16 Febbraio, 2021 15:15

Sezione di Analisi

Symmetry and rigidity for composite membranes and plates

Eugenio Vecchi, Politecnico di Milano

Link: polimi-it.zoom.us/j/86140582546?pwd=U0hoTEhhalo1Q2NLeFAvTWd2TTdUUT09

abstract

abstract

The composite membrane problem is an eigenvalue optimization problem that can be formulated as follows:
Build a body of prescribed shape out of given materials (of varying densities) in such a way that the body has a prescribed mass and so that the basic frequency of the resulting membrane (with fixed boundary) is as small as possible.
In the first part of the talk we will review the known results and present a Faber-Krahn-type result obtained in collaboration with G. Cupini (Università di Bologna).
A natural extension of the above problem to the case of plates is the composite plate problem, which is an eigenvalue optimization problem involving the bilaplacian operator. The Euler-Lagrange equation associated to it is a fourth-order PDE that is coupled with Navier boundary conditions (for the hinged plate). In the second part of the talk we will focus on symmetry properties of optimal pairs. These results have been obtained in collaboration with F. Colasuonno (Università di Bologna).

10 Febbraio, 2021 15:00

Seminari FDS

Digital Storytelling e narrazione matematica: costruire competenze matematiche online

Giovannina Albano, Università degli Studi di Salerno

online

abstract

abstract

Il seminario presenta la progettazione e realizzazione di attività di insegnamento/apprendimento della matematica in ambienti digitali online. La metodologia soggiacente, sviluppata nell’ambito di un Progetto di Ricerca di Interesse Nazionale, sfrutta una duplice metafora di narrazione e un’opportuna organizzazione didattico-tecnologica, come elementi abilitanti lo sviluppo di competenze matematiche. Il progetto, nato per integrare e sfruttare le potenzialità dell'ambiente digitale nel contesto della didattica in presenza, assume particolare significatività nel nuovo contesto di didattica a distanza in cui la scuola si trova, a causa dell’attuale pandemia.

10 Febbraio, 2021 11:15

Sezione di Analisi

Some global results for homogeneous Hormander sums of squares

Stefano Biagi, Politecnico di Milano

Link: polimi-it.zoom.us/j/81317327136?pwd=S1VuZ1NQVmxmckZuUWVzbytlZVQrUT09

abstract

abstract

In this talk we present several global results concerning the class of the homogeneous Hörmander sums of squares. As the name suggests, the operators falling in this class are sums of squares of smooth vector fields which are homogeneous of degree 1 with respect to a family of non-isotropic diagonal maps (usually called dilations); moreover, these operators intervene in several contexts of interest (Lie group Theory, sub-Riemannian manifolds, Mathematical Finance, etc.).

After a brief introduction on general sub-elliptic operators (of which any homogeneous sum of squares is a particular case), we properly introduce the class of the homogeneous Hörmander sums of squares and we discuss some global qualitative aspects regarding these operators: global lifting on Carnot groups; existence/global estimates for the associated fundamental solution and heat kernel; maximum principles on unbounded domains.

The results presented in this talk are contained in several papers in collaboration with A. Bonfiglioli, M. Bramanti and E. Lanconelli.

8 Febbraio, 2021 15:00

Sezione di Fisica Matematica

New results on the Lieb-Thirring inequality

Mathieu Lewin, CEREMADE, Université Paris Dauphine, Paris

online

abstract

abstract

The Lieb-Thirring inequality is a generalization of the
Gagliardo-Nirenberg inequality, which plays a central role in the
analysis of large quantum systems. In this talk I will first introduce
the inequality and then focus the discussion on its best constant. After
reviewing what is believed, what is known and what is open, I will
present new results on the value of the best constant as well as
numerical simulations in 1D and 2D.
Collaboration with Rupert L. Frank (Caltech, USA) and David Gontier
(Paris-Dauphine, France).

2 Febbraio, 2021 11:15

Sezione di Analisi

Minimal time optimal control for the moon lander problem

Elsa Marchini, Politecnico di Milano

abstract

abstract

We study a variant of the classical safe landing optimal control problem in aerospace, introduced by Miele in the Sixties, where the target was to land a spacecraft on the moon by minimizing the consumption of fuel. Assuming that the spacecraft has a failure and that the thrust (representing the control) can act in both vertical directions, the new target becomes to land safely by minimizing time, no matter of what the consumption is. In dependence of the initial data (height, velocity, and fuel), we prove that the optimal control can be of four different kinds, all being piecewise constant. Our analysis covers all possible situations, including the nonexistence of a safe landing strategy due to the lack of fuel or for heights/velocities for which also a total braking is insufficient to stop the spacecraft.
This talk is based on a joint work with Filippo Gazzola

1 Febbraio, 2021 11:45

Seminario Matematico e Fisico di Milano

Mathematics vs Dementia

Alain Goriely, University of Oxford

polimi-it.zoom.us/j/83674264668

abstract

abstract

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no cure. Yet, a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages associated with various cognitive deficits and pathologies. How can we use mathematics to gain insight into this process and, doing so, gain understanding about how the brain works? In this talk, I will show that by linking new mathematical theories to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions.

