16 Marzo, 2023 15:15

Sezione di Analisi

Some results on the Stokes eigenvalue problem under Navier boundary conditions

Alessio Falocchi, Politecnico di Milano

Aula Seminari III piano

abstract

abstract

We study the Stokes eigenvalue problem under Navier boundary conditions in 2D or 3D bounded domains with connected boundary of class $ C^1 $. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple. We apply these results to show the validity/failure of a suitable Poincaré inequality. We then consider the general version of the problem in any space dimension with $ n\geq2 $, characterizing the kernel of the strain tensor for solenoidal vector fields with homogeneous normal trace. We conclude analyzing some similarities and differences with the Laplacian eigenvalue problem.

This is based on a joint work with Filippo Gazzola, Politecnico di Milano.

14 Marzo, 2023 11:30

Seminario Matematico e Fisico di Milano

On KPZ problem and statistics of stochastic flows

Konstantin Khanin, University of Toronto

Aula 3014 - Dipartimento di Matematica e Applicazioni - Edificio U5 (Milano-Bicocca)

abstract

abstract

In this talk we will discuss a geometrical approach to the problem of the KPZ Universality.
Instead of looking at the height (interface) function and Airy processes, we will focus on the statistics
of shocks and points of concentration of mass. We will also discuss the connection with the problem
of the coalescing Brownian motions and coalescing fractional Brownian motions.

13 Marzo, 2023 10:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

Hilbert series of matroid Chow rings and cohomologies

Luis Ferroni, KTH Stockholm

Aula seminari III piano

abstract

abstract

Starting with a matroid M one can construct three important algebraic objects: the Chow ring, the augmented Chow ring and the intersection cohomology. They play instrumental roles within the proofs of the log-concavity of the Whitney numbers of the first kind, and the top-heaviness of the Whitney numbers of the second kind. We will describe the combinatorics of their Hilbert series for matroids in general and uniform matroids in particular. Several aspects such as unimodality, gamma-positivity and real-rootedness will be discussed under different lenses. In particular, the Koszulness of the Chow ring and the augmented Chow ring seem to be a hint towards the real-rootedness conjectures.

8 Marzo, 2023 12:15

Seminari di Cultura Matematica

Il concetto di tempo nella filosofia della natura di Isaac Newton

Niccolò Guicciardini, Università Statale di Milano

In streaming tiny.cc/CM2023

Starting with a matroid M one can construct three important algebraic objects: the Chow ring, the augmented Chow ring and the intersection cohomology. They play instrumental roles within the proofs of the log-concavity of the Whitney numbers of the first kind, and the top-heaviness of the Whitney numbers of the second kind. We will describe the combinatorics of their Hilbert series for matroids in general and uniform matroids in particular. Several aspects such as unimodality, gamma-positivity and real-rootedness will be discussed under different lenses. In particular, the Koszulness of the Chow ring and the augmented Chow ring seem to be a hint towards the real-rootedness conjectures.

2 Marzo, 2023 15:15

Sezione di Analisi

Rotund Gâteaux smooth norms which are not locally uniformly rotund

Jacopo Somaglia, Politecnico di Milano

Aula seminari III piano

abstract

abstract

In a recent monograph on renorming techniques in Banach spaces, A.J. Guirao, V. Montesinos, and V. Zizler posed the following problem; does every infinite-dimensional separable Banach space admit a norm that is rotund and Gâteaux smooth but not locally uniformly rotund?
The aim of the talk is to provide a positive answer to the above question. All the presented results have been obtained in collaboration with C.A. De Bernardi.

1 Marzo, 2023 12:15

Seminari di Cultura Matematica

Event Horizon Telescope: risultati e prospettive

Mariafelicia De Laurentis, Università di Napoli Federico II

Sala Consiglio VII piano - tiny.cc/CM2023

In a recent monograph on renorming techniques in Banach spaces, A.J. Guirao, V. Montesinos, and V. Zizler posed the following problem; does every infinite-dimensional separable Banach space admit a norm that is rotund and Gâteaux smooth but not locally uniformly rotund?
The aim of the talk is to provide a positive answer to the above question. All the presented results have been obtained in collaboration with C.A. De Bernardi.

1 Marzo, 2023 

Sezione di Analisi

Successioni non nulle i cui infiniti momenti sono tutti nulli.

Enrico Laeng, Dipartimento di matematica, Politecnico di Milano

Aula seminari, III piano, Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Il teorema di Stone-Weierstrass implica che se tutti i momenti su un intervallo (integrali della funzione per una potenza intera della variabile) di una funzione continua sono nulli, allora anche tale funzione deve essere nulla. E` abbastanza ben noto che questo non e` piu` vero se prendiamo i momenti su una semiretta o su una retta. Noi affrontiamo il caso discreto, ovvero l'esistenza di successioni tali che le serie che hanno per addendi i termini della successione moltiplicati per una qualsiasi potenza intera dell'indice di convergono a zero grazie a un peculiare gioco di cancellazioni. E` un risultato generale che ha anche conseguenze per la teoria delle probabilita` perche` quantita` come media, varianza, curtosi, etc. sono legate ai momenti.

28 Febbraio, 2023 14:30

MOX Seminar

Uncertainty Quantification for spatially-extended neurobiological networks

Daniele Avitabile, VU-Amsterdam

Aula Saleri

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

23 Febbraio, 2023 14:00

MOX Colloquia

Applications of AAA rational approximation

Nick Trefethen, University of Oxford

Aula Consiglio VII piano - Dipartimento di Matematica

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

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.

20 Febbraio, 2023 17:30

Seminario Matematico e Fisico di Milano

Schrodinger operators with singular boundary conditions

Jan Derezinski, University of Warsaw

Sala Consiglio 7 piano, Edificio La Nave

abstract

abstract

In order to define a differential operator on a domain with boundary one often needs
to specify boundary conditions. In the presence of singular terms this can be quite tricky. It may lead to interesting and surprising phase diagrams.
I will illustrate this with the so-called perturbed Bessel operators,
that is one-dimensional Schrodinger operators on the half-line of the form
$$-\partial_x^2+(m^2-1/4) 1/x^2+ Q(x).$$
The talk will be based on my joint work with Jeremy Faupin.

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.