12 Aprile, 2021 15:00

Sezione di Fisica Matematica

Which magnetic fields support a zero mode?

Michael Loss, Georgia Institute of Technology, Atlanta, GA, US

online

abstract

abstract

I present some results concerning the size of magnetic fields
that support zero modes for the three dimensional Dirac equation and
related problems for spinor equations. The critical quantity, is the 3/2
norm of the magnetic field B. The point is that the spinor structure
enters the analysis in a crucial way.
This is joint work with Rupert Frank at Caltech.

8 Aprile, 2021 15:00

Laboratorio FDS

The Epistemic Route to Reflection

Marianna Antonutti Marfori, MCMP, LMU Munich

online: politecnicomilano.webex.com/politecnicomilano/j.php?MTID=md18a1c58416f6039dd546beedad7b9ad

abstract

abstract

Gödel’s incompleteness theorems show that any formal theory that contains a certain minimal amount of arithmetic is incomplete: there are statements in the language of the theory that the theory cannot either prove or disprove. Such statements include statements (called reflection principles) that formally express the soundness of the theory, i.e. that everything that is provable from the axioms is true. This seems to contrast with our pre-theoretical understanding of arithmetic in everyday life and in science, and with our readiness to accept that every statement that can be proved from our axioms is true. Many authors have therefore sought to justify the addition to our formal theories of such reflection principles by different means. This paper presents a new way of justifying the addition of reflection principles on the basis of epistemic considerations, by employing certain uncontroversial properties of the notion of informal or absolute provability for arithmetic. S!
tarting from the observation that the fundamental properties of this notion can be formally characterised by using the axioms of the modal logic S4 (along the lines already proposed by Gödel), I will show that the recognition that the axioms and rules of inference of Peano arithmetic (PA) correctly formalise informal arithmetical reasoning is sufficient to formally imply the local and uniform reflection schemes.

30 Marzo, 2021 11:15 oclock

Sezione di Analisi

Optimization of eigenvalues of partially hinged rectangular composite plates

Alessio Falocchi, Dipartimento di Scienze Matematiche, Politecnico di Torino

abstract

abstract

We study the spectrum of non-homogeneous partially hinged rectangular plates having structural engineering applications. A possible way to prevent instability phenomena is to optimize the frequencies of certain oscillating modes with respect to the density function of the plate. In striking contrast to what happens under Dirichlet boundary conditions, we prove a result of positivity preserving property for the biharmonic operator of the related problem through fine estimates of the Fourier expansion of the corresponding Green function. This is useful in order to get qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue.
This is a joint work with Elvise Berchio (Politecnico di Torino).

24 Marzo, 2021 12:15

Seminari di Cultura Matematica

Progetto COMPASS: Control for Orbit Manoeuvring Through Perturbations for Application to Space Systems

Camilla Colombo, Politecnico di Milano

online: tiny.cc/fdsCM2021

We study the spectrum of non-homogeneous partially hinged rectangular plates having structural engineering applications. A possible way to prevent instability phenomena is to optimize the frequencies of certain oscillating modes with respect to the density function of the plate. In striking contrast to what happens under Dirichlet boundary conditions, we prove a result of positivity preserving property for the biharmonic operator of the related problem through fine estimates of the Fourier expansion of the corresponding Green function. This is useful in order to get qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue.
This is a joint work with Elvise Berchio (Politecnico di Torino).

24 Marzo, 2021 15:15 oclock

Sezione di Analisi

Explicit bounds for the generation of a lift force exerted by steady-state Navier-Stokes flows over a fixed obstacle

Gianmarco Sperone, Dept. of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague

Url: polimi-it.zoom.us/j/87365124649?pwd=SjZOcDgrQU9qUGZCM3FZRmxCTUhlUT09

abstract

abstract

We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian. The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano)

22 Marzo, 2021 17:30 oclock

Sezione di Finanza Quantitativa

COVID Narrative Risk: A Computational Linguistic Approach to the Econometric Identification of Narrative Risk During the COVID-19 Pandemic

