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

Andrei Agrachev (SISSA): *Control of Diffeomorphisms*

ABSTRACT: Given a control system on a smooth manifold, any admissible control function generates a flow, i.e. a one-parametric family of diffeomorphisms. We give a sufficient condition for the system that guarantees the existence of an arbitrary good uniform approximation of any isotopic to the identity diffeomorphism by an admissible diffeomorphism and provide simple examples of control systems that satisfy this condition. This work is a joint work with A. Sarychev (Florence) motivated by the deep learning of artificial neural networks treated as an interpolation technique.

Michal Jozwikowski (University of Warsaw): *New second-order optimality conditions in sub-Riemannian geometry*

ABSTRACT: A sub-Riemannian geodesic problem is essentially a problem of minimizing a Riemannian distance on a manifold when the velocities are subject to linear constraints. Despite its simplicity, the question whether all sub-Riemannian geodesics are smooth/regular remains open for over 30 years. In the talk I will discuss newly-obtained second-order optimality conditions. In particular, I will prove that the class of minimizing abnormal geodesics splits into two subclasses: 2-normal, which are regular, and 2-abnormal, which require the analysis of order higher than two. Familiar Goh conditions of Agrachev-Sarychev follow as a corollary.

Peter Olver (University of Minnesota): *Fractalization and Quantization in Dispersive Systems*

ABSTRACT: The evolution, through spatially periodic linear dispersion, of rough initial data produces fractal, non-differentiable profiles at irrational times and, for asymptotically polynomial dispersion relations, quantized structures at rational times. Such phenomena have been observed in dispersive wave models, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number theory. Ramifications and recent progress on the analysis, numerics, and extensions to nonlinear wave models, both integrable and non-integrable, will be presented.

Richard Montgomery (UCSC): *Four open questions in the N-body problem*

ABSTRACT: The 333 year old classical N-body problem is alive and well. I begin with a pictorial survey of a few of its solution curves. I then describe four open questions within the problem and recent progress on these questions.

Włodzimierz Jelonek (Cracow University of Technology): *Generalized Calabi type Kahler surfaces*

ABSTRACT: pdf

Gabriel Paternain (University of Cambridge): *The non-Abelian X-ray transform*

ABSTRACT: I will discuss the problem of how to reconstruct a matrix-valued potential from the knowledge of its scattering data along geodesics on a compact non-trapping Riemannian manifold with boundary.

The problem arises in new experiments designed to measure magnetic fields inside materials by shooting them with neutron beams from different directions, like in a CT scan.

Towards the end of the lecture I will focus on the recent solution of the injectivity question on simple surfaces for any matrix Lie group.

Zohreh Ravanpak (IM PAN): *Discrete mechanics on octonions*

ABSTRACT: Discrete Lagrangian mechanics on Lie groups and Lie groupoids has been developed in many papers. Nevertheless, the generalization of the discrete mechanics to non-associative objects is still lacking and my talk is about that generalization. We will see the associativity assumption is not crucial for mechanics and this opens new perspectives.

I will briefly review the discrete Lagrangian mechanics on Lie groups and then I will show how the discrete mechanics can be constructed on non-associative objects, smooth loops. I will explain the process of the formulation of the discrete Lagrangian mechanics on unitary octonions, understood as an inverse loop in the algebra of octonions which as a manifold is the seven-sphere.

Thomas Mettler (Goethe-Universität, Frankfurt): *Deformations of the Veronese embedding and
Finsler 2-spheres of constant curvature*

ABSTRACT: A path geometry on a surface M prescribes a path for each direction in every tangent space. A path geometry may be encoded in terms of a line bundle P on the projectivised tangent bundle P(TM) of M. Besides P, the projectivised tangent bundle is also equipped with the vertical bundle L of the base-point projection P(TM) -> M. Interchanging the role of L and P leads to the notion of duality for path geometries. In my talk I will discuss joint work with Christian Lange (Cologne), where we investigate global aspects of the notion of duality for Finsler 2-spheres of constant curvature and with all geodesics closed. In particular, we construct new examples of such Finsler 2-spheres from suitable deformations of the Veronese embedding.

Jean Petitot (CAMS, Paris): *Why and how sub-Riemannian geometry can
be operational for visual perception*

ABSTRACT: Since the 1990s, new imaging methods have made it possible to visualize the « functional architecture » of the primary areas of the visual cortex. These intracortical very special connectivities explain how local cues can be integrated into geometrically well-structured global percepts. In particular, we can access neural correlates of well known psychophysical phenomena studied since Gestalt theory (illusory contours, etc). We have shown that the first visual area implements the contact structure and the sub-Riemannian geometry of the 1-jet space of plane curves. Illusory contours can then be interpreted as geodesics of the Heisenberg group or of the SE(2) group, which specifies previous models of David Mumford using the theory of elastica. These sub-Riemannian models have many applications, in particular for inpainting algorithms.

