14 Novembre, 2016 11:30

MOX Seminar

PDE regularized principal component analysis on bidimensional manifolds, with applications to neuroimaging data

Laura Sangalli, Politecnico di Milano, Mox - Dipartimento di Matematica

Aula Consiglio VII Piano Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Motivated by the analysis of high-dimensional neuroimaging signals over the cerebral cortex, we introduce a principal component analysis technique that can be used for exploring the variability and performing dimensional reduction of signals observed over two-dimensional manifolds. The proposed method is based on a PDE regularization approach, involving the Laplace-Beltrami operator associated to the manifold domain. The model introduced can be applied to data observed over any two-dimensional manifold topology, and can naturally handle missing data and signals evaluated in different grids of points. The proposed method is applied to the analysis of resting state functional magnetic resonance imaging data from the Human Connectome Project.

contact: laura.sangalli@polimi.it

7 Novembre, 2016 16:30

Lezioni Leonardesche

Convergence to equilibrium via quantitative stability

Alessio Figalli, ETH Zurich

Aula Chisini, via Saldini 50

abstract

abstract

Geometric and functional inequalities play a crucial role in several PDE problems.

Very recently there has been a growing interest in studying the stability for such inequalities. The basic question one wants to address is the following:

Suppose we are given a functional inequality for which minimizers are known. Can we prove, in some quantitative way, that if a function “almost attains the equality” then it is close to one of the minimizers?

Actually, in view of applications to PDEs, a even more general and natural question is the following: suppose that a function almost solve the Euler-Lagrange equation associated to some functional inequality. Is this function close to one one of the minimizers?

While in the first case the answer is usually positive, in the second case one has to face the presence of bubbling phenomena.

In this talk I’ll give a overview of these general questions using some concrete examples, and then present recent applications to some fast diffusion equation related to the Yamabe flow.

2 Novembre, 2016 15:00

Seminari FDS

L'evoluzione dei simboli matematici dalla loro nascita alla maturità

Joseph Mazur, Marlboro College, in Marlboro, Vermont

Aula Consiglio, 7° piano, edificio 14 Nave – Via Bonardi, 9 – Milano

abstract

abstract

L'evoluzione dei simboli matematici dalla loro nascita alla loro maturità
Prima del sedicesimo secolo, tutta la trasmissione del sapere matematico, in prosa o in poesia che fosse, si basava sulla parola. Passando dalle parole alle abbreviazioni ai simboli, la matematica si è evoluta fino a raggiungere la forma che ci è oggi familiare. Questa conferenza seguirà la trasformazione dell’algebra dallo stile retorico a quello simbolico, investigherà l'influenza che questi simboli esercitano su di noi, come ci guidano verso nuove idee attraverso associazioni inconsce tra l’esperienza e l’ignoto, e come contribuiscono alla comunicazione dei fondamenti della matematica.

The Evolution of Mathematical Symbols from Their Birth to Their Maturity
Before the sixteenth century, all communication of mathematics was rhetorical, written in either prose or poetic form. From words to abbreviations to symbols, mathematics has evolved to the familiar forms we use today. This talk will follow the transfigurations of algebra from its rhetorical style to its symbolic one. It will investigate how these symbols influence us and how they lead to new ideas by subconscious associations connecting experience and the unknown, and how they contribute to the communication of basic mathematics.

26 Ottobre, 2016 15:00

Seminari FDS

Dal pugnale di Tutankhamon ai quadri di Van Gogh: un viaggio affascinante nella scienza al servizio dell'arte

Gianluca Valentini, Politecnico di Milano

Aula Consiglio, 7° piano, edificio 14 Nave – Via Bonardi, 9 - Milano

abstract

abstract

La conservazione del patrimonio artistico richiede la conoscenza approfondita dei materiali di cui le opere d’arte sono costituite. La scienza può guidarci in un affascinante viaggio che svela i misteri del passato e ci aiuta a trasmettere l’eredità della nostra cultura alle generazioni future. Nel seminario verranno mostrate recenti applicazioni delle tecniche fisiche alla diagnostica di opere d’arte e reperti archeologici. Vedremo come la spettroscopia a raggi X ci ha permesso di svelare un enigma ancora celato nella del tomba del faraone Tutankhamon, a più di 90 anni dalla sua scoperta. Ricostruiremo la tavolozza dei colori di van Gogh e dei pittori futuristi studiando la fluorescenza indotta dalla luce laser ultravioletta e vedremo come la rivoluzione industriale ha permesso di sostituire le ricette antiche con i materiali di sintesi chimica. Comprenderemo infine come tecniche di immagine avanzate permettono di studiare e monitorare lo stato di conservazione dei capolavori di Michelangelo.

