**Organizers:** Bronisław Jakubczyk, Paweł Nurowski, Wojciech Kryński

Maciej Dunajski (University of Cambridge): *TBA*

ABSTRACT: -

Aleksandra Borówka (Jagiellonian University, Kraków): *TBA*

ABSTRACT: -

Omid Makhmali (IM PAN): *TBA*

ABSTRACT: -

Helene Frankowska (CNRS and Sorbonne Université): *Integral and Pointwise Second-order
Necessary Conditions in Deterministic Control Problems*

ABSTRACT: The first order necessary optimality conditions in optimal control are fairly well understood and were extended to nonsmooth, infinite dimensional and stochastic systems. This is still not the case of the second order conditions, where usually very strong assumptions are imposed on optimal controls.

In this talk I will first discuss the second-order optimality conditions in the integral form.

In the difference with the main approaches of the existing literature, the second order tangents and the second order linearization of control systems will be used to derive the second-order necessary conditions. This framework allows to replace the control system at hand by a differential inclusion with convex right-hand side. Such a relaxation of control system, and the differential calculus of derivatives of convex set-valued maps lead to fairly general statements. When the end point constraints are absent, the pointwise second order conditions will be stated : the second order maximum principle, the Goh and the Jacobson type necessary optimality conditions for general control systems (similar results in the presence of end point constraints are still under investigation).

The talk is intended to be introductory and elements of calculus of set-valued maps will be discussed at the very beginning.

Bronisław Jakubczyk (IM PAN): *Division of differential forms: Koszul complex, Saito's theorem
and Cartan's lemma with singularities*

ABSTRACT: Given two differential forms \alpha and \beta on a manifold M, it is often useful to know if \alpha one divides \beta (locally or globally). We will first answer the the question in the case when \alpha is a 1-form having singularities. The local problem is related to exactness of a Koszul complex. The global version uses H. Cartan Theorems A and B. Another question related to the above is a global version of E. Cartan Lemma, where the differential forms have singularities. We will show that it can be solved using an algebrac Saito's theorem.

Paweł Nurowski (CFT PAN): *Kerr's Theorem*

ABSTRACT: -

Wojciech Kryński (IM PAN): *(3,5,6)-distributions, bi-Hamiltonian systems and contact structures on 5-dimensional manifolds.*

ABSTRACT: I'll discuss geometry of the (3,5,6)-distributions, which are very interesting, non-generic, rank-3 distributions on 6-dimensional manifolds. The class of distributions naturally arise in the context of special bi-Hamiltonian systems and in the context of certain second order systems of PDEs. I'll also show how the distributions are connected to the contact geometry in dimension 5.

Paweł Nurowski (CFT PAN): *On optical structures in space-time physics*

ABSTRACT: I will elaborate on notions, motivations and results which were briefly mentioned by A Trautman in his talk at IMPAN on 22nd November 2017.

Henrik Winther (University of Tromso): *Submaximally Symmetric Quaternionic Structures*

ABSTRACT: The symmetry dimension of an almost quaternionic structure on a manifold is the dimension of its full automorphism algebra. Let the quaternionic dimension $n$ be fixed. The maximal possible symmetry dimension is realized by the quaternionic projective space $mathbb{H} P^n$, which has symmetry group $G=PGl(n+1,mathbb{H})$ of dimension $dim(G)=4(n+1)^2-1$. An almost quaternionic structure is called submaximally symmetric if it has maximal symmetry dimension amongst those with lesser symmetry dimension than the maximal case. We show that for $n>1$, the submaximal symmetry dimension is $4n^2-4n+9$. This is realized both by a quaternionic structure (torsion free) and by an almost quaternionic structure with vanishing Weyl curvature. Joint work with Boris Kruglikov and Lenka Zalabova.

Omid Makhmali (McGill University): *Local aspects of causal structures and related geometries*

ABSTRACT: In this talk the study of causal structures will be motivated, which are defined as a field of tangentially non-degenerate projective hypersurfaces in the projectivized tangent bundle of a manifold. They can be realized as a generalization of conformal pseudo-Riemannian structures. The solution of the local equivalence of causal structures on manifolds of dimension at least four reveals that these geometries are parabolic and the harmonic curvature (which is torsion) is given by the Fubini cubic forms of the null cones and a generalization of the sectional Weyl curvature. Examples of such geometries will be presented. In dimension four the notion of self-duality for indefinite conformal structures will be extended to causal structures via the existence of a 3-parameter family of surfaces whose tangent planes at each point rule the null cone. Finally, it will be shown how certain four dimensional indefinite causal structures give rise to G2/P12 geometries and rolling of Finsler surfaces, following the work of An-Nurowski.