Sergei Tabachnikov (PSU): *Flavors of bicycle mathematics*

ABSTRACT: This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems. Here is a sampler of problems that I hope to touch upon:

1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences.

2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence, which defines a discrete time dynamical system on the space of curves. This system is completely integrable and it is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation.

3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam's problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? This problem is also related to the motion of a charge in a magnetic field of a special kind. It turns out that the known solutions are solitons of the planar version of the filament equation.

Paweł Nurowski (CFT PAN): *Mathematics behind the Nobel Prize in Physics 2020*

ABSTRACT: -

Dennis The (UiT The Arctic University of Norway): *Simply-transitive CR real hypersurfaces in C^3*

ABSTRACT: Holomorphically (locally) homogeneous CR real hypersurfaces M^3 in C^2 were classified by Elie Cartan in 1932. A folklore legend tells that an unpublished manuscript of Cartan also treated the next dimension M^5 in C^3 (in conjunction with his study of bounded homogeneous domains), but no paper or electronic document currently circulates.

Over the past 20 years, significant progress has been made on the 5-dimensional classification problem. Recently, only the simply-transitive, Levi non-degenerate case remained. Kossovskiy-Loboda settled the Levi definite case in 2019, and Loboda announced a recent solution to the Levi indefinite case in June 2020, both implementing normal form methods.

In my talk, I will describe joint work with Doubrov and Merker in which we use an independent approach to settle the simply-transitive, Levi non-degenerate classification.

Marek Demiański (University of Warsaw): *Brief history of black holes*

ABSTRACT: Black holes are one of the most fascinating objects in the Universe. In my talk I will discuss history of the concept of black holes from early heuristic ideas to their observational discovery. I will present basic properties of black holes and results of recent observations of black holes with LIGO and VIRGO gravitational wave detectors.

Bronisław Jakubczyk (IM PAN): *Solving geometric PDEs for mathematical Nobel of 2019
(and Fields Medal of 1986)*

ABSTRACT: In 1954 C.N. Yang and R. Mills proposed a model for strong interactions in atomic nuclei. The main role in the classical version of the model was played by certain „physical fields” now called Yang-Mills fields. Mathematically, these were connections on certain vector (or principal) bundles which were supposed to satisfy a set of canonical PDEs (now Yang-Mills equations). The equations were Euler-Lagrange equations for the energy functional defined by the curvature of the connection. Almost three decades later mathematicians started to study solutions to such PDEs and got unexpected results.

We will give a gentle overwiew of results of Karen Uhlenbeck (Abel Prize 2019). These will include: existence and regularity of a connection given its curvature, solutions to Yang-Mills equations and their singularities, regulartity and singularities of harmonic maps. We will briefly mention how Uhlenbeck's results helped S. Donaldson to obtain his revolutionary results in topology of 4-manifolds. The gauge symmetry of the set of solutions to Yang-Mills PDEs was used for defining invariants of differentiable manifolds.

Paweł Nurowski (CFT PAN): *Homogeneous 5-dimensional para-CR structures with nongeneric Levi form*

ABSTRACT: -

Michael Eastwood (University of Adelaide): *Homogeneous hypersurfaces*

ABSTRACT: What's so great about the Archimedean screw? Well, for one thing, it's affine homogeneous as a surface in R^3. The Cayley surface is another classical example. Using a Lie algebraic approach, the affine homogeneous surfaces in R^3 were classified in 1996 by Doubrov, Komrakov, and Rabinovich. I shall describe a geometric approach of Vladimir Ezhov and myself, which provides an alternative classification in R^3 and some further classifications in R^4 and C^4.

Maciej Dunajski (University of Cambridge): *Conformal geodesics, and integrability*

ABSTRACT: I shall discuss the integrability of the conformal geodesic flow (also known as the conformal circle flow) on some gravitational instantons, and provide a first example of a completely integrable conf. geodesic flow on a four-manifold which is not a symmetric space. This is joint work with Paul Tod.

Adam Doliwa (UWM, Olsztyn): *Multidimensional consistency of (discrete) Hirota equation*

ABSTRACT: The notion of multidimensional consistency is an important element of the contemporary theory of integrable systems. It appeared first in the context of discrete/difference equations, but recently it has been applied to some geometrically meaningful PDEs, like the heavenly Plebański equations or the dispersionless Hirota equation. My goal is to present this notion on example of the non-commutative version of the original Hirota discrete KP equation. In particular, I will show how the multidimensional consistency of the system leads to the corresponding solutions of the Zamolodchikov equation (a multidimensional generalization of the Yang-Baxter equation). I will point out the importance of geometric understanding of the non- commutative Hirota system, which helps to construct the quantum version of the Zamolodchikov map and its classical/Poisson reduction. The talk is based on results obtained in collaboration with Sergyey Sergeev and Rinat Kashaev.

