**Organizers:**
Andrei Agrachev,
Maciej Dunajski,
Janusz Grabowski,
Bronisław Jakubczyk,
Wojciech Kryński,
Ben Warhurst,

Serhii Koval (Memorial University of Newfoundland): *Point symmetries of the heat equation revisited*

ABSTRACT: We derive a nice representation for point symmetry transformations of the (1+1)-dimensional linear heat equation and properly interpret them. This allows us to prove that the pseudogroup of these transformations has exactly two connected components. That is, the heat equation admits a single independent discrete symmetry, which can be chosen to be alternating the sign of the dependent variable. We introduce the notion of pseudo-discrete elements of a Lie group and show that alternating the sign of the space variable, which was for a long time misinterpreted as a discrete symmetry of the heat equation, is in fact a pseudo-discrete element of its essential point symmetry group. The classification of subalgebras of the essentialLie invariance algebra of the heat equation is enhanced. We also consider the Burgers equation because of its relation to the heat equation and prove that it admits no discrete point symmetries. The developed approach to point-symmetry groups whose elements have components that are linear fractional in some variables can directly be extended to many other linear and nonlinear differential equations.

Stefano Baranzini (Universita degli Studi di Torino): *Spectral properties of the Second Variation*

ABSTRACT: In this talk I will discuss some results concerning spectral properties of the Second Variation of an optimal control problem.

The first topic I will discuss is a formula to compute how the Morse Index changes under different boundary conditions. For instance, this result can be used to produce a certain type of discretization formulae to reduce the Morse Index computation to a nite dimensional problem. It can be specialized to the case of periodic extremals to get iteration formulae. Moreover, it is useful when dealing with Variational problems on graphs since it can be employed to reduce the complexity of the domain.

The second topic I will discuss is a possible definition of the determinant of the Second Variation for an optimal control problem with general smooth boundary conditions.

One of the technical points is a precise understanding of the asymptotic behaviour of the spectrum. It turns out that the second variation is not in general a trace class operator and the standard approach using nite rank approximations does not immediately apply. Instead of working with regularized determinants, we provide a formalism to compute the determinant using the symplectic structure of the problem.

This talk is based on joint works with A. Agrachev and I. Beschastnyi.

Irina Yegorchenko (IMPAN and National Academy of Sciences of Ukraine, Kyiv): *Solving reduction conditions for the Schroedinger equations by contact
transformations*

ABSTRACT: We consider a practical application of the direct method for finding exact solutions of PDE that requires finding solutions of seemingly more complicated overdetermined systems of PDE. We use some ansatzes (most often it is a generalised symmetry ansatz), and then find reduction conditions for the PDE to be reduced using this ansatz. These conditions in most cases are not easy to solve. However, as they are overdetermined systems, we often manage to find their parametric general solutions. I will present an algorithm to find such solutions using successive application of godograph and contact transformations. For the case of the Schroedinger equation with a general nonlinearity that is invariant under the Galilei transformations, we show that this method does not produce anything more than solutions that can be obtained using the classical Lie symmetry reduction. However, in some special cases we can obtain new exact (parametric) solutions.

Maciej Dunajski (University of Cambridge): *Legacy of Eisenhart*

ABSTRACT:-

Maciej Dunajski (University of Cambridge): *Four facets of geometry*

ABSTRACT: The study of geometry is at least 2500 years old, and it is within this field that the concept of mathematical proof - deductive reasoning from a set of axioms - first arose. The lecture will present four areas of geometry: Euclidean, non-Euclidean, projective geometry in Renaissance art, and geometry of space-time inside a black hole.

Artur Sergyeyev (Silesian Univarsity, Opava): *Multidimensional Integrable
Systems: New Insights from Contact Geometry*

ABSTRACT: Contact geometry is well known to play a prominent role in the general geometric theory of partial differential systems. In this talk we show that it also has an important application in the study of partial differential systems that are integrable in the sense of soliton theory. Namely, using a novel kind of Lax pairs involving three-dimensional contact vector fields, we present an explicit effective construction for a large new class of such systems in four independent variables, thus dispelling a long-standing impression that the systems of this sort are scarce. As a byproduct of the construction in question, we also present a first example of a nonisospectral Lax pair for an integrable partial differential system in four independent variables with the property that its Lax operators are algebraic, rather than rational, with respect to the variable spectral parameter.

