GEOGRAL · Geometry of Grassmannian Lagrangian manifolds and their submanifolds, with applications to nonlinear partial differential equations of physical interest
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2015-09-01 → 2017-08-31
- EU contribution
- €146,462
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
Lines connect the coordinator with its partners.
Results in brief
Geometry of Grassmannian Lagrangian manifolds and their submanifolds, with applications to nonlinear partial differential equations of physical interest
The project has been designed around an unique ambitious research objective, namely answering to a conjecture posed by E. Ferapontov and B. Doubrov in the paper “On the integrability of symplectic Monge–Ampère equations”, appeared 2010 in the Journal of Geometry and Physics, and to date still unanswered. Even if the main objective has not been achieved, the Researcher and his collaborators have been working hard on it, thus producing a lot of interesting stand-alone partial results and spin-offs. The aforementioned conjecture (referred to as “Ferapontov conjecture”) is an excellent example of a question involving nonlinear PDEs, whose solutions inevitably requires the exploitation of purely geometric methods. As such, it encourages deepening and developing the (relatively) scarcely explored area of geometric methods in nonlinear PDEs, to the benefit of both the pure mathematical society and physics/applied mathematics society. Nonlinear partial differential equations (PDEs) are indeed at the heart of all theories describing natural phenomena. Beside obtaining original research results, the Researcher has spread the rudiments of the geometric theory of nonlinear PDEs to a wider public through a dedicated Ph.D course. The research results themselves have been duly spread within the mathematical community through a dedicated workshop, and several seminars in European institutions. Various papers, both of research and review character, have appeared or have been submitted to internationally recognised journals. More detail can be found in the section “WORK PERFORMED” below.
Data: CORDIS, © European Union
Project objective
The aim of GEOGRAL is to strengthen the bonds of the geometric theory of nonlinear PDEs (and, in particular, integrable systems and equations of Monge-Ampère type) with the geometry of Lagrangian Grassmannians and their submanifolds. In spite of the evident parallelism between these two disciplines, attempts have been rare, yet sophisticated, to cast a bridge between them, and the Applicant himself already gave his own contribution in this direction: he clarified the structure of the space of non-maximal integral elements of the contact planes in jet spaces and studied 3rd order Monge-Ampère equations (which turn out to be of key relevance in topological field theories) through the so-called meta-symplectic structure on the 1st prolongation of a contact manifold.GEOGRAL has a wide applicative scope, as its theoretical results can be tested on equations and variational problems of key importance for Natural Sciences, Technology and Economy. Tailored to the Applicant's scientific profile and designed in continuity with his previous and current research activities, GEOGRAL consists of four research lines:[MOV] Regard Lagrangian Grassmannians as homogeneous spaces and and use Cartan's method of moving frame to classify their submanifolds, as in D. The's work, and characterise the corresponding invariant equations, in continuity with D. Alekseevsky's work.[HYD] Continue the study of certain rational normal curve bundles on Lagrangian Grassmannians, and their bisecant varieties, which are associated with integrable systems of hydrodynamic type, discovered by E. Ferapontov.[HMA] Geometric study of multi-dimensional and higher-order Monge-Ampère equations, initiated by G. Manno and the Applicant.[FBV] Study some examples of Cauchy problems and variational problems with free boundary values by exploiting the geometric structures on the spaces of isotropic flags and non-maximal isotropic elements of a meta-symplectic space, in continuity with the Applicant's own work.
Original text from CORDIS.
Participants
- INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK · WARSZAWACoordinatorPoland
Links
Data: CORDIS, © European Union