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Documents : Multi angle  Conférences Vidéo | enregistrements trouvés : 200

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Research talks

Generalized descriptive set theory has mostly been developed for uncountable cardinals satisfying the condition $\kappa ^{< \kappa }=\kappa$ (thus in particular for $\kappa$ regular). More recently the case of uncountable cardinals of countable cofinality has attracted some attention, partially because of its connections with very large cardinal axioms like I0. In this talk I will survey these recent developments and propose a unified approach which potentially could encompass all possible scenarios (including singular cardinals of arbitrary cofinality). Generalized descriptive set theory has mostly been developed for uncountable cardinals satisfying the condition $\kappa ^{< \kappa }=\kappa$ (thus in particular for $\kappa$ regular). More recently the case of uncountable cardinals of countable cofinality has attracted some attention, partially because of its connections with very large cardinal axioms like I0. In this talk I will survey these recent developments and propose a unified approach w...

03E15 ; 03E55 ; 54A05

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Research talks

By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this statement isconsistent at a weakly compact cardinal $\kappa$. We show using stacking mice that the existence of a non-domestic mouse (which yields a model with a proper class of Woodin cardinals and strong cardinals) is a lower bound. Moreover, we study variants of this statement involving sealed trees, i.e. trees with the property that their set of branches cannot be changed by certain forcings, and obtain lower bounds for these as well. This is joint work with Yair Hayut. By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this ...

03E45 ; 03E35 ; 03E55 ; 03E05

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Research talks

I give a survey of some recent results on set mappings.

03E05 ; 03E35

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Research talks

The Galvin-Prikry theorem states that Borel partitions of the Baire space are Ramsey. Thus, given any Borel subset $\chi$ of the Baire space and an infinite set $N$, there is an infinite subset $M$ of $N$ such that $\left [M \right ]^{\omega }$ is either contained in $\chi$ or disjoint from $\chi$ . In their 2005 paper, Kechris, Pestov and Todorcevic point out the dearth of similar results for homogeneous relational structures. We have attained such a result for Borel colorings of copies of the Rado graph. We build a topological space of copies of the Rado graph, forming a subspace of the Baire space. Using techniques developed for our work on the big Ramsey degrees of the Henson graphs, we prove that Borel partitions of this space of Rado graphs are Ramsey. The Galvin-Prikry theorem states that Borel partitions of the Baire space are Ramsey. Thus, given any Borel subset $\chi$ of the Baire space and an infinite set $N$, there is an infinite subset $M$ of $N$ such that $\left [M \right ]^{\omega }$ is either contained in $\chi$ or disjoint from $\chi$ . In their 2005 paper, Kechris, Pestov and Todorcevic point out the dearth of similar results for homogeneous relational structures. We have ...

05D10 ; 03C15 ; 03E75

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Research talks

In this talk I will present a couple of results for the existence of solutions to the one-dimensional Euler, Navier-Stokes and multi-dimensional Navier-Stokes systems. The purpose of the talk is to focus on the role of the pressure in the compressible fluid equations, and to understand whether or not it can be replaced by the nonlocal attraction-repulsion terms arising in the models of collective behaviour.

76N10 ; 35Q35

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Research talks

A common way to prove global well-posedness of free boundary problems for incompressible viscous fluids is to transform the equations governing the fluid motion to a fixed domain with respect to the time variable. An elegant and physically reasonable way to do this is to introduce Lagrangian coordinates. These coordinates are given by the transformation rule

$x(t)=\xi +\int_{0}^{t}u(\tau ,\xi ) d\tau $

where $u(\tau ,\xi )$ is the velocity vector of the fluid particle at time $\tau$ that initially started at position $\xi$. The variable $x(t)$ is then the so-called Eulerian variable and belongs to the coordinate frame where the domain that is occupied by the fluid moves with time.The variable $\xi$ is the Lagrangian variable that belongs to time fixed variables. In these coordinates the fluid only occupies the domain $\Omega_{0}$ that is occupied at initial time t = 0.
To prove a global existence result for such a problem, it is important to guarantee the invertibility of this coordinate transform globally in time. By virtue of the inverse function theorem, this is the case if