Valerio Potì, University College Dublin

abstract

abstract

In this paper, we study the role in financial markets of narratives related to the ongoing COVID-19 pandemic. The pandemic represents a natural setting for the development of narratives that may effects or be affected by financial markets. We thus treat the pandemic as a natural experiment on the relation between prevailing narratives and financial markets. We adopt the SIR model and natural language processing on financial newspaper news and Twitter tweets that deal at the same time with financial market topics and COVID-19 to study the dynamics and determinants of coronavirus narrative epidemics. Our aim is to establish whether there an “infodemic” develops, and whether the prevailing narrative, whether resemnbling an infodemic or otherwise, drives or is driven by financial markets developments, controlling for developments regarding the COVID-19 pandemic. We find associations between narratives about the epidemic, stock market dynamics (both regarding returns and volatility) and government responses to COVID-19. We also find that the narrative spread does resemble an infodemic, since it is well described by the SIR epidemic model. Our estimates of the shape of the narrative infodemic curves in different countries depend on whether the COVID-19 outbreak occurred early and on its severity. Negative tones and tones communicating uncertainty are prevalent in the growing stage of the infodemic. We find preliminary evidence of a causality relation between the negative and uncertainty tones of coronavirus tweets and both market return and volatility change.

18 Marzo, 2021 14:00

MOX Colloquia

Deep Learning and Computations of high-dimensional PDEs

Siddhartha Mishra, ETH Zurich (CH)

Online seminar: mox.polimi.it/elenco-seminari/?id_evento=2038&t=763724

abstract

abstract

Partial Differential Equations (PDEs) with very high-dimensional state and/or parameter spaces arise in a wide variety of contexts ranging from computational chemistry and finance to many-query problems in various areas of science and engineering. In this talk, we will survey recent results on the use of deep neural networks in computing these high-dimensional PDEs. We will focus on two different aspects i.e., the use of supervised deep learning, in the form of both standard deep neural networks as well as recently proposed DeepOnets, for efficient approximation of many-query PDEs and the use of physics informed neural-networks (PINNs) for the computation of forward and inverse problems for PDEs with high-dimensional state spaces.

Siddhartha Mishra is currently a Professor for Applied Mathematics at ETH Zurich, Switzerland. He received his PhD degree in Mathematics in 2005, from the Indian Institute of Science and Tata Institute of Fundamental Research, Bangalore, India. Mishra's research interests are in the areas of numerical analysis, scientific computing, nonlinear PDEs, computational fluid and plasma dynamics, computational geosciences and in machine learning. His work on the design and analysis of efficient numerical methods for hyperbolic systems of conservation laws, their implementation on state of the art HPC platforms and applications in science and engineering have been recognized by many awards and honors. These include ERC starting grant (2012), Richard von Mises prize (2015), ERC consolidator grant (2017), Jacques Louis Lions medal (2018), Collatz Prize (2019) and Infosys prize (2019). He has also been an invited speaker at the International congress of Mathematicians (2018)

17 Marzo, 2021 12:15

Seminari di Cultura Matematica

Buchi neri e campi gravitazionali estremi

Francesco Belgiorno, Politecnico di Milano

online: tiny.cc/fdsCM2021

Partial Differential Equations (PDEs) with very high-dimensional state and/or parameter spaces arise in a wide variety of contexts ranging from computational chemistry and finance to many-query problems in various areas of science and engineering. In this talk, we will survey recent results on the use of deep neural networks in computing these high-dimensional PDEs. We will focus on two different aspects i.e., the use of supervised deep learning, in the form of both standard deep neural networks as well as recently proposed DeepOnets, for efficient approximation of many-query PDEs and the use of physics informed neural-networks (PINNs) for the computation of forward and inverse problems for PDEs with high-dimensional state spaces.

Siddhartha Mishra is currently a Professor for Applied Mathematics at ETH Zurich, Switzerland. He received his PhD degree in Mathematics in 2005, from the Indian Institute of Science and Tata Institute of Fundamental Research, Bangalore, India. Mishra's research interests are in the areas of numerical analysis, scientific computing, nonlinear PDEs, computational fluid and plasma dynamics, computational geosciences and in machine learning. His work on the design and analysis of efficient numerical methods for hyperbolic systems of conservation laws, their implementation on state of the art HPC platforms and applications in science and engineering have been recognized by many awards and honors. These include ERC starting grant (2012), Richard von Mises prize (2015), ERC consolidator grant (2017), Jacques Louis Lions medal (2018), Collatz Prize (2019) and Infosys prize (2019). He has also been an invited speaker at the International congress of Mathematicians (2018)

17 Marzo, 2021 17:00

Sezione di Geometria, Algebra e loro applicazioni

A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps

Oliver Roth, University of Wuerzburg

On line

abstract

abstract

We establish several invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphic selfmaps of strongly convex domains in CN in the spirit of the boundary Schwarz lemma of Burns-Krantz.

