Renaud Coulangeon

Designs in Grassmannian Spaces and Lattices

AbstractWe introduce the notion of a t-design on the Grassmann manifold

Energy Minimization, Periodic Sets and Spherical Designs

\mathcal{G}_{m,n}

Lenstra's Constant and Extreme Forms in Number Fields

of the m-subspaces of the Euclidean space

Perfect lattices over imaginary quadratic number fields

\mathbb{R}

Codes and designs in Grassmannian spaces

n. It generalizes the notion of antipodal spherical design which was introduced by P. Delsarte, J.-M. Goethals and J.-J. Seidel. We characterize the finite subgroups of the orthogonal group which have the property that all its orbits are t-designs. Generalizing a result due to B. Venkov, we prove that, if the minimal m-sections of a lattice L form a 4-design, then L is a local maximum for the Rankin function γn,m.

Computing in arithmetic groups with Voronoï's algorithm☆

We set up a connection between the theory of spherical designs and the question of minima of Epsteins zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epsteins zeta function, at least at any real s>n/2. We deduce from this a new proof of Sarnak and Strombergssons theorem asserting that the root lattices D4 and E8, as well as the Leech lattice, achieve a strict local minimum of the Epsteins zeta function at any s>0. Furthermore, our criterion enables us to extend their theorem to all the so-called extremal modular lattices(up to certain restrictions) using a theorem of Bachoc and Venkov, and to other classical families of lattices (e.g. the Barnes-Wall lattices).