21 Ottobre, 2016 14:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

Tropical compactifications, Mori Dream Spaces and Minkowski bases

Stefano Urbinati, Università di Padova

Aula Seminari del terzo piano

abstract

abstract

Given a Mori Dream Space X we construct via tropicalization a model dominating all the small Q-factorial modifications. Via this construction
we recover a Minkowski bases for the Newton-Okounkov bodies on X and hence
the movable cone for X. This is a work in progress with Elisa Postinghel.

Nota dell'organizzatore (E. Schlesinger): Stefano Urbinati ha vinto una Polimi International Fellowship e prenderà presto servizio presso il nostro dipartimento; lavora nell'ambito della geometria algebrica e ultimamente si è interessato a problemi di geometria tropicale.

20 Ottobre, 2016 11:00

MOX Seminar

3D microphysiological systems: from modeling, to drug discovery, developmental biology, and space research

Riccardo Gottardi, Center for Cellular and Molecular Engineering, University of Pittsburgh, PA, USA

Aula Seminari "Saleri" VI Piano Mox Dipartimento di Matematica - Politecnico di Milano

abstract

abstract

A growing research in recent years focused on the development of next generation bioreactors capable of generating engineered constructs that mimic at least some of the physiological functions of native organ systems. Such microphysiological systems rely on the combination of microfluidic and tissue engineering to generate a high number of identical 3D organoid models that can be used for drug and toxicological screening.
Moreover, the ultimate aim is to be able to connect multiple tissues within an organ system and from there multiple organ systems among themselves to better recapitulate the complexity and interconnection of human physiology, going from organ-on-a-chip to humans-on-a-chip. The effective development of these devices requires a solid understanding of their interconnected fluidics to predict the transport of nutrients and waste through the constructs and improve the design accordingly. In this lecture, the focus will be on a specific model of bioreactor developed at the Center for Cellular and Molecular Engineering (CCME), with multiple input/outputs, aimed at generating osteochondral constructs, i.e., a biphasic construct in which one side is cartilaginous in nature, while the other is osseous. Some of the challenges at the level of computational modelling of the system will be described, such as addressing the multi-physics nature of the problem that combines free flow in channels with hindered flow in porous media, and coupling of fluid dynamics with advection-diffusion-reaction equations that model the transport of biomolecules throughout the system and their interaction with living constructs. The same design and modelling approach is applicable to multi-chamber, interconnected system. In particular, it may be applied to human-on-chip devices. To this end, a lumped parameter approach to predict the behavior of multi-unit bioreactor systems with modest computational effort, will also be described. Furthermore, the lecture will present a comparison of the modeling outcomes with the experimental results and will introduce some of the opportunities opened by the possibility of using microphysiological systems to understand human physiology in challenging environments, such as microgravity on board the International Space Station, which is the object of a recent grant obtained by CCME. Finally, the lecture will outline some open problems where a modelling and computational approach can help guide the experimental development of microphysiological systems, such as understanding the processes of limb morphogenesis, digit patterning, and segmentation, and their translation to a regenerative approach.

contact: christian.vergara@polimi.it

17 Ottobre, 2016 10:30

MOX Seminar

Biologically inspired formulation of optimal transportation problems. An unexpected branching source

Enrico Facca, Franco Cardin, Mario Putti, Dipartimento di Matematica, Universita’ di Padova

