16 Marzo, 2022 12:15

Seminari di Cultura Matematica

The quantum women - le scienziate della meccanica quantistica

Gabriella Greison, Fisica e divulgatrice scientifica

Politecnico di Milano Campus Bonardi Edificio 14 Aula B 2.1 - Collegamento webex su tiny.cc/fdsCM2022

16 Marzo, 2022 17:00

Sezione di Geometria, Algebra e loro applicazioni

Horospheres in several complex variables

Leandro Arosio, Università di Roma 2

On line

abstract

abstract

A horocycle in the unit disk of the complex plane is a euclidean disk which is internally tangent to a point p of the boundary of the disk. Horocycles are limits of Poincaré balls as the center moves towards the point p while the radius grows suitably. The classical Julia lemma is a boundary version of the Schwarz lemma which shows that horocycles are useful to understand the behaviour of a holomorphic self-map of the disk near a point of the boundary. In this talk we deal with the generalization of this concept to several complex variables: horospheres. The existence of horospheres in bounded strongly convex domains of C^n was proved by Abate in 1988 using Lempert’s theory of complex geodesics. It is difficult to generalize such proof to bounded strongly pseudoconvex domains, which is the natural class of domains to study in this context.

In this talk I will show how to obtain this generalization following a different route, that is, proving that the horofunction compactification of the domain is topologically equivalent to its Gromov compactification. This is a joint work with Matteo Fiacchi, Sébastien Gontard, and Lorenzo Guerini.

14 Marzo, 2022 09:30

Laboratorio FDS

PiDay 2022

Laboratorio FDS, Politecnico di Milano

in streaming su www.facebook.com/effediesse.polimi

A horocycle in the unit disk of the complex plane is a euclidean disk which is internally tangent to a point p of the boundary of the disk. Horocycles are limits of Poincaré balls as the center moves towards the point p while the radius grows suitably. The classical Julia lemma is a boundary version of the Schwarz lemma which shows that horocycles are useful to understand the behaviour of a holomorphic self-map of the disk near a point of the boundary. In this talk we deal with the generalization of this concept to several complex variables: horospheres. The existence of horospheres in bounded strongly convex domains of C^n was proved by Abate in 1988 using Lempert’s theory of complex geodesics. It is difficult to generalize such proof to bounded strongly pseudoconvex domains, which is the natural class of domains to study in this context.

In this talk I will show how to obtain this generalization following a different route, that is, proving that the horofunction compactification of the domain is topologically equivalent to its Gromov compactification. This is a joint work with Matteo Fiacchi, Sébastien Gontard, and Lorenzo Guerini.

14 Marzo, 2022 17:30 in punto

Sezione di Probabilità e Statistica Matematica

No repulsion between critical points for random plane wave and planar Gaussian random fields.

Valentina Cammarota, Università di Roma La Sapienza

abstract

abstract

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. This is known to be true on average. We discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius. Joint work with Dmitry Beliaev and Igor Wigman.

1 Marzo, 2022 10:00 in punto

Sezione di Probabilità Quantistica

Perturbation Analysis of Quantum Reset Models

Alain Joye, Université Joseph Fourier, Grenoble

Aula seminari (piano III)

abstract

abstract

We consider Lindblad operators of Quantum Reset Models that describe the effective dynamics of systems subject to stochastic resets. We analyse in particular weakly coupled tripartite systems such that for zero coupling two subsystems are driven by independent reset Lindbladians, while the third system is driven by a Hamiltonian. Under generic assumptions on the coupling, we prove the existence of a unique steady state, and derive large time dynamical consequences.
This is joint work with Géraldine Haack.

28 Febbraio, 2022 17:30 in punto

Sezione di Probabilità e Statistica Matematica

Zero-sum mean-field Dynkin games.