Mikołaj Rotkiewicz (MIM UW): *Higher order algebroids and representations (up to homotopy) of Lie algebroids*

ABSTRACT: Higher order algebroids are generalizations of higher order tangent bundles and Lie algebroids at the same time. They appear naturally in the context of geometric mechanics when higher order derivatives and symmetry are in the game. In the approach of M. Jóźwikowski and M. Rotkiewicz they are introduced by means of a vector bundle comorphism of a special kind. Natural examples come from reductions of higher order tangent bundles of groupoids. I will explain the algebraic structure staying behind higher order Lie algebroids, at least in order two. It turned out that they lead to representations up to homotopy of Lie algebroids, a fundamental notion in the theory of algebroids discovered by C. A. Abad and M. Crainic.

Anton Alexeev (University of Geneva): *Large toric charts on coadjoint orbits*

ABSTRACT: A toric chart is a product U x T^n of an open subset U \subset R^n and a torus T^n endowed
with the standard symplectic structure. We consider toric charts on coadjoint orbits of compact Lie groups.
The standard example is given by Gelfand-Zeitlin integrable systems which provide dense toric charts on coadjoint orbits of U(n).

We suggest a new method of constructing large (covering the part of sympletic volume arbitrarily close
to 1) toric charts on coadjoint orbits. Our main tools are the theory of Poisson-Lie groups,
cluster algebra techniques, tropicalization and the Berenstein-Kazhdan potential.

As an application, we prove an exact bound on the Gromov width of coadjoint orbitrs in some new situations.

The talk is based on a joint work with B. Hoffman, J. Lane and Y. Li.

Paweł Nurowski (CFT PAN): *Another PDE system in 5 variables*

ABSTRACT: -

Antoni Pierzchalski (University of Łódź): *Some natural differential operators: the ellipticity and the
ellipticity at the boundary*

ABSTRACT: We will discuss some natural linear differential operators for different geometric structures.

For a Riemannian manifold of dimension n, an interesting family consist of operators of form S*S, where S* is the
operator formally adjoint to S and where S is the the gradient in the sense of Stein and Weiss,
i.e., S is an $O(n)$-irreducible summand of the covariant derivative.

We will discuss the ellipticity and the boundary properties such operators. In particular, we will discuss natural
boundary conditions for the elliptic operators and the ellipticity of these conditions at the boundary.
One of the consequences of such the ellipticity for a given boundary condition is the existence of a basis
for L^2 composed of smooth sections that are the eigenvectors of the operator and satisfy the boundary condition.

We will also discuss the Laplace type operators of form div grad acting in tensor bundles on a Riemannian or symplectic
manifold. Here the operator grad is a natural generalization of the classic gradient operator acting on vector fields.
The negative divergence -div is the operator formally adjoint to grad.
The second order operator –div grad relates to the Lichnerowicz Laplacian which acts on tensors (forms) of any symmetry.
The relation involves the curvature.

We will also mention the problem of restriction of differential operators (so the Stein-Weiss gradients in particular)
to submanifolds or to the leaves of a foliation.

Daniel Ballesteros-Chavez (University of Durham): *A C^2 estimate for the prescribed curvature problem in de Sitter space*

ABSTRACT: We will introduce the setting of the prescribed k-curvature problem for compact spacelike hypersurfaces in de Sitter space. Then we give an interior a priori curvature estimate for the solution of the associated fully non-linear elliptic problem.

Giovanni Moreno (University of Warsaw): *Lagrangian Grassmanians, nonlinear
second order differential equations and chracteristics (part II)*

ABSTRACT: I will introduce the framework for studying nonlinear second order differential equations based on the concept of Lagrangian Grassmanian. Lagrangian Grassmanian is the manifold of all n-dimensional vector subspaces of a 2n-dimensional symplectic space such that symplectic form vanishes on them. In particular I will discuss the case n=2, especially Monge-Ampere equation and its characteristics.

Katja Sagerschnig (CFT PAN): *Parabolic geometries and the exceptional group G_2*

ABSTRACT: I will give an introduction to parabolic geometries: these are Cartan geometries modelled on homogeneous spaces of the form G/P, where G is a semisimple Lie group and P is a parabolic subgroup. As a main example of a parabolic geometry, I will discuss the geometry of (2,3,5) distributions, which is related to the exceptional simple Lie group G=G_2. I will review some history, explain some of the key methods, and discuss recent developments in the field.

