Geometric methods in computational complexity

Le théorème de Chow

Dans cet exposé, je vais présenter les lieux de dégénérescence orbitaux.
Ces lieux généralisent les lieux de dégénérescence classiques; étant
donnés des fibrés principaux (ou fibrés vectoriels avec une certaine
structure), ils permettent de construire des variétés projectives
“spéciales”.
Deux outils majeurs permettent de controler les propriétés intrinsèques
des variétés construites (e.g. la positivité du fibré canonique):
l’existence de désingularisations explicites de ces lieux et l’existence
de résolutions localement libres de leurs idéaux. Selon le temps
disponible, je vais présenter ces outils à  l’aide d’exemples concrets et
significatifs.
Il s’agit d’un projet en commun avec Sara Angela Filippini, Laurent
Manivel et Fabio Tanturri.

Théorème de décomposition en cellules (II)

The Willmore energy arises naturally as a measure of how curved an immersed surface in $mathbb{R}^3$ is, with applications in relativity (the Hawking mass). Willmore immersions are critical points of this energy. We will study sequences of Willmore surfaces, which are subject to concentration-compactness i.e. : bubbling phenomena. We will focus on simple minimal bubbles, and detail consequences on the compactness below certain thresholds.

Let $s_{1}, dots, s_{k}$ be the elementary symmetric functions of the complex variables $x_{1}, dots, x_{k}$.
We say that $F in C[s_{1}, dots, s_{k}]$ is a {trace function} if their exists $f in C[z]$ such that
$F(s_{1}, dots, s_{k}] = sum_{j=1}^{k} f(x_{j})$ for all $s in C^{k}$.
We give an explicit finite family of second order differential operators in the Weyl algebra
$W_{2}:= C[s_{1}, dots, s_{k}]langle frac{partial}{partial s_{1}}, dots, frac{partial}{partial s_{k}}rangle $
which generates the left ideal in $W_{2}$ of partial differential operators killing all trace functions.
The proof uses a theorem for symmetric differential operators analogous
to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer $k$
a regular holonomic system which is a quotient of $W_{2}$ by an explicit left ideal whose local solutions are given by linear
combinations of the branches of the multivalued root of the universal equation of degree $k$:
$z^{k} + sum_{h=1}^{k} (-1)^{h}.s_{h}.z^{k-h} = 0$.

Théorème de décomposition en cellules (I)

Soit Y une hypersurface lisse dans une variété hyper-kählérienne irréductible projective X de dimension 2n et $sigma$ une forme holomorphiquement symplectique sur X. Le feuilletage caractéristique F sur une hypersurface Y est le noyau de la forme symplectique $sigma$ restreinte à  Y. Supposons qu’il existe une fibration lagrangienne $pi:Xto mathbb{P}^n$ et $Y=pi^{-1}D$ pour une hypersurface $D$ dans $mathbb{P}^n$. Je montre que une feuille générale de $F$ est Zariski dense dans une fibre de $pi$.

In 2008 Campana conjectured that a smooth projective family of canonically polarized manifolds over a special manifold (being opposed to general type manifolds) is isotrivial, i.e. any two fibers are isomorphic. This conjecture was proven by Taji in 2016. In this talk I will present a hyperbolic version of Campana’s isotriviality conjecture: a smooth family of canonically polarized or polarized Calabi-Yau manifolds over a hyperbolically special complex manifold (i.e. its Kobayashi pseudo distance vanishes identically) is necessarily isotrivial. This result is indeed inspired by another conjecture of Campana: a complex manifold is special if and only if it is hyperbolically special, and thus provides some (indirect) evidence to this conjecture.

This talk is about a joint work, still in progress, with Friedrich Tomi (Heidelberg University, Germany) where one investigates the existence of solutions which are invariant by a Lie subgroup of the isometry group of a Riemannian manifold $M$; acting freely and properly on $M$, to the Dirichlet problem of a certain class of partial differential equations on $M$: Typical examples of this class are the $p$-Laplacian PDE and the minimal surface equation. This approach may reduce the study of the Dirichlet problem in unbounded to bounded domains and also allows to prove the existence of solutions on domains which are not necessarily mean convex in the case of the minimal surface equation for certain boundary data.