Dipartimento di Matematica

Prossimi Seminari

• Seminar
  Evolution equations driven by dissipative operators in Wasserstein spaces
  Giulia Cavagnari, Politecnico di Milano
  mercoledì 20 gennaio 2021 alle ore 11:15
  Link:
  ABSTRACT
  ABSTRACT

  In this talk we present new results framing into the recent theory of Measure Differential Equations introduced by B. Piccoli (Rutgers University-Camden). The state space where these evolution equations are set is the Wasserstein space of probability measures, hence tools of Optimal Transport are essential. The key point here is that the vector field itself maps into the space of probability measures lying on the tangent bundle, in a way compatible with the projection on the state space. We give a stronger definition of solution which indeed “selects” only one of the (not unique) solutions in the sense of Piccoli. In addition to uniqueness, we are also able to prove stability results. To do so, we borrow ideas from the theory of evolution equations driven by dissipative operators on Hilbert spaces, giving a notion of solution in terms of a so called Evolution Variational Inequality.
  
  This is a joint work with G. Savaré (Bocconi University) and G. E. Sodini (TUM-IAS).
  
• Seminar
  The spectral theorem for a normal operator on a Clifford module
  David P. Kimsey, Newcastle University
  mercoledì 20 gennaio 2021 alle ore 17:00
  On line
  ABSTRACT
  ABSTRACT

  In this talk we will consider the problem of obtaining a spectral resolution for a densely defined closed normal operator on a Clifford module $\mathcal{H}_n := \mathcal{H} \otimes \mathbb{R}_n$, where $\mathcal{H}$ is a real Hilbert space and $\mathbb{R}_n := \mathbb{R}_{0, n}$ is the Clifford algebra generated by the units $e_1, \ldots, e_n$ with $e_i e_j = -e_j e_i$ for $i \neq j$ and $e_j^2 = -1$ for $j=1,\ldots, n$. We shall see that any densely defined closed normal operator on a Clifford module admits an integral representation which is analogous to the integral representation for a densely defined closed normal operator on a quaternionic Hilbert space (which one may think of as a Clifford module $\mathcal{H}_2$) discovered by Daniel Alpay, Fabrizio Colombo and the speaker in 2014. However, the Clifford module setting sketched above with $n > 2$ presents a number of technical difficulties which are not present in the quaternionic Hilbert space case.
  
  In order to prove this result, one needs to slightly generalise the notion of $S$-spectrum to allow for operators which are not necessarily paravector operators, i.e., operators of the form $T =T_0 + \sum_{j=1}^n T_j e_j$. This observation has implications on a generalisation of the $S$-functional calculus and some related function theory which we shall briefly highlight.
  
  The main thrust of this talk is based on joint work with Fabrizio Colombo. The work on the $S$-functional calculus is joint work with Fabrizio Colombo, Jonathan Gantner and Irene Sabadini. The work on the related function theory is joint work with Fabrizio Colombo, Irene Sabadini and Stefano Pinton.
• MOX Colloquia
  Patient-specific hemodynamics simulations for interventional planning of congenital and acquired cardiac diseases
  Irene Vignon-Clementel,  Inria - France
  giovedì 21 gennaio 2021 alle ore 14:00
  Online Seminar: mox.polimi.it/elenco-seminari/?id_evento=2011&t=763724
  ABSTRACT

  

  Irene Vignon-Clementel

  Irene Vignon-Clementel is directrice de recherche (prof. equiv.) at Inria, the French National Institute for Research in Digital Science and Technology. She holds a 'habilitation' degree in Applied Mathematics (Sorbonne U., formerly U. Pierre & Marie Curie) and a PhD in Mechanical Engineering (Stanford U.). Her research focuses on modeling and numerical simulations of physiological flows to better understand a number of pathophysiologies and their treatment (surgical planning, medical device design), especially related to blood circulation and breathing. This requires developing models of different complexities, coupling them, that their numerical implementation is robust, and that their parameters are based on medical or experimental data specific to a subject. Applications include congenital and acquired cardiovascular diseases, respiratory diseases and liver pathophysiology, and more recently the interpretation of non-invasive dynamic imaging. Irene VC is member of several conference committees, of the Int. J. Num. Methods Biomed. Eng. editorial board, of the VPHi board, of the scientific advisory committee for the 3DS-FDA ENRICHMENT Project, and was co-chair of the international conference VPH2020. She received the top recipient award of the western states American Heart Association fellowship (2004-2006), the student award at the World Congress of Computational Mechanics by the USACM and the USACM Executive Committee (2006), Inria excellence awards (2012 and 2016), and has been awarded an ERC consolidator grant (2019). She has been working with companies and clinicians as a PI in a number of national and international grants such as a Leducq transatlantic network of excellence, and actively promoting the computational bioengineering and medicine interface through co-supervision of MD-PhDs, joint research projects, conference organization and interface articles with clinicians.
  ABSTRACT

