In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains and Mahler’s conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer–Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

We present new examples of complete embedded self-similar surfaces under mean curvature flow by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of nonrotationally symmetric self-shrinkers in ${\mathbf{R}}^3$. Although our initial approximating surfaces are asymptotic to a plane at infinity, the constructed self-similar surfaces are asymptotic to cones at infinity.

Soit $A$ une variété abélienne définie sur un corps de nombres $k$. Nous démontrons qu’il existe un faisceau inversible ample et symétrique sur $A$ dont le degré est borné par une constante explicite qui dépend seulement de la dimension de $A$, de sa hauteur de Faltings et du degré du corps de nombres $k$. Nous établissons également des versions explicites du théorème de Bertrand relatif au théorème de réductibilité de Poincaré et des théorèmes d’isogénies de Masser et Wüstholz entre variétés abéliennes. Les preuves reposent sur des arguments de géométrie des nombres dans les réseaux euclidiens constitués des morphismes entre variétés abéliennes munis des métriques de Rosati. Nous majorons les minima successifs de ces réseaux grâce au théorème des périodes que nous avons démontré dans un article précédent.

Let $A$ be an abelian variety over a number field $k$. We prove that $A$ admits a polarization of degree explicitly bounded in terms of the Faltings height of $A$, its dimension and the degree of $k$. The fact that this could be done effectively was known only in special cases, thanks to the work of Masser and Wüstholz. We also provide sharpened, explicit versions of their isogeny and factorization estimates, as well as of a result of Bertrand about almost complements of abelian subvarieties. One crucial tool is our recent period theorem and the proofs proceed through the detailed study of lattices of morphisms of abelian varieties, endowed with euclidean metrics deduced from suitable Rosati involutions.

We study asymptotics of traces of (noncommutative) monomials formed by images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge to (commutative) moments of a Gaussian process that can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian free fields. The limiting process has previously arisen via the global scaling limit of spectra for submatrices of Wigner Hermitian random matrices.

Motivated by studying the unitary dual problem, a variation of Kazhdan–Lusztig polynomials was defined in a previous publication by the author which encodes signature information at each level of the Jantzen filtration. These so-called signed Kazhdan–Lusztig polynomials may be used to compute the signatures of invariant Hermitian forms on irreducible highest weight modules. The key result of this paper is a simple relationship between signed Kazhdan–Lusztig polynomials and classical Kazhdan–Lusztig polynomials: signed Kahzdan–Lusztig polynomials are shown to equal classical Kazhdan–Lusztig polynomials evaluated at $-q$ rather than $q$ and multiplied by a sign. A simple signature character inversion formula follows from this relationship. These results have applications to finding the unitary dual for real reductive Lie groups since Harish–Chandra modules may be constructed by applying Zuckerman functors to the highest weight modules.