Higher signature Delaunay decompositions

E-print/Working paper (2016)

A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay ... [more ▼]

A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles. [less ▲]

Small circulant complex Hadamard matrices of Butson type

in European Journal of Combinatorics (2016), 51

We study the circulant complex Hadamard matrices of order nn whose entries are llth roots of unity. For n=ln=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for ... [more ▼]

We study the circulant complex Hadamard matrices of order nn whose entries are llth roots of unity. For n=ln=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n=p+q,l=pqn=p+q,l=pq with p,qp,q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n,ln,l. [less ▲]

Anti-de Sitter space: from physics to geometry

in CMS Notes (2015), 47(3), 14-15

A survey of recent developments in anti-de Sitter geometry.

Three applications of anti-de Sitter geometry

Speeches/Talks (2015)

Polyèdres inscrits dans des quadriques

Speeches/Talks (2015)

Productivity and Mobility in Academic Research: Evidence from Mathematicians

in Scientometrics (2014), 98(3), 1669-1701

Recovering the geometry of a flat spacetime from a background radiation

in Annales Henri Poincare (2014), 15(9), 1733-1799

Analytic aspects of the circulant Hadamard conjecture

in Annales mathématiques Blaise Pascal (2014), 21

Symplectic Wick rotations between moduli spaces of 3-manifolds

in Annali della Scuola Normale Superiore di Pisa: Classe di Scienze (2014)

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S ... [more ▼]

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called ``Wick rotations'', defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic. [less ▲]

A cyclic extension of the earthquake flow I

in Geometry and Topology (2013), 17(1), 157--234