  Hemodynamics modeling has become mature enough to simulate local fluid dynamics changes due to a surgery or device implantation. However taking into account their interactions with the rest of the circulation, or even the downstream vascular bed not accessible by imaging remains challenging. We will present multi-fidelity models and computational methods that have been developed to tackle this issue for patient-specific image-based modeling. We will demonstrate through examples of congenital heart disease and coronary vascular disease how such simulations can be performed by including morphological and functional data. Finally we will discuss the advantage of combining these simulations with supervised machine learning as a tool to predict abdominal aneurysm growth risk and palliate the lack of mechanistic growth equations.
  
  Contatto: paolo.zunino@polimi.it

  

  Irene Vignon-Clementel

  Irene Vignon-Clementel is directrice de recherche (prof. equiv.) at Inria, the French National Institute for Research in Digital Science and Technology. She holds a 'habilitation' degree in Applied Mathematics (Sorbonne U., formerly U. Pierre & Marie Curie) and a PhD in Mechanical Engineering (Stanford U.). Her research focuses on modeling and numerical simulations of physiological flows to better understand a number of pathophysiologies and their treatment (surgical planning, medical device design), especially related to blood circulation and breathing. This requires developing models of different complexities, coupling them, that their numerical implementation is robust, and that their parameters are based on medical or experimental data specific to a subject. Applications include congenital and acquired cardiovascular diseases, respiratory diseases and liver pathophysiology, and more recently the interpretation of non-invasive dynamic imaging. Irene VC is member of several conference committees, of the Int. J. Num. Methods Biomed. Eng. editorial board, of the VPHi board, of the scientific advisory committee for the 3DS-FDA ENRICHMENT Project, and was co-chair of the international conference VPH2020. She received the top recipient award of the western states American Heart Association fellowship (2004-2006), the student award at the World Congress of Computational Mechanics by the USACM and the USACM Executive Committee (2006), Inria excellence awards (2012 and 2016), and has been awarded an ERC consolidator grant (2019). She has been working with companies and clinicians as a PI in a number of national and international grants such as a Leducq transatlantic network of excellence, and actively promoting the computational bioengineering and medicine interface through co-supervision of MD-PhDs, joint research projects, conference organization and interface articles with clinicians.
• Seminar
  Symmetric solutions to supercritical elliptic problems
  Benedetta Noris, Politecnico di Milano
  martedì 26 gennaio 2021 alle ore 11:15
  Link:
  ABSTRACT
  ABSTRACT

  When searching for solutions to Sobolev-supercritical elliptic problems, a major difficulty is the lack of Sobolev embeddings, that entrains a lack of compactness. In this talk, I will discuss how symmetry and monotonicity properties can help to overcome this obstacle. In particular, I will present a recent result concerning the existence of axially symmetric solutions to a semilinear equation, in collaboration with A. Boscaggin, F. Colasuonno and T. Weth.
• Seminar
  Dalla dinamica delle popolazioni all'epidemiologia: alcuni semplici modelli differenziali
  Gianmaria Verzini , Politecnico di Milano
  mercoledì 27 gennaio 2021 alle ore 15:00
  online
  ABSTRACT
  ABSTRACT