$\nabla_{\xi }x(t)=Id+\int_{0}^{t}\nabla_{\xi }u(\tau ,\xi )d\tau $

is invertible. By using a Neumann series argument, this is invertible, if the integral termon the right-hand side is small in $L^{\infty }(\Omega _{0})$. Thus, it is important to have a global control of this $L^{1}$-time integral with values in $L^{\infty }(\Omega _{0})$. If the domain is bounded, this can be controlled by exponential decay properties of the corresponding semigroup operators that describe the motion of the linearized fluid equation. On unbounded domains, however, these decay properties are not true anymore. While there are technical possibilities to fix these problems if the boundary is compact, these fixes cease to work if the boundary is non-compact.
As a model problem, we consider the case where $\Omega _{0}$ is the upper half-space. To obtain estimates of the $L^{1}$-time integral we use the theorem of Da Prato and Grisvard of 1975 about maximal regularity in real interpolation spaces. The need of global in timecontrol, however, makes it necessary to work out a version of this theorem that involves “homogeneous” estimates only (this was also done in the book of Markus Haase). In the talk, we show how to obtain this global Lagrangian coordinate transform from this theorem of Da Prato and Grisvard.
A common way to prove global well-posedness of free boundary problems for incompressible viscous fluids is to transform the equations governing the fluid motion to a fixed domain with respect to the time variable. An elegant and physically reasonable way to do this is to introduce Lagrangian coordinates. These coordinates are given by the transformation rule

$x(t)=\xi +\int_{0}^{t}u(\tau ,\xi ) d\tau $

where $u(\tau ,\xi )$ is ...

35Q35 ; 76D05

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Research talks

In this talk, I will present a recent study on traveling waves solutions to a 1D biphasic Navier-Stokes system coupling compressible and incompressible phases. With this original fluid equations, we intend to model congestion (or saturation) phenomena in heterogeneous flows (mixtures, collective motion, etc.). I will first exhibit explicit partially congested propagation fronts and show that these solutions can be approached by profiles which are solutions to a singular compressible Navier-Stokes system. The last part of the talk will be dedicated to the analysis of the stability of the approximate profiles. This is a joint work with Anne-Laure Dalibard. In this talk, I will present a recent study on traveling waves solutions to a 1D biphasic Navier-Stokes system coupling compressible and incompressible phases. With this original fluid equations, we intend to model congestion (or saturation) phenomena in heterogeneous flows (mixtures, collective motion, etc.). I will first exhibit explicit partially congested propagation fronts and show that these solutions can be approached by profiles which ...

35Q35 ; 35L67

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Research talks

In this joint work with Athanasios Tzavaras (KAUST) and Corrado Lattanzio (L’Aquila) we develop a relative entropy framework for Hamiltonian flows that in particular covers the Euler-Korteweg system, a well-known diffuse interface model for compressible multiphase flows. We put a particular emphasis on extending the relative entropy framework to the case of non-monotone pressure laws which make the energy functional non-convex.The relative entropy computation directly implies weak (entropic)-strong uniqueness, but we will also outline how it can be used in other contexts. Firstly, we describe how it can be used to rigorously show that in the large friction limit solutions of Euler-Korteweg converge to solutions of the Cahn-Hilliard equation. Secondly, we explain how the relative entropy can be used for obtaining a posteriori error estimates for numerical approximation schemes. In this joint work with Athanasios Tzavaras (KAUST) and Corrado Lattanzio (L’Aquila) we develop a relative entropy framework for Hamiltonian flows that in particular covers the Euler-Korteweg system, a well-known diffuse interface model for compressible multiphase flows. We put a particular emphasis on extending the relative entropy framework to the case of non-monotone pressure laws which make the energy functional non-convex.The relative ...

35Q31 ; 76D45 ; 76T10

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Research talks

The Euler-Korteweg system corresponds to compressible, inviscid fluids with capillary forces. It can be used to model diffuse interfaces. Mathematically it reads as the Euler equations with a third order dispersive perturbation corresponding to the capillary tensor.

In dimension one there exists traveling waves with equal or different limit at infinity, respectively solitons and kinks. Their stability is ruled by a simple criterion a la Grillakis-Shatah-Strauss. This talk is devoted to the construction of multiple traveling waves, namely global solutions that converge as $t\rightarrow \infty $ to a profile made of several (stable) traveling waves. The waves constructed have both solitons and kinks. Multiple traveling waves play a peculiar role in the dynamics of dispersive equations, as they correspond to solutions that follow in some sense a purely nonlinear evolution.
The Euler-Korteweg system corresponds to compressible, inviscid fluids with capillary forces. It can be used to model diffuse interfaces. Mathematically it reads as the Euler equations with a third order dispersive perturbation corresponding to the capillary tensor.