1 Febbraio, 2021 15:00

Seminario Matematico e Fisico di Milano

JMGT [Jordan-Moore Gibson-Thompson] dynamics arising in non- linear acoustics - a view from the boundary

Irena Lasiecka, University of Memphis

polimi-it.zoom.us/j/83674264668

abstract

abstract

A third-order (in time) JMGT equation is a nonlinear (quasi-linear) Partial Differential Equation (PDE) model introduced to describe a non-linear propagation of high frequency acoustic waves. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences-including high intensity focused ultrasound [HIFU] technologies, lithotripsy, welding and others. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second order in time equation referred to as Westervelt equation. Replacing a classical heat transfer by heat waves gives rise to the third order in time derivative scaled by a small parameter $\tau > 0$, the latter represents the thermal relaxation time parameter and is intrinsic to the properties of the medium where the dynamics occurs.
The aim of the present lecture is to provide a brief overview of recent results in the area which are pertinent to both linear and non-linear dynamics. From the mathematical point of view JMGT, can be seen as a nonlinear perturbation of a third order strictly hyperbolic system, which however has a characteristic boundary. This feature has, of course, strong implications on boundary behavior [both regularity and controllability] which can not be patterned after classical hyperbolic systems theory [as it is the case for the wave equation]. As a consequence, the analysis of regularity [both forward and inverse estimates] is particularly challenging-even in the linear case. Several recent results pertaining to boundary stabilization, optimal control and asymptotic analysis of the solutions with vanishing time relaxation parameter will be presented and discussed. In all these case, peculiar features associated with the third order dynamics leads to novel phenomenological behaviors.

1 Febbraio, 2021 16:15

Seminario Matematico e Fisico di Milano

Counting minimal surfaces in negatively curved manifolds

Andrè Neves, University of Chicago

polimi-it.zoom.us/j/83674264668

abstract

abstract

After presenting some of the recent progress on existence of minimal surfaces, I will talk about my recent work with Calegari and Marques where we introduce a quantity that counts some minimal surfaces in negatively curved manifolds and which is minimized by the hyperbolic metric

1 Febbraio, 2021 10:30

Seminario Matematico e Fisico di Milano

When can solutions of polynomial equations be algebraically parametrized?

Olivier Debarre, Sorbonne Université - Université de Paris

polimi-it.zoom.us/j/83674264668

abstract

abstract

The description of all the solutions of the equation $x^2+y^2=z^2$ in integral numbers (a.k.a. Pythagorean triples) is a very ancient problem: a Babylonian clay tablet from about 1800BC may contain some solutions, Pythagoras (about 500BC) seems to have known one infinite family of solutions, and so did Plato... This gives a first example of a rational variety: the rational points on the circle with equation $x^2+y^2=1$ can be algebraically parametrized by one rational parameter. More generally, one says that a variety, defined by a system of polynomial equations, is rational if its points (the solutions of the system) can be algebraically parametrized, in a one-to-one fashion, by independent parameters. I will begin with easy standard examples, then explain and apply some (not-so-recent) techniques that can be used to prove that some varieties (such as the set of rational solutions of the equation $x^3+y^3+z^3+t^3=1$) are not rational.

29 Gennaio, 2021 17:30

Sezione di Analisi

A conservative correction procedure for Level Set methods in evolving interfaces

Élise Delphine Le Mélédo, University of Zurich

online

abstract

abstract

Evolving interfaces is a key feature of many practical applications as e.g. pattern dynamics, multi-phase flows, material design, and their approximation raise tremendous interesting numerical issues.

In particular, incompressible two-phase flows require an accurate description of the interface in order to select the proper phase properties and dynamic to consider on each subdomain. Indeed, while the volume of each phase remains constant in time, the interface may develop heavy distortion and change topology across time. There, a small error in the interface location may have dreadful repercussions on the state dynamic (e.g. when vortices are encountered).

Therefore, the designed numerical scheme has to pay a particular attention to the representation of the interface(s) and the description of its motion.

Using an implicit representation of the interface, the Level Set method is a popular technique that allows a complex interface shape and is suited to topology changes in time. However, it is not natively conservative, that is the quantity of each fluid is not preserved as the interface is evolved.

In this talk, we propose a correction technique to the Level Set method that enforces mass conservation while preserving the continuity of each connected component of the interface. Flexible, it can be used in combination with any meshed scheme updating the non-corrected Level Set field.