Lenka Zalabova (University of South Bohemia, České Budějovice): *On automorphisms with natural tangent action for Cartan geometries*

ABSTRACT: We study automorphisms with natural tangent action on Cartan and parabolic geometries. We introduce the concept of automorphisms with natural tangent action. We study consequences of the existence of such morphisms for particular cases of morphisms and Cartan/parabolic geometries (affine geometry, partially integrable almost CR structures)

Aleksandra Borówka (Jagiellonian University, Kraków): *Armstrong cones and generalized Feix--Kaledin construction*

ABSTRACT: One can observe that a maximal totally complex submanifold of a quaternionic manifold is naturally equipped with a real-analytic c-projective structure with type (1,1) Weyl curvature. A Generalized Feix--Kaledin construction provides a way to invert this in a special case, i.e. starting from any real-analytic c-projective 2n-manifold S with type (1,1) Weyl curvature, additionally equipped with a holomorphic line bundle with a compatible connection with type (1,1) curvature we get a twistor space of quaternionic 4n-manifold with quaternionic S^1 action such that S is the fixed point set of the action. Moreover, locally in this way we can obtain a twistor space of any quaternionic $4n$ manifold with S^1 action provided that it has a fixed point set of dimension 2n with no triholomorphic points.

In this talk we will overview the construction and show how it is related to c-pojective and quaternionic projective cones constructions by S. Armstrong (note that in quaternionic case the cone is called Swann bundle). Finally we will discuss the role of the line bundle and investigate its relation with Haydys--Hitchin quaternion-Kahler - hyperkahler correspondence.

Arman Taghavi-Chabert (University of Turin): *Twistor geometry of null foliations*

ABSTRACT: We give a description of local null foliations on an odd-dimensional complex quadric Q in terms of complex submanifolds of its twistor space defined to be the space of all linear subspaces of Q of maximal dimension.

Travis Willse (University of Vienna): *Curved orbit decompositions and the ambient metric construction*

ABSTRACT: Given a geometric structure on encoded as a Cartan geometry on a smooth manifold $M$, the curved orbit decomposition formalism describes how a holonomy reduction of the Cartan connection determines a partition of $M$ along with, on each of the constituent sets, a geometric structure encoded as some "reduced" Cartan geometry. The resulting descriptions can reveal new relationships among the involved types of structure.

A simple but instructive example is an (oriented) projective manifold $(M, p)$, $\dim M \geq 3$, whose normal Cartan connection is equipped with a reduction $H$ of holonomy to the orthogonal group, equivalently, a tractor metric parallel with respect to the normal tractor connection. Such a reduction determines a partition of the original manifold into three "curved orbits": Two are open submanifolds, each equipped with a Einstein metric, which is asymptotically equivalent to hyperbolic space in a way that can be made precise. The third is a separating hypersurface, equipped with a conformal structure $\mathbf{c}$; it can be regarded as a projective infinity and hence a natural compactifying structure for each of those Einstein metrics.

One can pose a natural Dirichlet problem for this situation: Given a conformal structure $(M_0, \mathbf{c})$, find a collar equipped with a projective structure and holonomy reduction for which the hypersurface geometry is $(M_0, \mathbf{c})$ itself. The solution turns out to be equivalent to the classical Fefferman-Graham ambient construction.

Applications of these ideas include new results in projective geometry, special Riemannian geometries, and exceptional pseudo-Riemannian holonomy.

Christoph Harrach (University of Vienna): *Poisson transforms for differential forms adapted to homogeneous
parabolic geometries*

ABSTRACT: We present a construction of Poisson transforms between differential forms on homogeneous parabolic geometries and differential forms on Riemannian symmetric spaces tailored to the exterior calculus. Moreover, we show how their existence and compatibility with natural differential operators can be reduced to invariant computations in finite dimensional representations of reductive Lie groups.

Shin-Young Kim (Masaryk University, Brno): *Geometric structures modeled on some horospherical varieties*

ABSTRACT: To prove Hwang-Mok's deformation rigidity problems modeled on projective complex parabolic manifolds, we studied geometric structures arising from varieties of minimal rational tangents. To generalize these rigidity results to quasihomogeneous complex manifolds, we study a smooth projective horospherical variety of Picard number one and their geometric structures. Using Cartan geometry, we prove that a geometric structure modeled on a smooth projective horospherical variety of Picard number one is locally equivalent to the standard geometric structure when the geometric structure is defined on a Fano manifold of Picard number one. In this seminar, we also briefly introduce the origin of this specific problem and horospherical varieties which are completely different from horospheres.

Sean Curry (University of California, San Diego): *Compact CR 3-manifolds, and obstruction flatness*

ABSTRACT: We motivate and consider the problem of determining whether the vanishing of the Fefferman ambient metric obstruction implies local flatness for compact CR 3-manifolds, possibly embedded in C^2.