AULA CONSIGLIO VII Piano DIPARTIMENTO DI MATEMATICA, POLITECNICO DI MILANO

abstract

abstract

We have recently developed an approach, based on an extension of a model proposed by Tero et al (2007) for the simulation of the dynamics of a slime mold (Physarium Polycephalum). We conjecture that this model is an original formulation of the PDE-based OT problem. This new formulation assumes that the potential and the diffusion
coefficient (the latter yielding the transportation plan) are time dependent. The classical constraint on the norm of the gradient is then replaced by an ODE describing transient dynamics of the diffusion coefficient.
Analytical results in the case of the Monge-Kantorovich problem, although yet largely incomplete, suggest that indeed the conjecture is true. This is supported by several numerical experiments showing that at large times the solution to this problem is equivalent to the solution of the classical Monge-Kantorovich PDE based OT. One of the most important advantages of the proposed formulation is that its numerical solution is very efficient and well-defined
using simple numerical approaches. Moreover, this dynamical extension allows the reconstruction of the time-history of the process, thus enlarging the applicability of the OT model to wider sets of processes. The proposed model can also be easily adapted to branched and congested transport problems. Preliminary numerical simulations show that the proposed formulation is efficient in finding solutions also of congested transport and branched transport problems.
While the interpretation via $p$-Laplacians allows a straight forward interpretation of the new formulation in the simulation of congested transport, only numerical evidence is currently available for applications to branched transport problems. Although obvious limitations are present in the numerical solution of highly discontinuous problems, we present some experimental convergence results showing robustness of the scheme for sufficiently regular solutions. Finally, we will present models and related numerical results of diverse applications of this formulation ranging from slime-mold dynamics to geomorphological applications, and discuss current and future progress.

contact: christian.vergara@polimi.it

13 Ottobre, 2016 11:30

MOX Seminar

Divergence free virtual elements for Stokes equations and H1-conforming virtual elements for Darcy and Brinkman equations

Giuseppe Vacca, Universita' degli Studi di Milano-Bicocca

Aula Seminari "Saleri" VI Piano Mox-Dipartimento di Matematica, POLITECNICO DI MILANO

abstract

abstract

The Virtual Element Method is a generalization of the classical Finite Element Method to arbitrary element-geometry. In the first part of the talk we present a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is pointwise divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced
problem with fewer degrees of freedom, thus yielding a very efficient scheme. The focus of the second part of the talk is on developing a Virtual Element Method for Darcy and Brinkman equations. We use introduce a slightly different Virtual Element space having two fundamental properties: the L2-projection onto the space of polynomials of degree k
(being k the order of the method) is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy problem has optimal order of convergence and H1 conforming velocity solution. In particular we obtain a Virtual Element scheme that is accurate for both Darcy and Stokes equations. Then we can apply the same approach to develop a stable virtual element method for Brinkman equations (that is combination of Stokes and Darcy equations). We provide a rigorous error analysis of the method and several numerical tests, including a
comparison with a different Virtual Element choice.

contact: paola.antonietti@polimi.it

12 Ottobre, 2016 16:15 in punto

Sezione di Analisi

Absence of critical points of solutions to the Helmholtz equation in 3D

Giovanni Alberti, ETH, Zurigo

Aula seminari 3° piano

abstract

abstract

In this talk I will discuss a method to prove the absence of critical points for the Helmholtz equation in 3D. The key element of the approach is the use of multiple frequencies in a fixed range, and the proof is based on the spectral analysis of the associated problem. This question is strictly connected with the Rado-Kneser-Choquet theorem, whose direct extension to the Helmholtz equation or to three dimensions is not possible.

12 Ottobre, 2016 15:00

Seminari FDS

Data Science, il futuro della Statistica

Piercesare Secchi, Politecnico di Milano

Aula Consiglio, 7° piano, edificio Nave, via Bonardi, 9 - Milano

abstract

abstract

Nell’era dei Big Data, la statistica dovrà vincere una sfida epocale adattando il proprio approccio epistemologico, i propri paradigmi, metodi e algoritmi all’accresciuta complessità degli oggetti dei quali esplora la variabilità per estrarne informazione. Nelle applicazioni scientifiche, i Big Data non sono solo “big” in termini di volume, velocità e varietà, ma sempre più spesso sono dati intrinsecamente complessi, come curve o superfici, immagini, matrici, reti, testi. In questo seminario illustrerò alcuni esempi in cui gli aspetti teorici e computazionali relativi all’analisi dei Big Data si integrano con specifiche conoscenze di ambito medico o biologico, geologico, finanziario o industriale.

Il Prof.Piercesare Secchi, Professore Ordinario di Statistica, è Direttore del Dipartimento di Matematica e ricopre numerosi importanti incarichi nel Politecnico