10 Aprile, 2024 15:00

Sezione di Analisi

Some Examples of Persistence of Superoscillations

Elodie Pozzi, St. Louis University, USA

Aula Seminari - III Piano

abstract

abstract

When combining N bandlimited functions with low frequencies and allowing N to approach infinity, one would typically anticipate obtaining a function with low frequency. However, for certain bandlimited functions with low frequency, the resulting function exhibits high frequencies. Such functions are termed superoscillating functions, and the phenomenon described is known as super oscillation. In this presentation, we will delve into the behavior of superoscillating functions under the time-dependent Schrödinger equation and its variations. Specifically, we will explore existing methods for demonstrating the persistence of superoscillations, applying them to illustrate the persistence of superoscillations under certain partial differential equations. Our focus will be on the Schrödinger equation governing spin particles and the Schrödinger equation pertaining to the Aharonov-Bohm effect. It is based on joint works with Fabrizio Colombo, Irene Sabadini and Brett D. Wick.

This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

4 Aprile, 2024 11:30

Sezione di Probabilità e Statistica Matematica

Global Well-Posedness and Interior Regularity of 2D Navier-Stokes Equations with Stochastic Wind Driven Boundary Conditions

Eliseo Luongo, Scuola Normale Superiore, Pisa

Aula Seminari MOX - VI piano

abstract

abstract

Partial differential equations with boundary noise have been introduced by Da Prato and Zabczyck. They showed that, also in the one dimensional case, the solutions of the heat equation with white noise Dirichlet or Neumann Boundary conditions have low regularity compared to the case of noise diffused inside the domain. In particular, in the case of Dirichlet boundary conditions the solution is only a distribution. Some improvements in the analysis of the interior regularity of the solutions of these problems and some nonlinear variants have been obtained exploiting specific properties of the heat kernel and of suitable nonlinearities. All these issues make the problem of treating non-linear partial differential equations with boundary noise coming from fluid dynamical models an, almost untouched, field of open problems. In this talk we discuss the global well-posedness and the interior regularity of a system of 2D Navier-Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere-ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force. The talk is based on a joint work with A. Agresti.

4 Aprile, 2024 15:30

MOX Seminar

Combining noisy well data and expert knowledge in a Bayesian calibration of a flow model under uncertainties: an application to solute transport in the Ticino basin

Lorenzo Tamellini, CNR-IMATI Pavia

Aula Saleri

abstract

abstract

In this talk we present the results of a case-study aimed at providing a UQ analysis of solute travel times in groundwater in the basin of the Ticino River (northern Italy), starting from well data collected over a month in summer 2014. We consider a steady-state groundwater flow model (developed in MODFLOW) and perform a sensitivity analysis using the Morris method to discard uninfluential parameters. We then employ Bayesian inversion (with Gaussian approximation) to obtain a data-informed posterior pdf for the remaining parameters, and propagate these pdfs to travel times computed by particle tracking (MODPATH). The likelihood function employed in the Bayesian inversion takes into account both well measurements and expert knowledge about the extent of the springs in the domain under study.

Contatto:
alessio.fumagalli@polimi.it

25 Marzo, 2024 15:30 in punto

Sezione di Geometria, Algebra e loro applicazioni

Automorphisms of OG10 towards Enriques manifolds

Annalisa Grossi, Université Paris-Saclay

Aula seminari III piano

abstract

abstract

Automorphisms of hyperkähler manifolds have been studied for many different reasons: construct symplectic quotients or study fixed loci in order to find examples of irreducible symplectic varieties, define maps among different deformation families of HK manifolds, find new examples of Enriques manifolds, that are higher dimensional analogue of Enriques surfaces, and for which Pacienza and Sarti recently proved the Morrison-Kawamata cone conjecture. In the first part of the talk I will show a recent result about symplectic rigidity of hyperkähler manifolds of OG10 type. Then I will show how to construct examples of Enriques manifolds considering nonysmplectic automorphisms of a Laza—Saccà—Voisin manifold that are induced by a nonysmplectic automorphism of the underlying cubic fourfold. The talk is based on a joint work with L. Giovenzana, Onorati and Veniani and on a joint work in progress with Billi, F. and L. Giovenzana.