Giovanni Moreno (University of Warsaw): *Lagrangian Grassmanians, nonlinear
second order differential equations and chracteristics*

ABSTRACT: I will introduce the framework for studying nonlinear second order differential equations based on the concept of Lagrangian Grassmanian. Lagrangian Grassmanian is the manifold of all n-dimensional vector subspaces of a 2n-dimensional symplectic space such that symplectic form vanishes on them. In particular I will discuss the case n=2, especially Monge-Ampere equation and its characteristics.

Omid Makhmali (Masaryk University, Brno): *On integrability conditions for (2,3,5) distributions*

ABSTRACT: Inspired by the classical Godlberg-Sachs theorem in general relativity, we find conditions that guarantee the existence of a null surface foliation for a (2,3,5) disitribution with respect to the Nurowski conformal structure and study path geometries that can be obtained from such foliation. We give an inverse construction that can be used for a larger class of Cartan geometries.

Andriy Panasyuk (UWM, Olsztyn): *On linear-quadratic Poisson pencils on central extensions of semisimple Lie algebras*

ABSTRACT: In a recent paper Vladimir Sokolov introduces a three-parametric family
of quadratic Poisson structures on gl(3) each of which is compatible
with the canonical linear Poisson bracket. The complete involutive
family of polynomial functions related to these bi-Poisson structures
contains the hamiltonian
of the so-called elliptic Calogero-Moser system, the quantum version
of which is also discussed in the same paper.

We show that there exists a 10-parametric family of quadratic Poisson
structures on gl(3) compatible with the canonical linear Poisson bracket
and containing the Sokolov family. Possibilities of generalization to
other Lie algebras and quantization matters will be also touched in this
talk.

(The joint work with Vsevolod Shevchishin.)

Paweł Nurowski (CFT PAN): *Parabolic geometry of a car*

ABSTRACT: We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type (SO(2; 3) P_12), where P_12 is a Borel parabolic subgroup in SO(2; 3). We discuss relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sphere geometry), the geometry of 3-dimensional conformal Minkowski spacetime, the geometry of 3-rd order ODEs, the projective contact geometry in three dimensions, and the corresponding twistor fibrations. We indicate how all these classical geometries can be interpreted in terms of nonholonomic movements of a car.

Vsevolod Shevchishin (UWM, Olsztyn): *Polynomially superintegrable surface metrics admitting a linear integral*

ABSTRACT: In my talk I give a complete local classification of superintegrable metrics on surfaces admitting two independent polynomial integrals one of which is linear. I also describe the structure of the Poisson algebra of polynomial invariants of such a superintegrable metric: a set of natural generators, polynomial relations between those generators, and expressions of Poisson brackets of the generators as polynomials in the generators.

Zohreh Ravanpak (IM PAN): *Bi-Hamiltonian systems on Poisson-Lie groups and underlying geometric structures*

ABSTRACT: In this talk, I will introduce the notion of a Nijenhuis-Lie bialgebra as a Nijenhuis endomorphism $n: {\frak g} \to {\frak g}$ on a Lie algebra ${\frak g}$ which is compatible, in a suitable sense, with a Lie bialgebra structure on ${\frak g}$. An interesting example (the Euler top) that motivates the previous definition and some results on the algebraic structure of a Nijenhuis-Lie bialgebra will be presented. I will also consider the Nijenhuis-Lie bialgebra in the case that Lie bialgebras are coboundary which turns to the $r$-$n$ structures. The Nijenhuis-Lie bialgebra structures are a starting point to get a deeper insight into the underlying geometric structures of the bi-Hamiltonian systems on Poisson-Lie groups.

Dmitri Alekseevsky (IITP, Moscow): *Non-compact Homogeneou Chern-Einstein Almost Kaehler Manifolds
of a Semisimple Lie Group*

ABSTRACT: We proved that any homogeneous symplectic manifold (M = G/L,omega) of a semisimple group G
with compact stabilizer L admits a unique extension to a homogeneous almost Kaehler manifold (M = G/L,omega,J)
and we classify all invariant almost Kaehler structures on the regular adjoint orbits M=G/T of classical
semisimple group which satisfy the Chern-Einstein equation.

It is a joint work with Fabio Podesta.

Lenka Zalabova (University of South Bohemia): *Notes on conformal circles*

ABSTRACT: We study circles in conformal geometry. We present a method to find equations of conformal circles using tractor calculus and symmetry algebras. We ask when are conformal circles metric geodesics. Finally we give a short discussion of examples. Joint work with M. Eastwood.