Curriculum Vitae for Marius Crainic

The broad research interests of Marius Crainic: Geometry and Topology
Geometry is the oldest branch of mathematics. It is concerned with questions of shape, size,
relative position of figures, and the properties of space. Its empirical origins were put into an
axiomatic form by Euclid (3rd century BC ), and the resulting Euclidean geometry has been
standard for many centuries. Questions from astronomy, on the position and the movement
of stars and planets, served as an important source of geometric problems during one and a
half millennia. The discovery of coordinates by Ren Descartes played an essential role in the
development of algebra (geometric figures, such as plane curves, could now be represented by
equations) and, later on, in the emergence of infinitesimal calculus in (17th century). For a
long time, there was no clear distinction between physical space and geometrical space; starting
with the discovery of non-Euclidean geometry (19th century) the notion of “geometric space”
has undergone a radical transformation (up to the point that even notions like “space”, “point”,
“line” etc lost their original intuitive content). The central question became: which “abstract”
geometrical space best fits (models) the physical space.
Differential Geometry: Contemporary geometry considers manifolds- which are spaces that
approximately resemble the familiar Euclidean space only at small scales (e.g. spheres). On
these, one considers various “geometric structures”, depending on the aspects one is interested
in. Differential Geometry is the resulting study of manifolds endowed with such geometric
structures. For instance, to talk about “lengths”, the geometric structure that one has to
consider is known under the name of “Riemannian metric” and the resulting field is known as
“Riemannian Geometry” (a subfield of Differential Geometry). Differential geometry is also the
language in which Einstein’s general theory of relativity is expressed: the universe appears as a
smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of
space-time; understanding this curvature is essential for the positioning of satellites into orbit
around the earth. One may even say that each physical theory comes with its own (subfield
of) geometry. Another example comes from the Hamiltonian formulation of classical mechanics,
which gives rise to Poisson Geometry. In this theory, the objects that are studied are manifolds
endowed with the geometric structures that allows one to write down Hamilton’s equations
(“Poisson structures”).
We also have to mention here that role played by symmetries in geometry. When looking at most
of Eschers drawings, one realizes the presence of symmetry, which allows one to reconstruct the
complete drawing from a smaller part of it. More generally, one of the central ideas in geometry
is to understand (and even classify) the geometric objects via their ”symmetries”; this originates
in Klein’s Erlanger Programm (1872). The precise mathematical notion that allows us to talk
about symmetries in Differential Geometry is that of “Lie group”. Its birth goes back to the
work of Sophus Lie on symmetries of differential equations (at the end of the nineteenth century)
and that of Elie Cartan on symmetries of geometric structures (at the beginning of the twentieth
century). Nowadays one talks about “Lie Theory”, which pervades modern mathematics and
theoretical physics.
Topology: A close relative of Geometry is the field of “Topology”, which is concerned with
properties that are preserved under continuous deformations of objects, such as deformations
that involve stretching, but no tearing or gluing. For instance, a circle and an ellipsis, although
geometrically distinct, have the same topology: one can be obtained from the other by a continuous
transformation- realise the ellipsis as the shadow of a non-horizontal circle and consider the
transformation along the rays of light. Similarly, spheres and convex polyhedra are topologically
similar; as a consequnce, one arrises at the famous theorem of Euler which says that, for any
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