Vladimir Matveev (Friedrich-Schiller-Universität, Jena): *Applications of Nijenhuis Geometry:
finite-dimensional reductions and integration in quadratures of certain non-diagonalizable systems of hydrodynamic type.
*

ABSTRACT: Nijenhuis Geometry is a recently initiated research program, I will recall its philosophic motivation and fundamental results. New part of my talk is related to applications of these results in the theory of infinite-dimensional integrable systems and includes the following topics

(1) Construction of a large (the freedom is a number of functions of one variable) family of integrable systems of hydrodynamic type. Different from most previously known examples, the corresponding generators are not diagonalizable.

(2) Finite-dimensional reductions of such systems. The commuting functions of the corresponding finite-dimensional integrable systems are quadratic in momenta and can be viewed as a metric and its (commuting) Killing tensors.

(3) Integration of such systems in quadratures.

This is a work in progress in collaboration with Alexey Bolsinov and Andrey Konyaev.

Omid Makhmali (CFT PAN): *On a class of cone structures with an infinitesimal symmetry*

ABSTRACT: We interpret the property of having an infinitesimal symmetry as a variational property in certain geometric structures. This is achieved by establishing a one-to-one correspondence between a class of cone structures with an infinitesimal symmetry and geometric structures arising from certain systems of ODEs that are variational. Such cone structures include conformal pseudo-Riemannian structures and distributions of growth vectors (2,3,5) and (3,6). In this talk we will primarily focus on conformal structures. The correspondence is obtained via symmetry reduction and quasi-contactification. Subsequently, we provide examples of each class of cone structures with more specific properties, such as having a null infinitesimal symmetry, being foliated by null submanifolds, or having reduced holonomy to the appropriate contact parabolic subgroup. As an application, we show that chains in integrable CR structures of hypersurface type are metrizable. This is a joint work with Katja Sagerschnig.

Marcin Zubilewicz (Warsaw University of Technology): *On local invariants of divergence-free webs*

ABSTRACT: The aim of the talk is to highlight some features of the theory of non-singular webs in the geometry of volume-preserving transformations. The local structure of these "divergence-free webs" is far richer than that of their classical counterparts due to the presence of an ambient volume form which interacts with the web. This is reflected in the existence of several local invariants which can be non-trivial even for webs which are parallelizable. They range from curvature invariants derived from the canonical connection of a divergence-free web (which was first defined by S. Tabachnikov in his work on Lagrangian and Legendrian 2-webs) to purely geometric ones inspired by the results of W. Blaschke, G. Bol and G. Thomsen on planar 3-web holonomy. We will construct and characterize these invariants, show how their triviality relates to the triviality of the corresponding divergence-free web, and discuss potential applications of the underlying theory in numerical relativity. Joint work with Wojciech Domitrz.

Benjamin Warhurst (MIM UW): *Energy and Contact maps on the Heisenberg Group*

ABSTRACT: The talk will discuss notions of energy of contact mappings and the properties of critical points. More specifically I will briefly mention the difficulties that arise using Korevaar-Schoen energy and in contrast, what can be said if the energy is the L^2 norm of the horizontal gradient.

Marek Grochowski (UKSW, Warsaw): *On the dimension of the algebras of local infinitesimal
isometries of 3-dimensional special sub-Riemannian manifolds*

ABSTRACT: Suppose that we are given a contact sub-Riemannian manifold (M,H,g) of dimension 3 such that the Reeb vector field is an infinitesimal isometry (such manifolds will be referred to as special). For a point q\in M denote by i(q) the Lie algebra of germs at q of infinitesimal isometries of (M,H,g). I will prove that for a generic point q in M, dim i(q) can only assume the values 1,2,4. Moreover dim i(q) = 4 if and only if the curvature function determined by the canonical sub-Riemannian connection is constant.