Firstly, we focus on the case of the unit disk and prove a general boundary rigidity theorem for conformal pseudometrics with variable curvature. In its simplest cases this result already includes new types of boundary versions of the lemmas of Schwarz-Pick, Ahlfors-Schwarz and Nehari-Schwarz. The proof is based on a new Harnack-type inequality as well as a boundary Hopf lemma for conformal pseudometrics which extend earlier interior rigidity results of Golusin, Heins, Beardon, Minda and others. Secondly, we prove similar rigidity theorems for sequences of conformal pseudometrics, which even in the interior case appear to be new. For instance, a first sequential version of the strong form of Ahlfors' lemma is obtained.

As an auxiliary tool we establish a Hurwitz-type result about preservation of zeros of sequences of conformal pseudometrics.

Thirdly, we apply the one-dimensional sequential boundary rigidity results together with a variety of techniques from several complex variables to prove a boundary version of the Schwarz-Pick lemma for holomorphic maps of strongly convex domains in $\C^N$ for $N>1$.

Siddhartha Mishra is currently a Professor for Applied Mathematics at ETH Zurich, Switzerland. He received his PhD degree in Mathematics in 2005, from the Indian Institute of Science and Tata Institute of Fundamental Research, Bangalore, India. Mishra's research interests are in the areas of numerical analysis, scientific computing, nonlinear PDEs, computational fluid and plasma dynamics, computational geosciences and in machine learning. His work on the design and analysis of efficient numerical methods for hyperbolic systems of conservation laws, their implementation on state of the art HPC platforms and applications in science and engineering have been recognized by many awards and honors. These include ERC starting grant (2012), Richard von Mises prize (2015), ERC consolidator grant (2017), Jacques Louis Lions medal (2018), Collatz Prize (2019) and Infosys prize (2019). He has also been an invited speaker at the International congress of Mathematicians (2018)

17 Marzo, 2021 09:15

MOX Seminar

Composition-on-function regression model for the analysis of spectroscopic data

Mara Bernardi, MOX, Dipartimento di Matematica - Politecnico di Milano

Online seminar: mox.polimi.it/elenco-seminari/?id_evento=2039&t=763721

abstract

abstract

Silicate glasses are widely present in natural volcanic rocks and terrestrial planets are largely covered by such products. The study of the spectral response of glasses is of great importance in the context of planetary investigation to better interpret available and future remotely sensed spectra. In this work, we consider the problem of retrieving the chemical composition of a silicate glass based on its spectral response. This problem poses challenges from a modelling perspective as it involves heterogeneous and complex data whose structure and properties should be appropriately accounted for. We propose a regression model with compositional response and functional predictor. The model allows to account for the geometry of the space of compositions and for the inherent continuous nature of the spectrum. Moreover, we propose a nonparametric framework for inference and prediction. The dataset analysed concerns the spectral response of different series of silicate glasses presenting a range of chemical compositions that covers most rocks on planet Earth.

Contact: Laura Sangalli, laura.sangalli@polimi.it

Siddhartha Mishra is currently a Professor for Applied Mathematics at ETH Zurich, Switzerland. He received his PhD degree in Mathematics in 2005, from the Indian Institute of Science and Tata Institute of Fundamental Research, Bangalore, India. Mishra's research interests are in the areas of numerical analysis, scientific computing, nonlinear PDEs, computational fluid and plasma dynamics, computational geosciences and in machine learning. His work on the design and analysis of efficient numerical methods for hyperbolic systems of conservation laws, their implementation on state of the art HPC platforms and applications in science and engineering have been recognized by many awards and honors. These include ERC starting grant (2012), Richard von Mises prize (2015), ERC consolidator grant (2017), Jacques Louis Lions medal (2018), Collatz Prize (2019) and Infosys prize (2019). He has also been an invited speaker at the International congress of Mathematicians (2018)