In dimension one there exists traveling waves with equal or different limit at infinity, respectively solitons and kinks. Their stability is ruled by a simple criterion a la ...

35Q35 ; 35C07 ; 35Q53 ; 35Q31 ; 35B35

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Research talks

Both multigrid and domain decomposition methods are so called optimal solvers for Laplace type problems, but how do they compare? I will start by showing in what sense these methods are optimal for the Laplace equation, which will reveal that while both multigrid and domain decomposition are iterative solvers, there are fundamental differences between them. Multigrid for Laplace’s equation is a standalone solver, while classical domain decomposition methods like the additive Schwarz method or Neumann-Neumann and FETI methods need Krylov acceleration to work. I will explain in detail for each case why this is so, and then also present modifications so that Krylov acceleration is not necessary any more. For overlapping methods, this leads to the use of partitions of unity, while for non-overlapping methods, the coarse space can be a remedy. Good coarse spaces in domain decomposition methods are very different from coarse spaces in multigrid, due to the very aggressive coarsening in domain decomposition. I will introduce the concept of optimal coarse spaces for domain decomposition in a sense very different from the optimal above, and then present approximations of this coarse space. Together with optimized transmission conditions, this leads to a two level domain decomposition method of Schwarz type which is competitive with multigrid for Laplace’s equation in wallclock time. Both multigrid and domain decomposition methods are so called optimal solvers for Laplace type problems, but how do they compare? I will start by showing in what sense these methods are optimal for the Laplace equation, which will reveal that while both multigrid and domain decomposition are iterative solvers, there are fundamental differences between them. Multigrid for Laplace’s equation is a standalone solver, while classical domain ...

65N55 ; 65N22 ; 65F10

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Research talks

I will present an efficient implementation of the highly robust and scalable GenEO preconditioner in the high-performance PDE framework DUNE. The GenEO coarse space is constructed by combining low energy solutions of local generalised eigenproblems using a partition of unity. In this talk, both weak and strong scaling for the GenEO solver on over 15,000 cores will be demonstrated by solving an industrially motivated problem with over 200 million degrees of freedom in aerospace composites modelling. Further, it will be shown that for highly complex parameter distributions in certain real-world applications, established methods can become intractable while GenEO remains fully effective. In the context of multilevel Markov chain Monte Carlo (MLMCMC), the GenEO coarse space also plays an important role as an effective surrogate model in PDE-constrained Bayesian inference. The second part will therefore focus on the approximation properties of the GenEO coarse space and on a high-performance parallel implementation of MLMCMC.
This is joint work with Tim Dodwell (Exeter), Anne Reinarz (TU Munich) and Linus Seelinger (Heidelberg).
I will present an efficient implementation of the highly robust and scalable GenEO preconditioner in the high-performance PDE framework DUNE. The GenEO coarse space is constructed by combining low energy solutions of local generalised eigenproblems using a partition of unity. In this talk, both weak and strong scaling for the GenEO solver on over 15,000 cores will be demonstrated by solving an industrially motivated problem with over 200 million ...

65F08 ; 65N22 ; 65N30 ; 65N55

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Research talks

We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation. This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated.
The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple computational results will be shown for the heat equation and the wave equation which indicate the potential as a parallel-in-time method.
This is joint work with Elle McDonald (CSIRO, Australia), Jennifer Pestana (Strathclyde University, UK) and Anthony Goddard (Durham University, UK)
We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation. This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated.
The underlying linear algebra includes circulant matrix approximations of Toeplitz-structured matrices and allows for effective parallel implementation. Simple co...

