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

Barbara Opozda (Jagiellonian University, Kraków): *Statistical structures*

ABSTRACT: A statistical structure on a manifold $M$ is a pair $(g,\nabla)$, where $g$ is a metric tensor field and $\nabla$ is a torsion-tree connection such that the cubic form $\nabla g$ is symmetric. Some basic information will be provided and some problems, like completeness, realization and rigidity will be discussed.

Katja Sagerschnig (CFT PAN): *Parabolic contactification*

ABSTRACT: In a series of papers Andreas Cap and Tomas Salac introduced and strudied a class of geometric structures that they called parabolic almost conformally symplectic structures. Any such structure determines a unique linear connection on the tangent bundle whose torsion satisfies a certain normalization condition. Via contactification, parabolic conformally symplectic structures can be related to parabolic contact structures with a transversal infinitesimal symmetry (and the authors use this relationship to descend sequences of invariant differential operators). In this talk we will review (some of) these results.

Marek Grochowski (UKSW): *Canonical connection in sub-pseudo-Riemannian geometry*

ABSTRACT: Given a contact sub-pseudo-Riemannian manifold (M,H,g), I develop the theory of connections on the bundle of horizontal frames associated with it and construct a canonical covariant differentiation which is compatible with the metric (H,g).

Asahi Tsuchida (IM PAN): *A generalized front and its generic singularities*

ABSTRACT: As a intrinsic expression of wave fronts, a concept of coherent tangent bundle was introduced by Saji, Umehara and Yamada in 2012. A coherent tangent bundle is a bundle homomorphism from a tangent bundle to a vector bundle of the same rank endowed with an inner product with certain properties. A point on which the rank of the bundle homomorphism drops is called a singular point. In the paper by Saji, Umehara and Yamada, differential geometric invariants of singularities of bundle homomorphisms are defined and investigated. On the other hand, topological properties of singular sets of bundle homomorphisms without metric are studied by them. In this talk, we consider a generalization of coherent tangent bundle by considering distribution instead of tangent bundle.

This talk is based on a joint work with Kentaro Saji.

Mikołaj Rotkiewicz (MIM UW): *On the structure of higher algebroids*

ABSTRACT: In a recent paper with Michał Jóźwikowski we have introduced a concept of a higher algebroid generalizing the notion of an algebroid and a higher tangent bundle. Our ideas based on the description of a (Lie) algebroid as a vector bundle comorphism - a relation of a special kind. In a special case of a Lie algebroid of a Lie groupoid $G$ such a relation is obtained as a natural reduction of the canonical involution $\kappa_G: T T G \to T T G$. In our approach, a higher algebroid is a vector bundle comorphism between certain graded-linear bundles satisfying some natural axioms. An important example is given by the reduction of a natural isomorphism $\kappa_G^k: T^k T G \to T T^k G$. In my talk I will describe the notion of a higher algebroid in terms of some bracket operations and vector bundle morphisms.

Aleksandra Lelito (AGH, Kraków): *Symmetries, exact solutions and nonlocal conservation laws for PDEs*

ABSTRACT: The objective of the talk is to give an overview of my results – obtained under the supervision of Oleg I. Morozov – concerning geometrical structures associated to nonlinear partial differential equations (pdes). On the example of the Gibbons-Tsarev equation it will be showed how to use a Lie group of local symmetries of a pde to find its exact solutions. The procedure is a classical tool in the theory of applications of Lie groups to differential equations. The Khokhlov-Zabolotskaya (KhZ) equation was previously subjected to this procedure. In the talk it will be illustrated on the example of the KhZ equation how the method can still yield new solutions, if coupled with a Miura-type transformation.

A distinguishing feature of integrable pdes is that they admit rich symmetry structures, but this can be often revealed only after examining them in nonlocal setting. The framework of differential coverings is particularly useful in this context. Within this framework, a Lie algebra of nonlocal symmetries of the second heavenly equation will be discussed. Another example of the strength of this framework will be presented in a review of the results concerning nonlocal conservation laws of several pdes, related to each other via Backlund transformations. The presented results formed the core of my Ph.D. thesis.

Omid Makhmali (IM PAN): *Half-flat causal structures and related geometries*

ABSTRACT: Half-flat causal structures are defined as a field of ruled projective surfaces over a manifold satisfying certain integrability condition. We extend conformal notions such as principal null planes and ultra-half-flatness to the causal setting. After showing that the unique submaximal model that does not descend to a conformal structure is Cayley-isotrivially flat, we will focus on Cayley structures and explore several geometries arising from them. Finally we formulate such structures in terms of a dispersionless Lax pair and study the resulting system of PDEs. This work is partly joint with W. Kryński.