  Lo scopo di questo seminario è di introdurre alcuni semplici modelli matematici nell'ambito della dinamica delle popolazioni. Dopo aver considerato i primi modelli per una o più popolazioni, dovuti a Verhulst, Lotka e Volterra, ci soffermeremo in particolare su uno dei modelli di base per lo studio della propagazione delle epidemie: il modello SIR di Kermack e McKendrick.
• MOX Seminar
  Nowcasting the Italian epidemic outbreak of SARS-CoV-2
  Alessio Farcomeni, Università Roma Tor Vergata
  giovedì 28 gennaio 2021 alle ore 14:00 precise
  Online seminar: mox.polimi.it/elenco-seminari/?id_evento=2001&t=763724
  ABSTRACT
  ABSTRACT

  In the talk, we briefly discuss the main epidemiological features of SARS-CoV-2 one year into the pandemic, giving also a short account of the public data available for Italy and of the main limits of lay analyses.
  We then discuss an accurate method for short-term forecasting ICU occupancy at local level. Our approach is based on an optimal ensemble of two simple methods: a generalized linear mixed regression model which pools information over different areas, and an area-specific non-stationary integer autoregressive methodology. Optimal weights are estimated using a leave-last-out rationale.
  Daily predictions between February 24th and November, 27th 2020 have a median error of 3 beds (third quartile: 8) at regional level, with coverage of 99% prediction intervals that exceeds the nominal one.
  Finally we present a different method based on a modified non-linear GLM for each indicator, including the potential effect of exogenous variables, based on appropriate distributional assumptions and a logistic-type growth curve. This allows us to accurately predict important characteristics of the epidemic (e.g., peak time and height).
  Based on joint works with Pierfrancesco Alaimo di Loro, Fabio Divino, Giovanna Jona Lasinio, Gianfranco Lovison, Antonello Maruotti, Marco Mingione
• Seminar
  Mathematics vs Dementia
  Alain Goriely, University of Oxford
  lunedì 1 febbraio 2021 alle ore 11:45
  On line
  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.
• Seminar
  JMGT [Jordan-Moore Gibson-Thompson] dynamics arising in non- linear acoustics - a view from the boundary
  Irena Lasiecka, University of Memphis
  lunedì 1 febbraio 2021 alle ore 15:00
  On line
  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.
  
• Seminar
  Counting minimal surfaces in negatively curved manifolds
  Andrè Neves, University of Chigaco
  lunedì 1 febbraio 2021 alle ore 16:15
  On line
  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
• Seminar
  When can solutions of polynomial equations be algebraically parametrized?
  Olivier Debarre, Sorbonne Université - Université de Paris
  lunedì 1 febbraio 2021 alle ore 10:30
  On line
  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.
• Seminar
  Minimal time optimal control for the moon lander problem
  Elsa Marchini, Politecnico di Milano
  martedì 2 febbraio 2021 alle ore 11:15
  Link:
  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
• Seminar
  New results on the Lieb-Thirring inequality
  Mathieu Lewin, CEREMADE, Université Paris Dauphine, Paris
  lunedì 8 febbraio 2021 alle ore 15:00
  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).
• Seminar
  Digital Storytelling e narrazione matematica: costruire competenze matematiche online
  Giovannina Albano, Università degli Studi di Salerno
  mercoledì 10 febbraio 2021 alle ore 15:00
  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.
• Seminar
  Some global results for homogeneous Hormander sums of squares
  Stefano Biagi, Politecnico di Milano
  mercoledì 10 febbraio 2021 alle ore 11:15
  Link:
  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.
  
• Seminar
  Symmetry and rigidity for composite membranes and plates
  Eugenio Vecchi, Politecnico di Milano
  martedì 16 febbraio 2021 alle ore 11:15
  Link:
  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).
  
• Seminar
  The Dunkl intertwining operator
  Hendrik De Bie, Ghent University
  mercoledì 17 febbraio 2021 alle ore 17:00
  On line
  ABSTRACT
  ABSTRACT

  There are two crucial operators in the theory of Dunkl harmonic analysis. The first is the Dunkl transform, which generalizes the Fourier transform. The second is the intertwining operator, which maps ordinary partial derivatives to Dunkl operators. Although some abstract statements are known about the intertwining operator, the explicit formula for classes of reflection groups is generally not known. In recent work Yuan Xu proposed a formula in the case of dihedral groups and a restricted class of functions. We extend his formula to all functions and give a general strategy on how to obtain similar formulas for other reflections groups. This is based on joint work with Pan Lian, available under arXiv:2002.09065 and to appear in J. Funct. Anal.
• MOX Colloquia
  Modeling Dementia
  Ellen Kuhl, Stanford University
  giovedì 18 febbraio 2021 alle ore 14:00 precise
  Online Seminar: mox.polimi.it/elenco-seminari/?id_evento=1981&t=763724
  ABSTRACT