21 Marzo, 2024 14:30

Seminario Matematico e Fisico di Milano

Periodic surfaces and deterministic random walks: dynamics between geometry and probability

Corinna Ulcigrai, Institut für Mathematik - Universität Zürich

aula U5 RATIO-3014 Università Milano - Bicocca

abstract

abstract

Many deterministic systems display chaotic features: the stronger the chaotic features, the better the system can be by approximated by a probabilistic model, an idea that can be traced back to Boltzmann and explains the success of the branch of dynamical systems known as ergodic theory. In this talk we will discuss systems which display only 'mild’ chaotic features, such as the geodesic flow on surfaces which a flat geometry or the Ehrenfest model in mathematical physics.
Recent breakthroughs on our understanding of the latter model, introduced more than a century ago, were made possible by the powerful tools exploiting moduli spaces of surfaces and Teichmueller dynamics, an area which has attracted the work of several Fields medallists. We will in particular highlight some results of probabilistic flavor that can still be proven for these deterministic systems, hidden in the fractal structure of trajectories and the 'deterministic random walks' that describe them.

19 Marzo, 2024 15:00

Sezione di Analisi

A Strong Form of the Quantitative Wulff Inequality for Crystalline Norms

Kenneth DeMason, The University of Texas at Austin

Aula seminari - III piano

abstract

abstract

The anisotropic perimeter is a natural functional appearing in the mathematical framework for determining equilibrium states of crystals in media. As with the usual isotropic perimeter there is an analogous anisotropic isoperimetric inequality, known as the Wulff inequality, where minimizers of the volume constrained anisotropic perimeter problem, known as Wulff shapes, are characterized. In view of statistical mechanics, almost-minimizers are the most likely observable states; as such their identification is just as important as the absolute minimizers. In this talk we will explore a recent result by the speaker which proves quantitative control on almost-minimizers in an H^1 sense when the Wulff shape is a polytope, an upgrade from the previous L^2 control via the so-called Fraenkel asymmetry.

18 Marzo, 2024 15:30 in punto

Sezione di Geometria, Algebra e loro applicazioni

Translation surfaces with poles and their moduli spaces

Gianluca Faraco, Università Milano Bicocca

Aula seminari III piano

abstract

abstract

On a Riemann surface $X$, every meromorphic abelian differential determines a translation surface and a representation $\chi\colon\pi_1(X)\to \mathbb C$ called period character. In this seminar we shall talk about strata of meromorphic differential with prescribed zeros and poles. Strata admit a natural foliation in which every leaf comprises translation surfaces with the same period character. In the special case of H(1,1;-2) each leaf is homeomorphic to the Loch Ness monster and it carries itself a transition structure.

14 Marzo, 2024 14:00

MOX Colloquia

On Neighbourhood Cross Validation

Simon Wood, School of Mathematics, University of Edinburgh

Aula Consiglio - VII piano

abstract

abstract

Cross validation is a well known method for estimating hyperparameters in complex regression models. It comes in many varieties, but some of the more interesting flavours require multiple model fits with consequently high cost. This talk shows how the high cost can be side-stepped for a wide range of models estimated using a quadratically penalized smooth loss, with rather low approximation error. Once the computational cost has the same leading order as a single model fit, it becomes feasible to efficiently optimize the chosen cross-validation criterion with respect to multiple smoothing/precision parameters. Interesting applications include cross-validating smooth additive quantile regression models, and the use of leave-out-neighbourhood cross validation for dealing with nuisance short range autocorrelation. The link between cross validation and the jackknife can be exploited to obtain reasonably well calibrated uncertainty quantification in these case.

Simon Wood is Professor of Statistical Computing and the University of Edinburgh, (and previously Bristol, Bath, Glasgow and St Andrews). After training in physics he worked in mathematical biology, and moved into statistics as a result of an increasing conviction that biological dynamic models should be calibrated. This move resulted in a shift towards smoothing and modern regression models, the production of the quite widely used mgcv package in R for generalized additive modelling, and an associated textbook, alongside a number of journal articles related to smoothing, smoothing parameter estimation, GAM computation and inference. The Covid pandemic re-awakened his interest in the calibration of biological dynamic models, resulting in some work that proved oddly controversial.

