21 Marzo, 2018 15:15

Sezione di Analisi

A representation formula for the Laplacian of the distance function

Andrea Mondino, University of Warwick

Sala del Consiglio 7° piano

abstract

abstract

In the seminar I will present a recent work in collaboration with Fabio Cavalletti (SISSA) where, using techniques from optimal transportation, we prove a rather explicit representation formula for the Laplacian of the distance function in spaces with Ricci curvature bounded below. Even if the paper deals with rather general non-smooth spaces, since some results seem new even for smooth manifolds, the seminar will be mostly focused on the smooth framework.



This seminar is organized within the PRIN 2015 Research project «Variational methods, with applications to problems in mathematical physics and geometry» Grant Registration 2015KB9WPT_010, funded by MIUR – Project coordinator Prof. Gianmaria Verzini

21 Marzo, 2018 14:45 in punto

Sezione di Fisica Matematica

Collisions and chaos in the Boltzmann-Grad limit

Sergio Simonella, ENS Lione

Aula seminari MOX, VI piano

abstract

abstract

I will review our current state of knowledge on the mathematical derivation of the Boltzmann equation from Newtonian systems. This problem has been the objective of intensive effort over recent years. I will focus on the size of correlations in the dilute gas and discuss some delicate aspects of the convergence in the kinetic limit. A statistical analysis of collisions via analytical tools provides new insights on the scale transition.

20 Marzo, 2018 12:15

Sezione di Finanza Quantitativa

"The Quantization approach for estimating Exposures in Counterparty risk. Review with numerical Applications"

MICHELE BONOLLO, NUMERIX

Aula Seminari Sesto Piano

I will review our current state of knowledge on the mathematical derivation of the Boltzmann equation from Newtonian systems. This problem has been the objective of intensive effort over recent years. I will focus on the size of correlations in the dilute gas and discuss some delicate aspects of the convergence in the kinetic limit. A statistical analysis of collisions via analytical tools provides new insights on the scale transition.

20 Marzo, 2018 11:00

MOX Seminar

Isogeometric-analysis-based Multi-Index Stochastic Collocation for Elliptic PDEs with random data

Lorenzo Tamellini, Istituto "E. Magenes", Consiglio Nazionale delle Ricerche, Pavia

Aula Seminari 'Saleri' VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano - Edificio 14

abstract

abstract

In many engineering applications governed by PDEs, the parameters of the equations (coefficients, forcing terms, boundary and initial conditions, shape of the domain) are not known exactly but rather affected by a certain degree of uncertainties, and can be described by means of random variables (or random fields).

Uncertainty Quantification (UQ) aims at estimating how the randomness of these "input" parameters affects the "outputs" of the PDE, typically its solution or functionals thereof. UQ techniques are often based on repeatedly solving the PDE at hand for different combinations of the input parameters (i.e., a sampling approach), which requires a significant computational effort.
To reduce such effort, so-called "multi-level" and "multi-index" methods have recently been proposed. These methods explore the variability of the PDE outputs using a hierarchy of suitably chosen discretization levels to balance the PDE discretization error and the sampling error, and refine the discretization of the PDE only when needed. We emphasize in particular that these methodologies are completely "black-box", in the sense that they allow reuse of legacy PDE solvers, and are moreover
embarrassingly parallel.

In this talk we describe in detail one such method, i.e., the so-called Multi-Index Stochastic Collocation method (MISC), which is closely related to the quite popular Sparse-Grids Stochastic Collocation method for the approximation of PDE with random data.
In particular, this method relies on PDE solvers with tensor structure. To this end, we use Isogeometric Analysis (IGA), which is a technique introduced in the early 2000 to bridge the gap between Computer Aided Design (CAD) and PDE-based engineering analysis.
The core idea of IGA is to use the basis functions used by CAD designers to describe geometries (typically Cubic Splines or Non-Uniform Rational B-Splines) as a basis for the approximation of the solution of the PDE as well; the PDE is then solved a traditional Galerkin approach.
Beside the fact that its tensorized construction makes IGA very suitable in the MISC framework, attractive features of IGA are the simplified treatment of complex geometries, and the fact that basis functions with high-order and high-degree of regularity can be easily generated, thanks to the flexibility of the splines bases.

