  1) In 1976, looking at simple finite-dimensional complex Lie superalgebras,
J.~Bernstein and I, and independently M.~Duflo, observed that certain
divergence-free vectorial Lie superalgebras have deformations with odd
parameters and conjectured that other simple Lie superalgebras have no such
deformations (unpublished). Here, I prove this conjecture and overview the
known classification of simple finite-dimensional complex Lie superalgebras,
their presentations, realizations, and (very sketchily) relations with simple
Lie (super)algebras over fields of positive characteristic.
  2) Any supermanifold which is a ringed space of the form (a manifold $M$, the
sheaf of sections of the exterior algebra of a vector bundle over $M$) is
called split. Gaw\c{e}dzki (1977) and Batchelor (1979) proved that every smooth
supermanifolds is split. In 1982, P. Green and Palamodov showed that
a~complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a
split supermanifold. So far, researchers considered, mostly, even obstructions
to splitness. This lead them to the conclusion that any supermanifolds of
superdimension $m|1$ is split. I'll show that there are non-split
supermanifolds of superdimension $m|1$; for example, certain $1|1$-dimensional
superstrings, the obstructions to their splitness correspond to odd parameters.