14 Marzo, 2024 15:00

Sezione di Analisi

Sulla caratterizzazione armonica delle sfere: una disuguaglianza di stabilita' per domini C^1-pericentrici

Ermanno Lanconelli, Università di Bologna

Aula seminari MOX - VI piano

abstract

abstract

Nel 2002 Lewis e Vogel dimostrarono che le pseudo sfere armoniche, cioè le frontiere dei domini limitati sui quali vale, per le funzioni armoniche, la formula di media di superficie rispetto ad un loro punto interno, sono sfere euclidee se i domini sono Dirichlet-regolari e in più il loro bordo ha misura (n- 1)-dimensionale con crescita al più euclidea.
Il risultato di Lewis e Vogel, nelle stesse ipotesi, può essere riformulato nel modo seguente: se il nucleo di Poisson di un dominio, con polo in un suo punto interno x_0, è costante sul bordo, allora il dominio è una sfera euclidea di centro x_0.
Nel 2007 Preiss e Toro, assumendo le stesse ipotesi, dimostrarono che il risultato di Lewis e Vogel è stabile, nel senso seguente: se il nucleo di Poisson di un dominio, con polo in un suo punto interno, è quasi costante sul bordo, allora il bordo del dominio è geometricamente vicino ad una sfera centrata in quel punto.
Con Giovanni Cupini abbiamo dimostrato che il risultato di rigidità di Lewis e Vogel, e una disuguaglianza di stabilità alla Preiss e Toro, valgono assumendo ''soltanto'' una regolarità C^1 del bordo vicino ad almeno un suo punto pericentrale, cioè un punto del bordo avente distanza minima dal fissato centro della pseudosfera, senza nulla richiedere sul resto della frontiera, neppure l'esistenza del nucleo di Poisson.
Le nostre tecniche sono dirette, e non usano gli elevati metodi di Analisi armonica e di frontiera libera utilizzate da Lewis e Vogel e da Preiss e Toro.

Simon Wood is Professor of Statistical Computing and the University of Edinburgh, (and previously Bristol, Bath, Glasgow and St Andrews). After training in physics he worked in mathematical biology, and moved into statistics as a result of an increasing conviction that biological dynamic models should be calibrated. This move resulted in a shift towards smoothing and modern regression models, the production of the quite widely used mgcv package in R for generalized additive modelling, and an associated textbook, alongside a number of journal articles related to smoothing, smoothing parameter estimation, GAM computation and inference. The Covid pandemic re-awakened his interest in the calibration of biological dynamic models, resulting in some work that proved oddly controversial.

14 Marzo, 2024 17:30

Seminari di Cultura Matematica

Raccontare la scienza. Una sfida creativa

Massimo Polidoro, Divulgatore scientifico, Presidente del CICAP

Aula Rogers e in collegamento webex su tiny.cc/mathseminars

Nel 2002 Lewis e Vogel dimostrarono che le pseudo sfere armoniche, cioè le frontiere dei domini limitati sui quali vale, per le funzioni armoniche, la formula di media di superficie rispetto ad un loro punto interno, sono sfere euclidee se i domini sono Dirichlet-regolari e in più il loro bordo ha misura (n- 1)-dimensionale con crescita al più euclidea.
Il risultato di Lewis e Vogel, nelle stesse ipotesi, può essere riformulato nel modo seguente: se il nucleo di Poisson di un dominio, con polo in un suo punto interno x_0, è costante sul bordo, allora il dominio è una sfera euclidea di centro x_0.
Nel 2007 Preiss e Toro, assumendo le stesse ipotesi, dimostrarono che il risultato di Lewis e Vogel è stabile, nel senso seguente: se il nucleo di Poisson di un dominio, con polo in un suo punto interno, è quasi costante sul bordo, allora il bordo del dominio è geometricamente vicino ad una sfera centrata in quel punto.
Con Giovanni Cupini abbiamo dimostrato che il risultato di rigidità di Lewis e Vogel, e una disuguaglianza di stabilità alla Preiss e Toro, valgono assumendo ''soltanto'' una regolarità C^1 del bordo vicino ad almeno un suo punto pericentrale, cioè un punto del bordo avente distanza minima dal fissato centro della pseudosfera, senza nulla richiedere sul resto della frontiera, neppure l'esistenza del nucleo di Poisson.
Le nostre tecniche sono dirette, e non usano gli elevati metodi di Analisi armonica e di frontiera libera utilizzate da Lewis e Vogel e da Preiss e Toro.

Simon Wood is Professor of Statistical Computing and the University of Edinburgh, (and previously Bristol, Bath, Glasgow and St Andrews). After training in physics he worked in mathematical biology, and moved into statistics as a result of an increasing conviction that biological dynamic models should be calibrated. This move resulted in a shift towards smoothing and modern regression models, the production of the quite widely used mgcv package in R for generalized additive modelling, and an associated textbook, alongside a number of journal articles related to smoothing, smoothing parameter estimation, GAM computation and inference. The Covid pandemic re-awakened his interest in the calibration of biological dynamic models, resulting in some work that proved oddly controversial.