NILS Science and Sustainability aims to enhance human capital and knowledge base in Spain and in Norway, Iceland and Liechtenstein, by improving and deepening academic and research collaboration, promoting scientific research on strategic fields for Human well-being and Earth Sustainability. See Nils Mobility Project
Manifolds are mathematical objects that generalise curves and surfaces. A curve is a 1 dimensional manifold, because locally looks like a line, and a surface, which looks like a plane, is a 2 dimensional one. In general, we can talk of n dimensional manifolds. This concept is the core of modern geometry, but restricting to dimensions 1, 2 and 3 is not enough to understand the universe we live in. For instance, one of the brances of the celebrated String Theory suggests we are living in a 10 dimensional manifold, of which 6 dimensions are too small to be seen.
Sometimes, a manifold of even dimension, say 2n, has a richer geometric structure. Richer, in this case, means that it has more properties to take into account. In particular, we are interested in complex structures. This is easily seen in the canonical case of the plane: one can see it just like a plane or like the set of complex numbers, with their own way of sum and multiply.
In general, referring to a curve, we mean a real curve, however, we can speak of complex curves which locally looks like complex numbers. Namely, they are real surfaces with complex structure. In this way, makes sense to define a complex manifold of dimension n. Like the plane, they can be understood as 2n dimensional real manifolds as well.
A natural question arises here: is there something in between real and complex manifolds? And the answer is yes. Sometimes we have manifolds endowed with partial complex structures. They are called CR-manifolds of dimension (2n,r), locally they can be seen as a product of a n dimensional complex space and a r dimensional real space.
The project we are carrying out with the NILS Grant is devoted to the study of these objects.