## Periodic Reporting for period 1 - GEOGRAL (Geometry of Grassmannian Lagrangian manifolds and their submanifolds, with applications to nonlinear partial differential equations of physical interest)

Reporting period: 2015-09-01 to 2017-08-31

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.

"The output of the project consists of 6 major deliverables - research papers accepted by internationally recognised journals. A few minor deliverables (e.g. preprints, unfinished collaborations, published off-topic papers, etc.) were also produced.

- MAJOR DELIVERABLE 1. “Lowest degree invariant 2nd order PDEs over rational homogeneous contact manifolds” - an original research paper written in cooperation with D. Alekseevsky, G. Manno and J. Gutt, and accepted 29.09.2017 by the journal Communication in Contemporary Mathematics (2016 impact factor: 1.191) and submitted 02.02.2017. Description: In this paper the authors have found a modern way to solve an old problem: finding all nonlinear PDEs admitting a given group of symmetries. The main toolbox is the theory of representation of simple Lie algebras. The main idea is to regard second-order PDEs as hypersurfaces in the Lagrangian Grassmannian bundle over a contact manifold.

- MINOR DELIVERABLE 1. ""Contact manifolds, Lagrangian Grassmannians and PDEs"" - a review paper written in cooperation with Olimjon Eshkobilov, Gianni Manno and Katja Sagerschnig, submitted 30.08.2017 to the journal Complex Manifolds (Mathematical Citation Quotient (MCQ) 2016: 0.67).

- MAJOR DELIVERABLE 2. ""An introduction to completely exceptional 2nd order scalar PDEs"" - a review paper, accepted by the Banach Centre Publications (2013-2016 Polish ranking: 14 points). Description: This paper provides a concise but comprehensive introduction to a remarkable class of nonlinear second-order PDEs, which P. Lax described as “nonlinear PDEs whose solutions display a linear behaviour”. In particular, it casts a bridge between the original heuristic definition and a more geometric one, recently found by the author and his collaborator (see MAJOR DELIVERABLE 4 below).

- MINOR DELIVERABLE 2. ""On a Geometric Framework for Lagrangian Supermechanics"" - an original research paper written in cooperation with A. Bruce and K. Grabowska, accepted 29.04.2017 by the Journal of Geometric Mechanics (2016 Impact Factor: 0.857).

- MINOR DELIVERABLE 3. ""Generalised Weingarten Hypersurfaces"" - a joint project with G. Manno, J Gutt and D. Alekseevsky.

- MINOR DELIVERABLE 4. “Classification of curves in the G_2-homogeneous quadric” - a joint project with J. Buczyński and D. The.

- MAJOR DELIVERABLE 3. ""Geometry of the free–sliding Bernoulli beam"" - an original research paper written in cooperation with M. Stypa, accepted (12.12.2016) by Communications in Mathematics (Mathematical Citation Quotient (MCQ) 2016: 0.28) Description: In this case the authors derive the equation governing the shape of a beam, whose endpoints are free to slide along a prescribed contour. The main tool is the theory of free boundary values variational problems.

- MAJOR DELIVERABLE 4. ""Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution"" - an original research paper written in cooperation with J. Gutt and G. Manno, accepted 27.04.2016 by Journal of Geometry and Physics (5-Year Impact Factor:0.845). Description: The paper provides a new geometric characterisation of a special class of second-order nonlinear PDEs, originally described by P. Lax (see MAJOR DELIVERABLE 2 above). The main tools are conformal geometry and the BGG resolution.

- MAJOR DELIVERABLE 5. ""Meta-Symplectic Geometry of 3rd Order Monge–Ampere Equations and their Characteristics"" - an original research paper written in cooperation with G. Manno, appeared 26.03.2016 on SYMMETRY, INTEGRABILITY and GEOMETRY: METHODS and APPLICATIONS (Five-Year Impact Factor: 0.929). Description: The author provide a description of third-order Monge-Ampere equations by exploiting the notion of a hyperplane section of the meta-symplectic Lagrangian Grassmannian. This method allows to easily distinguish non-equivalent types of such equations.

- MAJOR DELIVERABLE 6. ""Workshop on Geometry of Lagrangian Grassmannians and nonlinear PDEs"" - an international meeting (see below) on the topics of the Researcher’s ongoing projects.

05—09.09.2016 work package. Managing the ""Workshop on Geometry of Lagrangian Grassmannians and nonlinear PDEs”, together with J. Gutt and G. Manno, at the Host Institution."

- MAJOR DELIVERABLE 1. “Lowest degree invariant 2nd order PDEs over rational homogeneous contact manifolds” - an original research paper written in cooperation with D. Alekseevsky, G. Manno and J. Gutt, and accepted 29.09.2017 by the journal Communication in Contemporary Mathematics (2016 impact factor: 1.191) and submitted 02.02.2017. Description: In this paper the authors have found a modern way to solve an old problem: finding all nonlinear PDEs admitting a given group of symmetries. The main toolbox is the theory of representation of simple Lie algebras. The main idea is to regard second-order PDEs as hypersurfaces in the Lagrangian Grassmannian bundle over a contact manifold.

- MINOR DELIVERABLE 1. ""Contact manifolds, Lagrangian Grassmannians and PDEs"" - a review paper written in cooperation with Olimjon Eshkobilov, Gianni Manno and Katja Sagerschnig, submitted 30.08.2017 to the journal Complex Manifolds (Mathematical Citation Quotient (MCQ) 2016: 0.67).