This is a joint work with Joakim Beck and Raul Tempone (KAUST), Abdul-Lateef Haji-Ali (Oxford) and Fabio Nobile (EPFL).

Contact: luca.formaggia@polimi.it

15 Marzo, 2018 14:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

One-Slice Preserving Functions of a Quaternionic Variable

Chiara de Fabritiis, Università Politecnica delle Marche

Aula seminari del terzo piano

abstract

abstract

Regular functions on the skew-field of quaternions were introduced by
Gentili and Struppa some 10 years ago in order to give an analogue of
holomorphic functions in a non commutative setting. After a (short)
introduction, I will give a formula which allows us to simplify the
understanding of the *-product, which corresponds to the pointwise
product of holomorphic functions. The peculiar structure of quaternions,
foliated in copies of complex plane, drives naturally to consider the
classes of functions which preserve either one or all complex slices.
The main part of the talk will be devoted to characterize the functions
whose sum, *-product or conjugate preserve a slice. At the end, time permitting, I will address to the case of *-powers which shows an unexpected connection with a problem of algebraic geometry studied by Causa and Re. (Joint work with A. Altavilla)

14 Marzo, 2018 16:30

Seminario Matematico e Fisico di Milano

The constant scalar curvature equation in some singular spaces

Gilles Carron, Laboratoire de Mathématiques Jean Leray (UMR 6629), Université de Nantes, CNRS

Sala Consiglio, 7 piano, Edificio La Nave, Via Bonardi 9

abstract

abstract

I will survey the recent results about the Yamabe problem on stratified spaces. I will first introduce the scalar curvature and the Yamabe equation for the constant scalar curvature equation and its variational formulation and the results of Obata, Trudinger, Aubin and Schoen for smooth compact manifold. Then I will describe the geometry of stratified space with some 2D and 3D examples. Eventually I will formulate the Yamabe problem for stratified space and explained some of the recent results and will explain some perspectives.

13 Marzo, 2018 15:00

MOX Seminar

Spectral distribution of sequences of structured matrices: GLT theory and applications

Debora Sesana, University of Insubria - Como -

Aula Seminari 'Saleri' VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano - Edificio 14

abstract

abstract

Any discretization of a given differential problem for some sequence of stepsizes h tending to zero leads to a sequence of systems of linear equations {Am xm = bm}, where the dimension of {Am} depends on h and tends to infinity for h going to 0. To properly face the solution of such linear systems, it is important to deeply understand the spectral properties of the matrices {Am} in order to construct efficient preconditioners and to study the convergence of applied iterative methods. The spectral distribution of a sequence of matrices is a fundamental concept. Roughly speak- ing, saying that the sequence of matrices {Am} is distributed as the function f means that the eigenvalues of Am behave as a sampling of f over an equispaced grid of the domain of f, at least if f is smooth enough. The function f is called the symbol of the sequence. Many distribution results are known for particular sequences of structured matrices: diagonal matrices, Toeplitz matrices, etc. and an approximation theory for sequences of matrices has been developed to deduce spectral distributions of “complicated” matrix sequences from the spectral distribution of “simpler” matrix sequences. In this respect, recently, has played a fundamental role the theory of Generalized Locally Toeplitz (GLT) sequences (introduced by Tilli (1998) and Serra-Capizzano (2002, 2006)), which allows to deduce the spectral properties of matrix sequences obtained as a combination (linear combinations, products, inversion) of Toeplitz matrices and diagonal matrices; to this category belong many stiffness matrices arising from the discretization, using various methods, of PDEs. We present the main concepts of this theory with some applications.
This is a joint work with Stefano Serra-Capizzano and Carlo Garoni.

References
[1] C. Garoni, C. Manni, S. Serra-Capizzano, D. Sesana, H. Speleers. Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods. Mathematics of Computation 85 (2016), pp. 1639–1680.
[2] C. Garoni, C. Manni, S. Serra-Capizzano, D. Sesana, H. Speleers. Lusin theorem, GLT sequences and matrix computations: an application to the spectral analysis of PDE discretization matrices. Journal of Mathematical Analysis and Applications, 446 (2017), pp. 365–382.
[3] C. Garoni, S. Serra-Capizzano. Generalized Locally Toeplitz Sequences: Theory and Applications. Springer 2017.
[4] C. Garoni, S. Serra-Capizzano, D. Sesana. Block Locally Toeplitz Sequences: Construction and Properties. Springer INdAM Series: proceeding volume of the Cortona 2017 meeting, submitted.
[5] C. Garoni, S. Serra-Capizzano, D. Sesana. Block Generalized Locally Toeplitz Sequences:
Topological Construction, Spectral Distribution Results, and Star-Algebra Structure.
Springer INdAM Series: proceeding volume of the Cortona 2017 meeting, submitted.

