L4

The Prandtl equation was derived in 1904 by Ludwig Prandtl in order to describe the behavior of fluids with small viscosity around a solid obstacle. Over the past decades, several results of ill-posedness in Sobolev spaces have been proved for this equation. As a consequence, it is natural to look for more sophisticated boundary layer models, that describe the coupling with the outer Euler flow at a higher order. Unfortunately, these models do not always display better mathematical properties, as I will explain in this talk. This is a joint work with Helge Dietert, David Gérard-Varet and Frédéric Marbach.

I will present a construction of large families of singularity-free stationary solutions of Einstein equations, for a large class of matter models including vacuum, with a negative cosmological constant. The solutions, which are of course real-valued Lorentzian metrics, are determined by a set of free data at conformal infinity, and the construction proceeds through elliptic equations for complex-valued tensor fields. One thus obtains infinite dimensional families of both strictly stationary spacetimes and black hole spacetimes.

A posteriori error estimators are a key tool for the quality assessment of given finite element approximations to an unknown PDE solution as well as for the application of adaptive techniques. Typically, the estimators are equivalent to the error up to an additive term, the so called oscillation. It is a common believe that this is the price for the `computability' of the estimator and that the oscillation is of higher order than the error. Cohen, DeVore, and Nochetto [CoDeNo:2012], however, presented an example, where the error vanishes with the generic optimal rate, but the oscillation does not. Interestingly, in this example, the local $H^{-1}$-norms are assumed to be computed exactly and thus the computability of the estimator cannot be the reason for the asymptotic overestimation. In particular, this proves both believes wrong in general. In this talk, we present a new approach to posteriori error analysis, where the oscillation is dominated by the error. The crucial step is a new splitting of the data into oscillation and oscillation free data. Moreover, the estimator is computable if the discrete linear system can essentially be assembled exactly.
 

The FEniCS system [1] allows the description of finite element discretisations of partial differential equations using a high-level syntax, and the automated conversion of these representations to working code via automated code generation. In previous work described in [2] the high-level representation is processed automatically to derive discrete tangent-linear and adjoint models. The processing of the model code at a high level eases the technical difficulty associated with management of data in adjoint calculations, allowing the use of optimal data management strategies [3].

This previous methodology is extended to enable the calculation of higher order partial differential equation constrained derivative information. The key additional step is to treat tangent-linear
equations on an equal footing with originating forward equations, and in particular to treat these in a manner which can themselves be further processed to enable the derivation of associated adjoint information, and the derivation of higher order tangent-linear equations, to arbitrary order. This enables the calculation of higher order derivative information -- specifically the contraction of a Kth order derivative against (K - 1) directions -- while still making use of optimal data management strategies. Specific applications making use of Hessian information associated with models written using the FEniCS system are presented.

[1] "Automated solution of differential equations by the finite element method: The FEniCS book", A. Logg, K.-A. Mardal, and  G. N. Wells (editors), Springer, 2012
[2] P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, "Automated derivation of the adjoint of high-level transient finite element programs", SIAM Journal on Scientific Computing 35(4), C369--C393, 2013
[3] A. Griewank, and A. Walther, "Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation", ACM Transactions on Mathematical Software 26(1), 19--45, 2000

In this talk I will present two recent findings concerning the preconditioning of elliptic problems. The first result concerns preconditioning of elliptic problems with variable coefficient K by an inverse Laplacian. Here we show that there is a close relationship between the eigenvalues of the preconditioned system and K. 
The second results concern the problem on mixed form where K approaches zero. Here, we show a uniform inf-sup condition and corresponding robust preconditioning. 

Standard quadratic programs have numerous applications and play an important role in copositivity detection. We consider reformulating a standard quadratic program as a mixed integer linear programming (MILP) problem. We propose alternative MILP reformulations that exploit the specific structure of standard quadratic programs. We report extensive computational results on various classes of instances. Our experiments reveal that our MILP reformulations significantly outperform other global solution approaches. 
This is joint work with Jacek Gondzio.

Amplituhedra are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed as a geometric construction encoding scattering amplitudes in the four-dimensional maximally supersymmetric Yang-Mills theory, they are mathematically interesting objects on their own. In my talk I strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. I focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional
conjugates. I show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron canonical differential form in all these cases. Notably, this also naturally exposes the rich combinatorial structures of amplituhedra, such as their regular triangulations

The Beilinson-Drinfeld Grassmannian, which classifies a G-bundle trivialised away from a finite set of points on a curve, is one of the basic objects in the geometric Langlands programme. Similar construction in higher dimensions in the algebraic and analytic settings are not very interesting because of Hartogs' theorem. In this talk I will discuss a differentiable version. I will also explain a theory of D-modules on differentiable spaces and use it
to define differentiable chiral and factorisation algebras. By linearising the Grassmannian we get examples of differentiable chiral algebras. This is joint work with Dennis Borisov.

We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety.

 We study codimension one holomorphic distributions on projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is an integrable phenomenon. This part of the  talk  is work joint with  M. Jardim(Unicamp) and O. Calvo-Andrade(Cimat).

We also study foliations by curves via the investigation  of their  singular schemes and  conormal  sheaves and we provide a classification  of foliations of degree at most 3 with  conormal  sheaves locally free.  Foliations of degrees  1 and 2 are aways given by a global intersection of two codimension one distributions. In the classification of degree 3 appear Legendrian foliations, foliations whose  conormal sheaves are instantons and other ” exceptional”
type examples. This part of the  talk   is  work joint with  M. Jardim(Unicamp) and S. Marchesi(Unicamp).