Witold Respondek (Institute of Automatic Control, Lodz University of Technology, Poland
and
INSA de Rouen Normandie, France): *Linearization of mechanical control systems*

ABSTRACT: For mechanical control systems we present the problem of linearization that preserves the mechanical structure of the system. We give necessary and sufficient conditions for the mechanical state-space-linearization and mechanical feedback-linearization using geometric tools, like covariant derivatives, symmetric brackets, and the Riemann tensor, that have an immediate mechanical interpretation. In contrast with linearization of general nonlinear systems, conditions for their mechanical counterpart can be given for both, controllable and noncontrollable, cases. We illustrate our results by examples of linearizable mechanical systems. The talk is based on joint research with Marcin Nowicki (Politechnika Poznanska, Poland).

Taras Skrypnyk (Bogolyubov Institute for Theoretical Physics, Kyiv): *Symmetric and asymmetric variable
separation in the Clebsch model: two solutions of the one hundred and fifty year problem*

ABSTRACT: The Clebsch model is one of the few classical examples of the dynamics of rigid bodies in the liquid where the equations of motion are integrable in the sense of Liouville. The explicit solution of the problem of the Hamilton-Jacobi separation of variables for this model is, however, particularly hard and has remained unsolved for more than a century. We have managed to solve this problem in several different ways. In this talk we will present two variable separations for the Clebsch model - symmetric and asymmetric ones. The asymmetric variable separation is very unusual: it is characterized by the quadratures containing differentials defined on two different curves of separation. In the case of symmetric SoV both curves of separation are the same. This case has a bonus: on a zero level set of one of the Casimir functions it yields the famous Weber-Neumann separated coordinates. We also find the explicit reconstruction formulae for the both sets of the constructed separated variables and explicitly write the corresponding Abel-type equations, completely resolving in such a way the long-standing problem of variable separation for the Clebsch model.

Daniel Ballesteros-Chavez (Silesian University of Technology): *On the Weyl problem in de Sitter space and a Weyl-type estimate*

ABSTRACT: The problem of isometric embedding of a positively curved 2-sphere in the Euclidean 3-space was considered by Hermann Weyl in 1916 and it's known as the classical Weyl problem. In this talk we consider (spacelike) isometric embeddings of a metric on the sphere into de Sitter space, with a suitable curvature restriction. We show a bound for the mean curvature H of such spacelike hypersurfaces in terms of the scalar curvature, its Laplacian, the dimension and a scaling factor of the ambient space. The proof uses geometric identities, and the maximum principle for a prescribed symmetric-curvature equation.

This is joint work with Ben Lambert and Wilhelm Klingenberg.

Organizational meeting & short communications:

**10:15 - 11:00 **

Wojciech Kryński (IM PAN): *3D path geometries and the dancing construction*

**11:00 - 11:45 **

Michail Zhitomirski (Technion): *On singular (3,5)-distributions*

Andriy Panasyuk (University of Warmia and Mazury): *Webs, Nijenhuis operators, and heavenly equations*

ABSTRACT: In 1989 Mason and Newman proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields $\lambda),Y(\lambda)$ which are divergence free with respect to some volume form. Earlier (in 1975) Plebański showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Plebański heavenly PDEs. Other PDEs leading to Mason--Newman vector fields are also known in the literature: Husain--Park (1992--94), Konopelchenko--Schief--Szereszewski (2021). In this talk I will discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular I will show how to apply this theory for constructing new "heavenly" PDEs.

Prim Plansangkate (Prince of Songkla University): *Einstein-Weyl structures and dispersionless equations*

ABSTRACT: In this talk, it is shown that, under a symmetry assumption the equations governing a generic anti-self-dual conformal structure in four dimensions can be explicitly reduced to the Manakov-Santini system, which determines a generic three-dimensional Lorentzian Einstein-Weyl structure, using a simple transformation. Then, motivated by the dKP Einstein-Weyl structure, two generalisations of the dKP (dispersionless Kadomtsev-Petviashvili) equation to higher dimensions are discussed. For one generalisation, its (non)integrability is investigated by constructing solutions constant on central quadrics. Another generalisation determines a class of Einstein-Weyl structures in n+2 dimensions, for which an explicit local expression for a subclass is obtained.

Mikołaj Rotkiewicz (MIM UW): *Linearization of supermanifolds*

ABSTRACT: Transformations in Grassmann coordinates on a supermanifold are non-linear, in general. They can be 'linearized' giving rise to a series of k-fold vector bundles Vb_k(M), k=1, 2, 3, 4..., associated with a supermanifold M which can be seen as linear approximations of M (up to order k). On the other hand we construct the cover functor F_k which takes a supermanifold M to a non-negatively Z-graded supermanifold. Both functors, Vb_k and F_k, are related by means of the diagonalization functor studied before in [BGR]. If M is a Lie supergroup then the cover of M is a Z-graded Lie supergroup the structure of which will be discussed. This work was inspired by a cooperation with E. Vishnyakova.