Boualem Djehiche, KTH, Stockholm

abstract

abstract

I will review recent results on a class of zero-sum game problems of mean-field type which extend the classical zero-sum Dynkin game problems. The results I will highlight include
sufficient conditions under which such a game admits a value and a saddle point and a characterization of the value of the game in terms of a specific class of doubly reflected backward stochastic differential equations (BSDEs) of mean-field type. Moreover, a corresponding system of weakly interacting values of zero-sum Dynkin games is introduced and is shown to converge to the value of the zero-sum mean-field Dynkin game. In particular, a propagation of chaos result is derived.
This is a joint work with Roxana Dumistrescu.

24 Febbraio, 2022 11:00

Sezione di Finanza Quantitativa

Experiments and computational models of trading in decentralized markets

Giulia Iori, City, University of London

Zoom virtual Room polimi-it.zoom.us/j/81675315372 and Aula Seminari III piano, Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

We present the results of experimental and computational analysis of decentralized markets with asymmetric information. We examine how market efficiency and agents profit distribution depend on the structure of the underlying trading network and on the number and location of insiders. We find that market efficiency and fairness are both affected by the network macro-structure and that the network that better supports price efficiency is not necessarily the one that leads to greater fairness in profits. By calibrating a behavioural Agent Based Model to the experimental data we characterize the traders’ learning rule and shed further lights on the dynamics of information diffusion.

24 Febbraio, 2022 14:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

On the strength of homogeneous polynomials

Alessandro Oneto, Università degli Studi di Trento

Aula seminari del terzo piano

abstract

abstract

The strength of a homogeneous polynomial is the smallest length of an additive decomposition as sum of reducible forms. It is called slice rank if we additionally require that the reducible forms have a linear factor. Geometrically, the slice rank corresponds to the smallest codimension of a linear space contained in the hypersurface defined by the form. Due to this relation, it is well-known and easy to compute the value of the general slice rank and also to show that the set of forms with bounded slice rank is Zariski-closed. In this talk, I will present the following results from recent joint works with A. Bik, E. Ballico and E. Ventura:
(1) the set of forms with bounded strength is not always Zariski-closed: this is an asymptotic result in the number of variables proved by using the theory of polynomial functors;
(2) for general forms, strength and slice rank are equal: this is proved by showing that the largest component of the secant variety of the variety of reducible forms is the secant variety of the variety of forms with a linear factor.

24 Febbraio, 2022 14:00

Sezione di Fisica Matematica

On Well-posedness for the Gross-Pitaevskii and Hartree Hierarchy Equations

Zied Ammari, IRMAR - Université de Rennes I

Seminario Online - Registrarsi sul sito nel link per avere le coordinate Zoom

abstract

abstract

Gross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs related to mean field theory of Bose gases. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this talk I will introduce a new approach based on a duality between hierarchies and Liouville equations. Several new results are obtained as an outcome of this approach.

23 Febbraio, 2022 17:00

Seminario Matematico e Fisico di Milano

Control of multiagent systems viewed as dynamical systems on the Wasserstein space

Marc Quincampoix, Université de Brest, France

Sala Consiglio 7 piano, Edificio La Nave e polimi-it.zoom.us/j/81969494860

abstract

abstract

This talk is devoted to an overview of recent results on the optimal control of dynamical systems on probability measures modelizing the evolution of a large number of agents.

The system is composed by a number of agents so huge, that at each time only a statistical description of the state is available. A common way to model such kind of system is to consider a macroscopic point of view, where the state of the system is
described by a (time-evolving) probability measure on $R^d$ (which the underlying space where the agents move). So we are facing to a two-level system where the mascroscopic dynamic concerns probability measure while the microscopic dynamic - which describes the evolution of an individual agent - is a controlled differential equation on $ R^d$.

Associated to this dynamics on the Wasserstein space, one can associate a cost which allows to define a value function. We discuss the characterization of this value function through a Hamilton Jacobi Bellman equation stated on the Wasserstein space. We also discuss the problem of compatibility of state constraints with a multiagent control system. Since the Wasserstein space can be also viewed as the set of the laws of random variables in a suitable $L^2$ space, one can hope to reduce our problems to $L^2$ analysis. We discuss when this is possible.
This overview talk is based on several works in collaboration with I. Averboukh, P. Cardaliaguet, G. Cavagnari, C. Jimenez and A. Marigonda.