- MAJOR DELIVERABLE 2. ""An introduction to completely exceptional 2nd order scalar PDEs"" - a review paper, accepted by the Banach Centre Publications (2013-2016 Polish ranking: 14 points). Description: This paper provides a concise but comprehensive introduction to a remarkable class of nonlinear second-order PDEs, which P. Lax described as “nonlinear PDEs whose solutions display a linear behaviour”. In particular, it casts a bridge between the original heuristic definition and a more geometric one, recently found by the author and his collaborator (see MAJOR DELIVERABLE 4 below).

- MINOR DELIVERABLE 2. ""On a Geometric Framework for Lagrangian Supermechanics"" - an original research paper written in cooperation with A. Bruce and K. Grabowska, accepted 29.04.2017 by the Journal of Geometric Mechanics (2016 Impact Factor: 0.857).

- MINOR DELIVERABLE 3. ""Generalised Weingarten Hypersurfaces"" - a joint project with G. Manno, J Gutt and D. Alekseevsky.

- MINOR DELIVERABLE 4. “Classification of curves in the G_2-homogeneous quadric” - a joint project with J. Buczyński and D. The.

- MAJOR DELIVERABLE 3. ""Geometry of the free–sliding Bernoulli beam"" - an original research paper written in cooperation with M. Stypa, accepted (12.12.2016) by Communications in Mathematics (Mathematical Citation Quotient (MCQ) 2016: 0.28) Description: In this case the authors derive the equation governing the shape of a beam, whose endpoints are free to slide along a prescribed contour. The main tool is the theory of free boundary values variational problems.

- MAJOR DELIVERABLE 4. ""Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution"" - an original research paper written in cooperation with J. Gutt and G. Manno, accepted 27.04.2016 by Journal of Geometry and Physics (5-Year Impact Factor:0.845). Description: The paper provides a new geometric characterisation of a special class of second-order nonlinear PDEs, originally described by P. Lax (see MAJOR DELIVERABLE 2 above). The main tools are conformal geometry and the BGG resolution.

- MAJOR DELIVERABLE 5. ""Meta-Symplectic Geometry of 3rd Order Monge–Ampere Equations and their Characteristics"" - an original research paper written in cooperation with G. Manno, appeared 26.03.2016 on SYMMETRY, INTEGRABILITY and GEOMETRY: METHODS and APPLICATIONS (Five-Year Impact Factor: 0.929). Description: The author provide a description of third-order Monge-Ampere equations by exploiting the notion of a hyperplane section of the meta-symplectic Lagrangian Grassmannian. This method allows to easily distinguish non-equivalent types of such equations.

- MAJOR DELIVERABLE 6. ""Workshop on Geometry of Lagrangian Grassmannians and nonlinear PDEs"" - an international meeting (see below) on the topics of the Researcher’s ongoing projects.

05—09.09.2016 work package. Managing the ""Workshop on Geometry of Lagrangian Grassmannians and nonlinear PDEs”, together with J. Gutt and G. Manno, at the Host Institution."

The outputs of the project have been accepted by important journals, such as:

1) Communication in Contemporary Mathematics (2016 impact factor: 1.191);

2) SIGMA (5-Year impact factor: 0.929)

3) Journal of Geometric Mechanics (2016 impact factor: 0.857);

4) Journal of Geometry and Physics (5-year impact factor: 0.845).

The editorials boards of these journals consist of the best experts in the field, on a global scale. Hence, acceptance means that the submitted paper represents a solid and ascertained step ahead with respect to the state of the art. A final remark. In a recently appeared preprint (25.07.2017) titled “Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property”, by E.V. Ferapontov, B. Kruglikov and V. Novikov (https://arxiv.org/abs/1707.08070) the authors acknowledge the paper “Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution” produced by the Researcher during the fellowship (see section “WORK PERFORMED” above). The authors - among whose there is the author of the conjecture at the heart of the whole fellowship - managed to prove the conjecture in a particular case (n=4), by using computer-algebra based methods and, as such, lacking a transparent geometric interpretation. The partial failure of the best experts in the world in solving the conjecture is an unmistakable evidence of its importance. And the fact that the one of the partial results obtained during the fellowship has been already exploited in such a remarkable attempt, indicates that the Researcher has indeed given his contribution in making a significant step ahead.

1) Communication in Contemporary Mathematics (2016 impact factor: 1.191);

2) SIGMA (5-Year impact factor: 0.929)

3) Journal of Geometric Mechanics (2016 impact factor: 0.857);

4) Journal of Geometry and Physics (5-year impact factor: 0.845).

The editorials boards of these journals consist of the best experts in the field, on a global scale. Hence, acceptance means that the submitted paper represents a solid and ascertained step ahead with respect to the state of the art. A final remark. In a recently appeared preprint (25.07.2017) titled “Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property”, by E.V. Ferapontov, B. Kruglikov and V. Novikov (https://arxiv.org/abs/1707.08070) the authors acknowledge the paper “Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution” produced by the Researcher during the fellowship (see section “WORK PERFORMED” above). The authors - among whose there is the author of the conjecture at the heart of the whole fellowship - managed to prove the conjecture in a particular case (n=4), by using computer-algebra based methods and, as such, lacking a transparent geometric interpretation. The partial failure of the best experts in the world in solving the conjecture is an unmistakable evidence of its importance. And the fact that the one of the partial results obtained during the fellowship has been already exploited in such a remarkable attempt, indicates that the Researcher has indeed given his contribution in making a significant step ahead.