Vignette d'image

Schlenk, Félix

Nom
Schlenk, Félix
Affiliation principale
Fonction
Professeure ordinaire
Email
felix.schlenk@unine.ch
Identifiants
https://libra.unine.ch/handle/123456789/461

Résultat de la recherche

Filtres

11
2
2
2
2
Réinitialiser les filtres

Paramètres

Voici les éléments 1 - 2 sur 2
  • Publication
    Accès libre
    Symplectic embeddings in dimension 4
    (2014)
    Frenkel, David
    ;
    Schlenk, Félix 
    La géométrie symplectique est la géométrie sous-jacente à la dynamique hamiltonienne. Depuis la démonstration du théorème de non-tassement de Gromov en 1985, les plongements symplectiques se trouvent au coeur de la géométrie symplectique. Cette thèse étudie certains problèmes de plonge- ments symplectiques en dimension 4. Nous commençons par résoudre com- plètement le problème des plongements d’ellipsoïdes dans des cubes. Ce résultat est un raffinement du théorème de Gromov, McDuff-Polterovich et Biran sur les plongements d’une union disjointe de boules égales dans un cube. Dans la deuxième partie de la thèse, nous construisons des plonge- ments explicites d’une union disjointe de boules dans certaines unions (non- disjointes) d’ellipsoïdes et de cylindres. Il découle des capacités ECH de Hutchings que ces plongements sont optimaux., Symplectic geometry is the underlying geometry of Hamiltonian dynamics. Since the proof of Gromov’s non-squeezing theorem in 1985, symplectic embeddings have been at the heart of symplectic geometry. This thesis studies some symplectic embedding problems in dimension 4. We start by completely solving the problem of embedding an ellipsoid into a cube. This result is a refinement of the theorem proved by Gromov, McDuff- Polterovich and Biran about embeddings of a disjoint union of equal balls into a cube. In the second part of the thesis, we construct explicit embed- dings of a disjoint union of balls into certain (non-disjoint) unions of an ellipsoid and a cylinder. It follows from Hutchings’ ECH capacities that these embeddings are optimal.
  • Publication
    Accès libre
    Symplectic embeddings of 4-dimensional ellipsoids into polydiscs
    (2012)
    Stylianou, Dorothee Cosima
    ;
    Schlenk, Félix 
    Recently, McDuff and Schlenk determined in [19] the function $c_{EB}(a)$, whose value at $a$ is the infimum of the size of a $4$-ball, into which the ellipsoid $E(1,a)$ symplectically embeds (here, $a > 1$ is the ratio of the area of the large axis to that of the smaller axis of the ellipsoid). This work is focused on the study of embeddings of $4$-dimensional ellipsoids $E(1,a)$ into four-dimensional polydiscs $P(\mu,\mu) = D^2(\mu) \times D^2(\mu)$, where $D^2(\mu)$ denotes the disc in $\mathbb{R}^2$ of area $\mu$, for $a\in [1,\sigma^2]$, where $\sigma^{2}=3+2\sqrt{2}\approx5.83$ is the square of the silver ratio $\sigma:=1+\sqrt{2}$. The embedding capacity function given by $c_{EC}(a):=\inf\left\{\mu \ \big\vert \ E(1,a) \ \rightarrow \ P(\mu,\mu)\right\}$ is defined for $a\in [1,\infty)$. As in the case of embeddings into balls, the structure of the graph of $c_{EC}(a)$ is very rich: since symplectic embeddings are volume preserving, we always have $c_{EC}(a) \geq \sqrt{\frac{a}{2}}$ and it is not hard to see that this lower bound is sharp for $a\ge 8$, that is the function $c_{EC}(a)$ is equal to the volume constraint given by $\sqrt{\frac{a}{2}}$ for $a\geq 8$. For $a$ less than the square of the silver ratio $\sigma:= 1 + \sqrt{2}$, the function $c_{EC}(a)$ turns out to be piecewise linear, having the form of a stair with an infinite number of steps, converging to $\sqrt{\frac{\sigma^2}{2}}$ for $a \rightarrow \sigma^2$. These ``stairs'' will be determined by the Pell numbers and we thus refer to them as the ``Pell stairs''. For the proof, we first translate the embedding problem $E(1, a) \rightarrow P(\mu,\mu)$ to a certain ball packing problem of the ball $B(2\mu)$. This embedding problem is then solved by adapting the method from McDuff and Schlenk in [19], which finds all exceptional spheres in blow-ups of the complex projective plane that provide an embedding obstruction. Furthermore we also prove the equivalence of symplectic embeddings of an ellipsoid into a polydisc and the embedding of its decomposition into disjoint balls into a ball, that is $E(a,b)\rightarrow P(c,d) \Leftrightarrow B(a,b)\sqcup B(c)\sqcup B(d)\rightarrow B(c+d)$.