Szymon Pliś (Cracow University of Technology): *Monge-Ampere equation on complex and almost complex manifolds*

ABSTRACT: First, I will survey the theory the complex Monge-Ampere equation and applications to Kahler geometry. In the second part of the talk, I will present some recent results about plurisubharmonic function and the Monge-Ampere equation on almost complex manifolds.

Marta Szumańska (MIM UW): Geodesic radius of curvature for horizontal curves in Heisenberg group
(based on work in progress with Katrin Faessler)

ABSTRACT: The intrinsic curvature of an Euclidean C^2 curve in Heisneberg group was introduced by Balogh, Tyson and Vecchi (It was obtained in a limiting process and is based on curvatures on Riemannian spaces approximating the Heisenberg group). For horizontal curves this curvature coincides with Euclidean curvature of its ortogonal projection onto XY-plane.

We define a notion of "global" curvature that can be considered for any horizontal curve (not necessarily C^2). The idea is based on the following fact: the image of the ortogonal projection into XY-plane of any geodesic in Heisenberg group is an arc of a circle. For any two points in Heisenberg group we define a geodesic radius of curvature which is the radius of the circle arc obtained by a the projection from the unique geodesic connecting those two points.

The aim of the talk is to show the similarities between the role played by the intrinsic curvature in Heisenberg group and "normal" curvature, and between the geodesic radius of the curvature and the Menger curvature in Euclidean space.

Michał Jóźwikowski (MIM UW): *A comparison of vakonomic and nonholonomic dynamics
for Chaplygin systems*

ABSTRACT: Given a mechanical system with a linear set of constraints there are two basic methods of generating the equations of motion: the nonholonomic dynamics obtained by means of the Chetaev-d'Alembert's principle and the vakonomic dynamics obtained from the constrained variational principle. It is well-known that these two methods give inconsistent results, and some researchers asked the question when one of the above-mentioned dynamics is a subset of another one. We show a simple method of adressing such a question based on the ideas of W. Tulczyjew. We provide a detailed answer for a relatively big class of non-invariant Chaplygin systems. The work is based on a joint paper with Witold Respondek to appear in J. Geom. Mech.

Arman Taghavi-Chabert (American University of Beirut): *Twisting shearfree congruences of
null geodesics in higher dimensions*

ABSTRACT: On a conformal Lorentzian 4-manifold, there are certain foliations of null geodesics, known as shearfree congruences of null geodesics (SCNG), which are of central importance in the study of solutions of Einstein's equations. It is well-known that their generators must be principal null directions of the Weyl tensor. What is more, their leaf space is endowed with the structure of a CR manifold.

In this talk I will give the integrability condition for the existence of SCNGs in dimension greater than four, and show that remarkably, in even dimension, the connection between SCNGs and (almost) CR structures still subsist under relatively mild curvature conditions on the Weyl tensor.

Finally, one can play a similar game in split signature: under suitable curvature prescriptions, SCNGs induce Lagrange contact structures and projective structures.

Janusz Grabowski (IM PAN): *Remarks on contact geometry*

ABSTRACT: We present an approach to contact (and Jacobi) geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key role is played by homogeneous symplectic (and Poisson) manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL(1;R)-bundle structure on the manifold and not just to a vector field. This allows for working with nontrivial line bundles that drastically simplifies the picture. Contact manifolds of degree 2 and contact analogs of Courant algebroids are studied as well. Based on a joint work with A. J. Bruce and K. Grabowska.

Konrad Lompert (Politechnika Warszawska): *Invariant Nijenhuis tensors and integrable geodesic flows on homogeneous
spaces*

ABSTRACT: We study invariant Nijenhuis (1,1)-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a hamiltonian system of differential equations with the $G$-invariant hamitonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $\Pi_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $\Pi_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of compact Lie groups $G$ for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of $G$ to two subgroups $G=G_1\cdot G_2$, where $G/G_i$ are symmetric spaces, $K=G_1\cap G_2$. (A joint work with Andriy Panasyuk.)