[BGR] A. Bruce, J. Grabowski, M. Rotkiewicz, Polarisation of graded bundles, SIGMA 12 (2016).

Maciej Dunajski (DAMTP, Cambridge): *Causal structures from path geometries*

ABSTRACT:-

Ivan Beschastnyi (CIDMA, Aveiro, Portugal): *Geometry and analysis on almost-Riemannian manifolds*

ABSTRACT: In this talk I will give the definitions and some results concerning the most simple non-equiregular sub-Riemannian manifolds which are called almost-Riemannian property. We will some of the unusual behaviour of their geodesics as well as some properties of the associated Laplace-Beltrami operator. This is a joint work with Ugo Boscain and Eugenio Pozzoli.

Juan Carlos Marrero (La Laguna University): *Some aspects of contact dynamics*

ABSTRACT: In this talk, I will introduce contact Hamiltonian and Lagrangian dynamics and I will discuss some aspects which are related with this topic. Particularly, I will consider the problem of the existence of an invariant measure for contact Hamiltonian dynamics and, if I have time, I will describe contact dynamics in terms of Legendrian submanifolds.

Michał Jóźwikowski (MIM UW): *Degree-two optimality conditions for sub-Riemannian geodesics*

ABSTRACT: In the talk I will present an enhancement of Agrachev-Sarychev theory which gives a set of algebraic equations that each abnormal minimizing sub-Riemannian geodesic should satisfy. The talk will be based on a preprint arXiv:2201.00041.

Alexey Podobryaev (Pereslavl-Zalesskiy,
Ailamazyan Program Systems Institute of RAS): *Homogeneous geodesics in sub-Riemannian geometry*

ABSTRACT: We consider homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. Homogeneous geodesics are the simplest geodesics in some sense. The natural questions are: how many homogeneous geodesics can there be? can all normal geodesics be homogeneous? We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We get conditions for an existence of at least one homogeneous geodesic. We discuss some examples of geodesic orbit sub-Riemannian manifolds (i.e., manifolds such that any geodesic is homogeneous) and prove that Carnot groups of step more than 2 can not be geodesic orbit. We prove that the geodesic flow for geodesic orbit sub-Riemannian manifold is itegrable in non-commutative sense.

Andrew J. Bruce (Swansea University, UK): *A hitchhiker's guide to supermanifolds*

ABSTRACT: Supermanifolds, as first proposed by F. A. Berezin, D. A. Leites (1975), are 'manifold-like' objects in which the coordinates are Z_2 graded commutative, also known as supercommutative. We will present a pedagogical review of the basic theory of supermanifolds as a 'species' of locally superringed space before describing the more familiar approach using local coordinates. We will also examine vector fields on supermanifolds and highlight some of the key novelties as compared with vector fields on manifolds.

Yannick Herfray (Université Libre de Bruxelles): *Gravitational radiations and their Cartan geometry*

ABSTRACT: Asymptotically flat spacetimes form a class of solutions to Einsteins equations which model isolated systems in General Relativity. In particular, gravitational radiations leaking away from these spacetimes are encoded by geometrical data "at infinity". These facts are technically well understood and form the conceptual bedrock for gravitational waves prediction. Despite this, many results typically appear as technical and seemingly coordinate dependent. However, as I will explain, conceptual clarity can be obtained through the use of Cartan geometry methods and Tractor geometry. From this perspective, gravitational characteristic data at null-infinity invariantly correspond to a choice of 3-dimensional Cartan geometry while the presence of radiation corresponds to curvature. The situation is in fact very similar to two dimensional conformal geometry where conformal Cartan geometries are not uniquely associated to a conformal geometry (Möbius structure need to be introduced) and one can draw an enlightening parallel, with holomorphic transformations playing the role of the BMS group. This also gives a precise geometrical meaning to the typical statement that "gravitational radiation is the obstruction to having a distinguished Poincaré group as asymptotic symmetries".