  

  Ellen Kuhl

  Ellen Kuhl is the Robert Bosch Chair of Mechanical Engineering at Stanford University. She is a Professor of Mechanical Engineering and, by courtesy, Bioengineering. She received her PhD from the University of Stuttgart in 2000 and her Habilitation from the University of Kaiserslautern in 2004. Her area of expertise is Living Matter Physics, the design of theoretical and computational models to simulate and predict the behavior of living structures. Ellen has published more than 200 peer-reviewed journal articles and edited two books; she is an active reviewer for more than 20 journals at the interface of engineering and medicine and an editorial board member of seven international journals in her field. Ellen is the current Chair of the US National Committee on Biomechanics and a Member-Elect of the World Council of Biomechanics. She is a Fellow of the American Society of Mechanical Engineers and of the American Institute for Mechanical and Biological Engineering. She received the National Science Foundation Career Award in 2010, was selected as Midwest Mechanics Seminar Speaker in 2014, and received the Humboldt Research Award in 2016. Ellen is an All American triathlete on the Wattie Ink. Elite Team, a multiple Boston, Chicago, and New York marathon runner, and a Kona Ironman World Championship finisher.
  ABSTRACT

  Neurodegeneration will undoubtedly become a major challenge in medicine and public health caused by demographic changes worldwide. More than 45 million people are living with dementia today and this number is expected to triple by 2050. Recent studies have reinforced the hypothesis that the prion paradigm, the templated growth and spreading of misfolded proteins, could help explain the progression of a variety of neurodegenerative disorders. However, our current understanding of prion-like growth and spreading is rather empirical. Here we show that a physics-based reaction-diffusion model can explain the growth and spreading of misfolded protein in a variety of neurodegenerative disorders. We combine the classical Fisher-Kolmogorov equation for population dynamics with anisotropic diffusion and simulate misfolding across representative sections of the human brain and across the brain as a whole. Our model correctly predicts amyloid-beta deposits and tau inclusions in Alzheimer's disease, alpha-synuclein inclusions in Parkinson's disease, and TDP-43 inclusions in amyotrophic lateral sclerosis. To reduce the computational complexity, we represent the brain through a connectivity-weighted Laplacian graph created from 418 brains of the Human Connectome Project. Our brain network model correctly predicts the key characteristic features of whole brain models at a fraction of their computational cost. Our results suggest that misfolded proteins in various neurodegenerative disorders grow and spread according to a universal law that follows the basic physical principles of nonlinear reaction and anisotropic diffusion. Our simulations can have important clinical implications, ranging from estimating the socioeconomic burden of neurodegeneration to designing clinical trials and pharmacological intervention.
  
  Contatto: alfio.quarteroni@polimi.it

  

  Ellen Kuhl

  Ellen Kuhl is the Robert Bosch Chair of Mechanical Engineering at Stanford University. She is a Professor of Mechanical Engineering and, by courtesy, Bioengineering. She received her PhD from the University of Stuttgart in 2000 and her Habilitation from the University of Kaiserslautern in 2004. Her area of expertise is Living Matter Physics, the design of theoretical and computational models to simulate and predict the behavior of living structures. Ellen has published more than 200 peer-reviewed journal articles and edited two books; she is an active reviewer for more than 20 journals at the interface of engineering and medicine and an editorial board member of seven international journals in her field. Ellen is the current Chair of the US National Committee on Biomechanics and a Member-Elect of the World Council of Biomechanics. She is a Fellow of the American Society of Mechanical Engineers and of the American Institute for Mechanical and Biological Engineering. She received the National Science Foundation Career Award in 2010, was selected as Midwest Mechanics Seminar Speaker in 2014, and received the Humboldt Research Award in 2016. Ellen is an All American triathlete on the Wattie Ink. Elite Team, a multiple Boston, Chicago, and New York marathon runner, and a Kona Ironman World Championship finisher.

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