Differential Geometry vs Higher Differential Geometry

Geometry is roughly the study of the shape of space. The notion of space has several incarnations which usually come from mathematical models to understand our physical world. As examples, one has the phase spaces of classical mechanics, the space of operators in quantum mechanics or curved manifolds arising in general relativity. In differential geometry, the basic notion of space is that of a smooth manifold, which is a locally euclidean topological space on which we can apply the machinery of differential calculus in order to understand its geometry.

Our research is mostly concerned with differentiable manifolds equipped with additional structures which are intricately linked with theoretical physics. Some examples include:

1. Symplectic manifolds: these correspond to the phase spaces of classical mechanics;

2. Poisson manifolds: as models for phase spaces with symmetries;

3. Dirac structures: originally introduced as a unified approach to classical mechanics with both symmetries and constraints;

4. Generalized complex structures: offer a unified setting to study symplectic and complex structures, hence useful to understand dualities in physics, e.g. mirror symmetry, T-duality.

Why higher differential geometry?

Higher differential geometry is the study of usual differential geometry by mean of higher categorical structures and homotopy methods. Higher structures encompass new algebraic and geometric structures which have become central tools in various domains of mathematics and theoretical physics. Higher categorical structures and homotopy methods have become focus of intense research in the last decade and are now a major topic in mathematics. One of the main reasons is the fact that they offer both, new notions of space and new tools in Geometry and hence permit to deal with more flexible notions of symmetries arising in different problems of modern geometry and mathematical physics. In particular, they provide the right framework to define and study most structures associated to geometric data while the classical structures are often too rigid to be defined or to determine invariants of geometric objects. Higher structures allow to deal with singularities that occur in nature, equivariant or gauge theoretic phenomenon and have also found tremendous applications in the study of algebraic geometry as well as of quantum field theories. A variety of examples of such structures can be found in algebra, geometry and physics, including: strongly homotopy algebras, differentiable stacks, higher stacks, Courant algebroids, representations up to homotopy, higher gauge theories, operads, among others.

Our interests

In our group we develop projects at several levels, going from undergraduate and master, to doctoral and postdoctoral research. The research done by the members of our group deals with the study and development of higher structures in differential geometry, having as main applications: on the one hand, new results in the differential geometry of singular quotients of manifolds known as differentiable stacks; on the other hand, the recognition of higher ``hidden'' symmetries of classical geometric structures, which are not detected in the more rigid setting of usual differential geometry. For that, we combine algebraic, geometric and topological tools often coming from homotopy theory and higher categories. For a more detailed information about our interests, please look at Our Team.