65F08 ; 15B05 ; 65M22

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Research talks

Linear matrix equations such as the Lyapunov and Sylvester equations and their generalizations have classically played an important role in the analysis of dynamical systems, in control theory and in eigenvalue computation. More recently, matrix equations have emerged as a natural linear algebra framework for the discretized version of (systems of) partial differential equations (PDEs), possibly evolving in time. In this new framework, new challenges have arisen. In this talk we review some of the key methodologies for solving large scale linear and quadratic matrix equations. We will also discuss recent matrix-based strategies for the numerical solution of time-dependent problems arising in control and in the analysis of spatial pattern formations in certain electrodeposition models. Linear matrix equations such as the Lyapunov and Sylvester equations and their generalizations have classically played an important role in the analysis of dynamical systems, in control theory and in eigenvalue computation. More recently, matrix equations have emerged as a natural linear algebra framework for the discretized version of (systems of) partial differential equations (PDEs), possibly evolving in time. In this new framework, new ...

65F10 ; 65M22 ; 15A24

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Research talks

We construct Skyrme fields from holonomy of the spin connection of multi-Taub-NUT instantons with the centres positioned along a line in R3. The domain of our Skyrme fields is the space of orbits of the axial symmetry of the multi-Taub-NUT instantons. We obtain an expression for the induced Einstein-Weyl metric on the space and its associated solution to the $SU(\infty )$-Toda equation.

35Q75 ; 81V17 ; 81T13 ; 81Q80 ; 35C08

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Research talks

83C60 ; 81R25

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Research talks

83CXX

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Twist and loop Rovelli, Carlo | CIRM H

Multi angle

Research talks

83C45 ; 83C47 ; 81V17

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Research talks

In geometric representation theory, one is interested in studying the geometry of affine Grassmannians of quasi-split simply-connected reductive groups. In this endeavor, one of the main techniques, introduced by Faltings in the split case, consists in constructing natural realisations of these ind-schemes over the integers. In the twisted case, this was done by Pappas and Rapoport in the tamely ramified case, i.e. over $\mathbb{Z}[1/e]$, where $e = 2$ or $3$ is the order of the automorphism group of the split form we are dealing with. We explain how to extend the parahoric group scheme that appeared in work of Pappas, Rapoport, Tits and Zhu to the polynomial ring $\mathbb{Z}[t]$ with integer coefficients and additionally how the group scheme obtained in char. $e$ can be regarded as a parahoric model of a basic exotic pseudo-reductive group. Then we study the geometry of the affine Grassmannian and also its global deformation à la Beilinson-Drinfeld, recovering all the known results in the literature away from $e = 0$. This also has some pertinence to the study of local models of Shimura varieties in wildly ramified cases. In geometric representation theory, one is interested in studying the geometry of affine Grassmannians of quasi-split simply-connected reductive groups. In this endeavor, one of the main techniques, introduced by Faltings in the split case, consists in constructing natural realisations of these ind-schemes over the integers. In the twisted case, this was done by Pappas and Rapoport in the tamely ramified case, i.e. over $\math...

20G44 ; 20C08

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Research talks

L’immeuble réduit de Bruhat-Tits de G (réductif connexe) se plonge dans l’analytifié $G^{an}$. Cela est dû à Berkovich et Rémy-Thuillier-Werner. Nous expliquerons cela puis nous expliquerons que l’on peut définir naturellement dans ce cadre des filtrations analytiques dont les points rationnels coïncident dans certains cas avec les groupes de Moy-Prasad.

20E42 ; 20G25

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Research schools

The aim is to give an introduction to the basic theory of affine Grassmannians and affine flag varieties. We put special emphasis on the utility of dynamic methods in sense of Drinfeld [D], and the utility of non-constant group schemes. We plan to adress the following aspects:
• Affine Grassmannians as moduli spaces of G-bundles, and as quotients of loop groups ;
• Cell decompositions of affine Grassmannians and affine flag varieties via dynamic methods: Iwahori, Cartan and Iwasawa decompositions ;
• Schubert varieties, Demazure resolutions, Convolution morphisms, Combinatorial structures ;
• Moduli spaces of G-bundles with level structure versus bundles under non-constant group schemes ;
• Beilinson-Drinfeld type deformations of affine Grassmannians ;
• Relation to the local geometry of moduli spaces of Drinfeld shtukas and Shimura varieties.
The aim is to give an introduction to the basic theory of affine Grassmannians and affine flag varieties. We put special emphasis on the utility of dynamic methods in sense of Drinfeld [D], and the utility of non-constant group schemes. We plan to adress the following aspects:
• Affine Grassmannians as moduli spaces of G-bundles, and as quotients of loop groups ;
• Cell decompositions of affine Grassmannians and affine flag varieties via dynamic ...

14M15 ; 14D24

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