## The spectrum of flexible polyhedra

**Goal:**
It has been known since Legendre and Cauchy that
convex polyhedra in Euclidean space are always rigid, that
is, they cannot be deformed without changing the ''shape''
or their faces. However, non-convex polyhedra can be
flexible, the first examples were discovered by R. Connelly
in 1968.
It was then conjectured that the interior volume of such
a flexible polyhedron would remain constant, and this well-known
''Bellows conjecture'' was proved by Sabitov in 2006 using
algebraic methods.
Recently, V. Alexandrov has conjectured another related
but stronger statement: all eigenvalues of the Laplace
operator, in a flexible polyhedron with Dirichlet boundary
conditions, remain constant.
The goal of the project would be to check experimentally
this conjecture, by computing numerically the first eigenvalues
of the Laplace operator in a few examples of flexible
polyhedra.

**Schedule:** to be determined.

**Persons involved:**
Advised by Jean-Marc Schlenker, NN.

**Difficulty level:**
medium.

**Tools:**
programs in python/sage, use of computational tools to compute
the first eigenvalues of the Laplace operator.

**Expected results:**
either disprove the conjecture, or give some evidence that
it might be correct by showing that, up to some approximation,
the first few eigenvalues of the Laplace operator remain
constant in an isometric deformation in a few simple examples.