journals in numerical analysis and editor-in-chief of Linear Algebra and its Applications. His research interests are in the areas of numerical mathematics/scientific computing, applied and numerical linear algebra, control theory, the theory and numerical solution of differential-algebraic equations, and recently also in energy based mathematical modeling.

17 Novembre, 2021 15:00

Seminari FDS

Le molteplici relazioni tra storia e didattica della matematica: il caso del calcolo infinitesimale

Laura Branchetti, Università di Milano

online tiny.cc/fds22webex

abstract

abstract

Nel seminario verrà affrontato il tema della relazione tra storia e didattica della matematica nel caso del calcolo infinitesimale, mostrando alcune delle tante possibilità che il dialogo tra questi due punti di vista offre agli insegnanti. Verranno presentati alcuni possibili approfondimenti per la scuola secondaria che prendono spunto dalla storia della matematica, ma anche riflessioni sulle ragioni per cui conoscere la storia può aiutare a capire le numerose difficoltà degli studenti nell'affrontare lo studio del Calculus e dell'Analisi.

Volker Mehrmann received his Diploma in mathematics in 1979, his Ph.D. in 1982, and his habilitation in 1987 from the University of Bielefeld, Germany. He spent research years at Kent State University in 1979-1980, at the University of Wisconsin in 1984-1985, and at IBM Research Center in Heidelberg in 1988--1989. After spending the years 1990-1992 as a visiting full professor at the RWTH Aachen, he was a full professor at TU Chemnitz from 1993 to 2000. Since then he is a full professor for Mathematics at TU Berlin.

He is a member of acatech (the German academy of engineering), academia europaea and the European Academy of Scienes, he was president of GAMM the (International association of Applied Mathematics and Mechanics), vice president of the European Mathematical Society (EMS), chair of MATHEON, the Research Center `Mathematics for key technologies´ and chair of the Einstein Center ECMath in Berlin. Since January 2019 he is president of the EMS.

He is SIAM Fellow, has held an ERC Advanced Grant and also was member of the ERC Mathematics Panel. He is editor of several journals in numerical analysis and editor-in-chief of Linear Algebra and its Applications. His research interests are in the areas of numerical mathematics/scientific computing, applied and numerical linear algebra, control theory, the theory and numerical solution of differential-algebraic equations, and recently also in energy based mathematical modeling.

17 Novembre, 2021 17:00

Sezione di Geometria, Algebra e loro applicazioni

Maximal Estimates in Several Complex Variables

Phillip S. Harrington, University of Arkansas

On line

abstract

abstract

Complex analysis in one variable is closely tied to the study of harmonic functions. In several complex variables, there is also a second-order PDE that is fundamental to the study of the holomorphic functions: the $\bar\partial$-Neumann problem. In contrast with the one variable case, the boundary condition for the $\bar\partial$-Neumann problem is non-coercive, so solution operators for the $\bar\partial$-Neumann problem gain at most one derivative in the Sobolev scale. Given this constraint, we say that a domain admits maximal estimates if the solution operator for the $\bar\partial$-Neumann problem gains two derivatives in every direction except one. We will see that a large class of domains admit maximal estimates, and many difficult problems in several complex variables are easier to study on such domains.

Volker Mehrmann received his Diploma in mathematics in 1979, his Ph.D. in 1982, and his habilitation in 1987 from the University of Bielefeld, Germany. He spent research years at Kent State University in 1979-1980, at the University of Wisconsin in 1984-1985, and at IBM Research Center in Heidelberg in 1988--1989. After spending the years 1990-1992 as a visiting full professor at the RWTH Aachen, he was a full professor at TU Chemnitz from 1993 to 2000. Since then he is a full professor for Mathematics at TU Berlin.

He is a member of acatech (the German academy of engineering), academia europaea and the European Academy of Scienes, he was president of GAMM the (International association of Applied Mathematics and Mechanics), vice president of the European Mathematical Society (EMS), chair of MATHEON, the Research Center `Mathematics for key technologies´ and chair of the Einstein Center ECMath in Berlin. Since January 2019 he is president of the EMS.

He is SIAM Fellow, has held an ERC Advanced Grant and also was member of the ERC Mathematics Panel. He is editor of several journals in numerical analysis and editor-in-chief of Linear Algebra and its Applications. His research interests are in the areas of numerical mathematics/scientific computing, applied and numerical linear algebra, control theory, the theory and numerical solution of differential-algebraic equations, and recently also in energy based mathematical modeling.