Vladimir Salnikov (CNRS, La Rochelle University): *Dirac dynamics in/for mechanics and numerics*

ABSTRACT: I will start this talk by recalling various instances of Dirac structures in mechanics. Motivated by them I will address the question of variational formulation of dynamics on Dirac structures, and in particular obstructions to it. I will also comment on possible application of these results to design numerical methods preserving Dirac structures, technical and conceptual difficulties that may appear in the process.

Madeleine Jotz Lean (University of Würzburg): *On the correspondence of VB-Courant algebroids with Lie 2-algebroids*

ABSTRACT: This talk begins with an introduction to Courant algebroids and Dirac structures. The direct sum of the tangent space and the cotangent space of a manifold carries the structure of a ``standard Courant algebroid'', which naturally extends the Lie algebroid structure of the tangent space.

Linear connections are useful for describing the tangent spaces of vector bundles, especially their Lie algebroid structure. Similarly, we introduce the notion of ``Dorfman connection'' and explain how the standard Courant algebroid structure over a vector bundle is encoded by a certain class of Dorfman connections. Then we explain how this is in fact a special case of a more general equivalence between Lie 2-algebroids and VB-Courant algebroids (its existence is due to Li-Bland).

The correspondence of Courant algebroids with symplectic Lie 2-algebroids is then explained as a special case of this result.

Wojciech Kryński (IM PAN): *Schwarzian derivative, conformal geodesics and the Euler-Lagrange equations*

ABSTRACT: Conformal geodesics are distinguished curves in the conformal geometry. They generalize the notion of geodesics well known in the Riemannian setting. However, unlike in the Riemannian case, the conformal geodesics are solutions to a third order system of equations which makes the variational approach problematic. I'll show a new approach to the conformal geodesics resulting in their interpretation as critical points of a functional.

Jan Derezinski (University of Warsaw): *From Heun class to Painleve*

ABSTRACT: Heun equations are 2nd order scalar linear equations with 4 regular-singular points, one of them at infinity. Heun class equations are obtained from Heun equations by confluence. Deformed Heun class equations have an additional non-logarithmic (apparent) singularity.

All types of Painleve equations can be derived by the method of isomonodromic deformations from deformed Heun class equations. In my talk will try to describe this derivation in a unified way. In particular, the "symbol" of the Heun equation turns out to be essentially equal to the corresponding "Painleve Hamiltonian".

Sebastiano Golo (University of Jyväskylä): *Horizontal jet spaces on Carnot groups*

ABSTRACT: Jet spaces are fiber bundles endowed with a contact structure. They have been invented to treat high order derivatives on manifolds and to apply Lie and Cartan methods to study PDEs. In addition, jet spaces on Rn have been shown to have a natural structure of Carnot groups. Starting from a Carnot group and working only with horizontal derivatives, we construct a certain type of jet space which we may call a horizontal jet space. We prove that horizontal jet spaces on abelian Carnot groups are the standard jet spaces, and that horizontal jet spaces are themselves Carnot groups. We also prove a Backlund type theorem regarding prolongation of contact mappings of horizontal jet spaces. Other applications will also be presented.

Benjamin Warhurst (University of Warsaw): *Schwarzians on the Heisenberg group*

ABSTRACT: In the conformal mapping theory of the complex plane, the Schwarzian arises as the differential equation that characterises Möbius transformations. In this talk I will discuss attempts to define a Schwarzian on the Heisenberg group and the consequences of rigidity.

Enrico Le Donne (University of Friburg): *Carnot groups and their geodesics*

ABSTRACT: Carnot groups are special metric spaces that are rich in structure: they are those Lie groups equipped with a geodesic distance function that is invariant by left-translation of the group and admit automorphisms that are dilations with respect to the distance. In the talk I will present the basic theory of Carnot groups equipped with Carnot-Carathéodory distances and discuss some results on their length-minimizing curves.

Ian Anderson (Utah State University): *What is the variational bicomplex and why is it useful?*

ABSTRACT: In this talk I will use the simplest problem in the calculus of variations to introduce the main ideas behind the formal mathematical structure of the variational bicomplex. Some basic results on the cohomology of the variational bicomplex to derive the global first variational formula for a general Lagrangian. Other applications will be briefly described.

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.