Settimana 08/01/2024 - 12/01/2024

 

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Seminari

 

Università degli Studi Roma Tor Vergata
Dipartimento di Matematica

 

Numerical Analysis Seminar

Date: 09 January 2024

Schedule: 14:30 - Rome Time

Where: Conference Room 1201 "R. Dal Passo"

Title: " Interpolation by weights: insights and challenges Part. 1 "

Speaker: Ludovico Bruni Bruno - Università di Padova

Abstract: Interpolation by weights: insights and challenges Interpolation of differential forms is a challenging aspect of modern approximation theory. Not only does it shed new light on some classical concepts of interpolation theory, such as the Lagrange interpolation and the Lebesgue constant, but it also suggests that they can be extended to a very general framework. As an extent of that, it is worth pointing out that the majority of classical shape functions commonly used in finite element methods, such as those involved in N´ed´elec or Raviar- Thomas elements, can be seen as a specialisation of this theory. Of course, this generality brings along the evident downside of an unfriendly level of abstraction. The scope of this series of two seminars is thus twofold: presenting the main challenges of this branch of approximation theory but in a concrete manner. The first seminar will hinge on a development of a convenient one dimensional toy model that enlightens parallelisms and differences with usual nodal interpolation. In the second seminar will extend these techniques to the multi-dimensional framework, motivating our choices by a geometrical flavour. Motivated by polynomial approximation of differential forms, we study analytical and numerical properties of a univariate polynomial interpolation problem that relies on function aver- ages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyse fundamental mathematical properties of this problem as existence, uniqueness and numerical con- ditioning of its solution. We provide concrete conditions for unisolvence and explicit Lagrange-type basis systems for its representation. To study the numerical conditioning, we offer a generalisation of the nodal Lebesgue constant, which we are able to characterise and estimate in some relevant scenarios. In the search for well-performing interpolation segments, a Fekete problem will naturally arise, forcing to carefully detail some theoretical aspects.

Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Organizing Committee:
Carla Manni (mail to contact)

Further Info: Click here for Numerical Analysis Page

Streaming Link (MS Teams): This seminar will be held in person

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Università degli Studi Roma Tor Vergata
Dipartimento di Matematica

 

Geometry Seminar

Date: 09 January 2024

Schedule: 14:30 - Rome Time

Where: Conference Room 1101 "C. D'Antoni"

Title: " Singular cscK metrics on smoothable varieties "

Speaker: Trusiani Antonio - Chalmers University of Technology

Abstract: The study of constant scalar curvature Kähler (cscK) metrics on compact complex manifolds is a classical topic that has attracted enormous interest since the 1950s. However, detecting the existence of cscK metrics is a difficult task, which in the projective integral case conjecturally amounts to proving an important algebro-geometric stability notion (K-stability). Recent significant advancements have established that the existence of unique cscK metric in a Kähler class is equivalent to the coercivity of the so-called Mabuchi functional. I will extend the notion of cscK metrics to singular varieties, and I will show the existence of these special metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. A key point in this variational approach is the lower semicontinuity of the coercivity threshold of Mabuchi functional along a degenerate family of normal compact Kähler varieties with klt singularities. The latter strengthens evidence supporting the openness of (uniform) K-stability for general families of normal compact Kähler varieties with klt singularities. This is a joint work with Chung-Ming Pan and Tat Dat Tô.

This talk is part of the activity of the MIUR Excellence Department Projects MathMod@TOV, and the PRIN 2022 Moduli Spaces and Birational Geometry

Organizing Committee:
Giulio Codogni (mail to contact) Guido Maria Lido (mail to contact)

Further Info: Click here for Geometry Page

Streaming Link (MS Teams): This seminar will be held in person

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Università degli Studi Roma Tor Vergata
Dipartimento di Matematica

 

Numerical Analysis Seminar

Date: 11 January 2024

Schedule: 14:30 - Rome Time

Where: Conference Room 1201 "R. Dal Passo"

Title: " Interpolation by weights: insights and challenges Part. 2 "

Speaker: Ludovico Bruni Bruno - Università di Padova

Abstract: Interpolation of differential forms is a challenging aspect of modern approximation theory. Not only does it shed new light on some classical concepts of interpolation theory, such as the Lagrange interpolation and the Lebesgue constant, but it also suggests that they can be extended to a very general framework. As an extent of that, it is worth pointing out that the majority of classical shape functions commonly used in finite element methods, such as those involved in N´ed´elec or Raviar- Thomas elements, can be seen as a specialisation of this theory. Of course, this generality brings along the evident downside of an unfriendly level of abstraction. The scope of this series of two seminars is thus twofold: presenting the main challenges of this branch of approximation theory but in a concrete manner. The first seminar will hinge on a development of a convenient one dimensional toy model that enlightens parallelisms and differences with usual nodal interpolation. In the second seminar will extend these techniques to the multi-dimensional framework, motivating our choices by a geometrical flavour. A geometrical class of degrees of freedom, called weights, is studied. A weight is the inte- gral of a differential form on a given support, usually called small simplex. These degrees of freedom generalise in a very natural fashion Lagrange interpolation to differential forms and allow physical measurements as coefficients of the expansion. Although the description and the characterisation of these degrees of freedom is neat, many theoretical aspects turn out to be very hard to handle: known conditions for unisolvence apply to few classes of small simplices and a general rule is only conjectural. Nevertheless, a meaningful concept of Lebesgue constant can be provided and turns out to be meaningful for a qualitative analysis of different families of supports. After having reviewed the main concepts of this theory, we thus offer numerical comparisons that entangle the theory of weights with nodal multivariate interpolation

Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Organizing Committee:
Carla Manni (mail to contact)

Further Info: Click here for Numerical Analysis Page

Streaming Link (MS Teams): This seminar will be held in person

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Eventi