Contact: franca.calio@polimi.it

8 Marzo, 2018 14:30 in punto

Sezione di Geometria, Algebra e loro applicazioni

Post-Quantum Group-based Cryptography

Delaram Kahrobaei, New York City College of Technology

Aula seminari, III piano, Dipartimento di matematica

abstract

abstract

The National Security Agency (NSA) in August 2015 announced plans to transition to post-quantum algorithms “Currently, Suite B cryptographic algorithms are specified by the National Institute of Standards and Technology (NIST) and are used by NSA’s Information Assurance Directorate in solutions approved for protecting classified and unclassified National Security Systems (NSS). Below, we announce preliminary plans for transitioning to quantum resistant algorithms.”

Shortly after the National Institute of Standardization and Technology (NIST) announced a call to select standards for post-quantum public-key cryptosystems.

The academic and industrial communities have suggested as the quantum-resistant primitives: Lattice-based, Multivariate, Code-based, Hash-based, Isogeny-based and group-based primitives.

In this talk I will focus on some ideas of (semi)group-based primitives. The one which was proposed to NIST is by SecureRF company based in Connecticut, among its founders there is a number theorist (Goldfeld) and two group theorists (Anshel and Anshel). They proposed a digital signature using a hard algorithmic problem in Braid groups, namely conjugacy problem.

I will then give a survey of some other suggested group-based cryptosystems that could be claimed as post-quantum cryptosystems.

6 Marzo, 2018 12:15

Sezione di Finanza Quantitativa

HARK the SHARK: Realized Volatility Modelling with Measurement Errors and Nonlinear Dependencies

Giuseppe Buccheri, Scuola Normale Superiore Pisa

Aula seminari terzo piano

The National Security Agency (NSA) in August 2015 announced plans to transition to post-quantum algorithms “Currently, Suite B cryptographic algorithms are specified by the National Institute of Standards and Technology (NIST) and are used by NSA’s Information Assurance Directorate in solutions approved for protecting classified and unclassified National Security Systems (NSS). Below, we announce preliminary plans for transitioning to quantum resistant algorithms.”

Shortly after the National Institute of Standardization and Technology (NIST) announced a call to select standards for post-quantum public-key cryptosystems.

The academic and industrial communities have suggested as the quantum-resistant primitives: Lattice-based, Multivariate, Code-based, Hash-based, Isogeny-based and group-based primitives.

In this talk I will focus on some ideas of (semi)group-based primitives. The one which was proposed to NIST is by SecureRF company based in Connecticut, among its founders there is a number theorist (Goldfeld) and two group theorists (Anshel and Anshel). They proposed a digital signature using a hard algorithmic problem in Braid groups, namely conjugacy problem.

I will then give a survey of some other suggested group-based cryptosystems that could be claimed as post-quantum cryptosystems.

1 Marzo, 2018 14:00

MOX Colloquia

Graphene from Molecular Mechanics

Ulisse Stefanelli, Faculty of Mathematics, University of Vienna, and CNR Pavia

Aula Consiglio VII Piano - Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO

abstract

abstract

I will review recent results on the analysis of graphene via variational methods. The setting is that of Molecular Mechanics: carbon atoms are identified with their nuclear positions and their bonds are described via classical two- and three-body interactions. I will discuss the crystallization problem in both two and three dimensions, the mechanics of the (planar) crystal under tension, the rolling-up of free graphene patches, and the emergence of ripples in suspended samples.

Contact: francesco.bonaldi@polimi.it

Ulisse Stefanelli received his PhD from the University of Pavia in 2003. He has been working at the Institute for Applied Mathematics and Information Technology 'E. Magenes' of the CNR since 2001. Since 2013 he holds the chair in 'Applied Mathematics and Modeling' at the University of Vienna. His activity focuses on calculus of variations and PDEs, especially in application to Material Science.