Paweł Goldstein (MIM UW): *Topologically nontrivial counterexamples to Sard's
theorem and approximation of $C^1$ mappings*

ABSTRACT: If a $C^2$ mapping $f$ from an $(n+1)$-sphere to an $n$-sphere is surjective, then its derivative must have rank $n$ on a set of positive measure. This follows easily from Sard's theorem: the set of critical values of $f$ has measure zero in $S^n$, thus the set of regular values is of full measure. Since $C^2$ mappings map sets of measure zero to sets of measure zero, the set of regular points of $f$ in the $(n+1)$-sphere must have positive measure. Sard's theorem does not apply to $C^1$ mappings, though, and one can construct a $C^1$ mapping $f$ from $S^{n+1}$ to $S^n$ with all points of $S^{n+1}$ critical for $f$; the known examples are, however, homotopically trivial. This leads to a natural question, due to Larry Guth: Assume $n\ge 2$ and $f\in C^1(S^{n+1},S^n)$ is not homotopic to a constant map. Can it happen that all the points of $S^{n+1}$ are critical for $f$ (i.e. the rank of the derivative of $f$ is less than $n$ everywhere)?

The cases of $n=2$ and $n=3$ have been solved in the negative (the first by M. Gromov, using estimates on the Hopf invariant, the second by L. Guth, using Steenrod squares). Together with Piotr Hajłasz and Pekka Pankka, we answer the question *in the positive* for all $n>3$, by constructing an explicite example. We also give a much simpler, direct proof of the case $n=3$, using the ideas behind the proof of Freudenthal's theorem.

Recently, Jacek Gałęski conjectured that a $C^1$ mapping from $R^n$ to $R^n$, with rank of the derivative less than $k < n$ everywhere, can be uniformly approximated by a smooth function satisfying the same constraint on the rank of the derivative. We use our construction to disprove this conjecture at least for some ranges of $n$ and $k$.

Paweł Nurowski (CFT PAN): *Hopf fibration 7 times in physics*

ABSTRACT: -

Wojciech Kryński (IM PAN): *Invariants and isometries of contact sub-Riemannian structures*

ABSTRACT: I will consider contact distributions endowed with sub-Riemannian (or sub-Lorentzian) metrics. I'll discuss results on sub-Riemannian isometries of the structures and present a simple construction of a canonical connection associated to the structures. The talk is based on a joint work with Marek Grochowski.

Ben Warhurst (University of Warsaw): *A canonical connection in Subriemannian contact geometry*

ABSTRACT: -

Omid Makhmali (IM PAN): *Causal structures from a microlocal viewpoint*

ABSTRACT: In this talk, a causal structure will be defined as a field of tangentially nondegenerate projective hypersurfaces over a manifold. Using Cartan's method, we will solve the local equivalence problem of causal structures and give a geometric interpretation of their fundamental invariants. We will mostly focus on special classes of causal geometries in dimension four, referred to as half-flat and locally isotrivial, and study several twistorial constructions arising from them.

Javier de Lucas Araujo (University of Warsaw): *Poisson-Hopf algebra deformations of a class of Hamiltonian systems*

ABSTRACT: This talk is devoted to the use the theory of deformation of Hopf-algebras to construct Hamiltonian systems on a symplectic manifold and to study their constants of the motion, multi-dimensional generalisations, and physical applications.

First, I will survey the theory of deformation of Hopf algebras by introducing co-algebras, bi-algebras, antipode mappings, Hopf and Poisson-Hopf algebras, the dual principle, and the deformation of Hopf algebras. I will detail some classical examples of Hopf algebras: the universal enveloping algebra and their associated quantum groups, or the Konstant-Kirillov-Souriau Poisson algebra and its quantum deformations.

In the second part of the talk, I will use representations of Poisson-Hopf algebras to construct Hamiltonian systems on a symplectic manifold. The representation of a universal enveloping algebra will give rise to a certain Hamiltonian system, a so-called Lie--Hamilton system, whereas its deformation will lead to a one-parametric deformation of the Lie--Hamilton system. The centers of Hopf algebras and their so-called antipodes will give rise to constants of motion of the Lie--Hamilton system and its deformations; the coalgebra structure will lead to multi-dimensional generalisations of the Lie--Hamilton system. As a final example, I will deform a t-dependent frequency Smorodinsky--Winternitz oscillator to obtain and to analyse a t-dependent frequency oscillator with a mass depending on the position and a Rosochatius-Winternitz potential term.

Jun-Muk Hwang (Korea Institute for Advanced Study): *Cone structures arising from varieties of minimal rational tangents
*

ABSTRACT: Varieties of minimal rational tangents are differential geometric structures arising from the algebraic geometry of uniruled projective manifolds. They are special cases of cone structures with conic connections. We give an overview of the subject, emphasizing the interaction of differential geometric methods and algebraic geometric methods.

Paweł Nurowski (CFT PAN): *Kerr's theorem revisited*

ABSTRACT: There is an abundance of congruences of null geodesics without shear in a conformally flat spacetime. In this talk I will try to describe how to determine if two given ones are locally nonequivalent.