Here are some recent preprints.
Recent publications are here,older
papers are there.
There are some other texts (surveys, etc) on another page.
Please send me an e-mail (or a s-mail) if you wish to receive a preprint
or a reprint.
Color code:
3-d hyperbolic geometry
Teichmueller theory
AdS or Lorentz geometry
manifolds with boundary
manifolds with "particles"
circle patterns, polyhedral geometry,
discrete geometry
Hadamard matrices, integration over O(n), etc
Outside mathematics
Recovering the geometry of a flat spacetime from a background radiation.Francesco Bonsante,
Catherine Meusburger,
Jean-Marc Schlenker.
arXiv:1302.6966.
We consider globally hyperbolic flat spacetimes in 2+1 and 3+1 dimensions,
where a uniform light signal is emitted on the $r$-level surface of the
cosmological time for $r\to 0$. We show that the intensity of this signal, as
perceived by a fixed observer, is a well-defined, bounded function which is
generally not continuous. This defines a purely classical model with
anisotropic background radiation that contains information about initial
singularity of the spacetime. In dimension 2+1, we show that this observed
intensity function is stable under suitable perturbations of the spacetime, and
that, under certain conditions, it contains sufficient information to recover
its geometry and topology. We compute an approximation of this intensity
function in a few simple examples.
Analytic aspects of the circulant Hadamard conjecture.Teodor Banica,
Ion Nechita,
Jean-Marc Schlenker.
arXiv:1212.3589.
We investigate the problem of counting the real or complex Hadamard matrices which are circulant,
by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$
the quantity $\Phi=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi\geq N^2$, with equality
if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix.
This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi$,
(2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$.
We explore here these questions, with some results and conjectures.
The renormalized volume and uniformisation of conformal structures.Colin Guillarmou,
Sergiu Moroianu,
Jean-Marc Schlenker.
arXiv:1211.6507.
We study the renormalized volume of asymptotically hyperbolic Einstein (AHE
in short) manifolds $(M,g)$ when the conformal boundary $\pl M$ has dimension
$n$ even. Its definition depends on the choice of metric $h_0$ on $\partial M$
in the conformal class at infinity determined by $g$, we denote it by ${\rm
Vol}_R(M,g;h_0)$. We show that ${\rm Vol}_R(M,g;\cdot)$ is a functional
admitting a "Polyakov type" formula in the conformal class $[h_0]$ and we
describe the critical points as solutions of some non-linear equation
$v_n(h_0)={\rm const}$, satisfied in particular by Einstein metrics. In
dimension $n=2$, choosing extremizers in the conformal class amounts to
uniformizing the surface, while in dimension $n=4$ this amounts to solving the
$\sigma_2$-Yamabe problem. Next, we consider the variation of ${\rm
Vol}_R(M,\cdot;\cdot)$ along a curve of AHE metrics $g^t$ with boundary metric
$h_0^t$ and we use this to show that, provided conformal classes can be
(locally) parametrized by metrics $h$ solving $v_n(h)=\int_{\pl M}v_n(h){\rm
dvol}_{h}$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to
Identity) can be viewed as a Lagrangian submanifold in the cotangent space to
the space $\mc{T}(\pl M)$ of conformal structures on $\pl M$. We obtain as a
consequence a higher-dimensional version of McMullen's quasifuchsian
reciprocity. We finally show that conformal classes admitting negatively curved
Einstein metrics are local minima for the renormalized volume for a warped
product type filling.
A cyclic extension of the earthquake flow II.Francesco Bonsante,
Gabriele Mondello,
Jean-Marc Schlenker.
arXiv:1208.1738.
The landslide flow, introduced in [5], is a smoother analog of the earthquake
flow on Teichmüller space which shares some of its key properties. We show
here that further properties of earthquakes apply to landslides. The landslide
flow is the Hamiltonian flow of a convex function. The smooth grafting map
$sgr$ taking values in Teichm\"uller space, which is to landslides as grafting
is to earthquakes, is proper and surjective with respect to either of its
variables. The smooth grafting map $SGr$ taking values in the space of complex
projective structures is symplectic (up to a multiplicative constant). The
composition of two landslides has a fixed point on Teichm\"uller space. As a
consequence we obtain new results on constant Gauss curvature surfaces in
3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow
has a satisfactory extension to the boundary of Teichmüller space.
Some questions on anti-de Sitter geometry.Thierry Barbot,
Francesco Bonsante,
Jeff Danciger, William M. Goldman, François Guéritaud,
Fanny Kassel, Kirill Krasnov, Jean-Marc Schlenker, Abdelghani Zeghib.
arXiv:1205.6103.
Text as pdf.
We present a list of open questions on various aspects of AdS geometry, that is, the geometry
of Lorentz spaces of constant curvature $-1$. When possible we point out relations
with homogeneous spaces and discrete subgroups of Lie groups, to Teichm\"uller theory, as well as
analogs in hyperbolic geometry.
Productivity and Mobility in Academic Research: Evidence from Mathematicians.Pierre Dubois,
Jean-Charles Rochet,
Jean-Marc Schlenker.
Working paper
TSE 10-160
and
IDEI 606
,
may 2010.
Using an exhaustive database on academic publications in mathematics, we
study the patterns of productivity by world mathematicians over the period
1984-2006. We uncover some surprising facts, such as the absence of age
related decline in productivity and the relative symmetry of international
movements, rejecting the presumption of a massive "brain drain" towards the
U.S. Looking at the U.S. academic market in mathematics, we analyze the
determinants of success by top departments. In conformity with recent
studies in other fields, we find that selection effects are much stronger
than local interaction effects: the best departments are most successful in
hiring the most promising mathematicians, but not necessarily at stimulating
positive externalities among them. Finally we analyze the impact of career
choices by mathematicians: mobility almost always pays, but early
specialization does not.
The convex core of quasifuchsian manifolds with particles.Cyril Lecuire, Jean-Marc Schlenker.
arXiv:0909.4182.
Text as pdf, ps.
We consider quasifuchsian manifolds with ``particles'', i.e., cone
singularities of fixed angle less than $\pi$ going from one connected component
of the boundary at infinity to the other. Each connected component of the
boundary at infinity is then endowed with a conformal structure marked by the
endpoints of the particles. We prove that this defines a homeomorphism from the
space of quasifuchsian metrics with $n$ particles (of fixed angle) and the
product of two copies of the Teichm\"uller space of a surface with $n$ marked
points. This is analoguous to the Bers theorem in the non-singular case.
Quasifuchsian manifolds with particles also have a convex core. Its boundary
has a hyperbolic induced metric, with cone singularities at the intersection
with the particles, and is pleated along a measured geodesic lamination. We
prove that any two hyperbolic metrics with cone singularities (of prescribed
angle) can be obtained, and also that any two measured bending laminations,
satisfying some obviously necessary conditions, can be obtained, as in [BO] in
the non-singular case.