Geometry and Differential Equations Seminar

Time and place: IMPAN, lecture room 106 / online, Wednesday, 10:15 AM (CET)

Organizers: Janusz Grabowski, Wojciech Kryński, Ben Warhurst


June 19, 2024

Gerd Schmalz (University of New England, Australia): Einstein manifolds with optical geometries of Kerr type

ABSTRACT: We classify the Ricci flat Lorentzian space times with shearfree congruences of null geodesics lifted as R^2-bundles from Riemann surfaces in a special way. We use an ansatz that is motivated by the Kerr and TaubNUT solutions. We obtain two series of solutions related to Kähler Riemann surfaces with positive or negative Gaussian curvature. The positive curvature series contains the rotating Kerr black hole solution.

This is joint work with Masoud Ganji, Cristina Giannotti and Andrea Spiro.


June 12, 2024

Prim Plansangkate (Prince of Songkla University, Thailand): Anti-self-dual Equations and Related Differential Equations

ABSTRACT: I will present two results in connection with anti-self-dual equations in four dimensions. Firstly, an affine sphere equation is shown to be a symmetry reduction of the anti-self-dual Yang-Mills equation, which confirms its integrability by twistor method. Secondly, a generalization of the dKP equation which determines a family of Einstein-Weyl structures in an arbitrary dimension will be discussed. The dKP equation itself is integrable, and can be realised as a reduction of the anti-self-dual conformal equation. Although, the generalised equation is not integrable in a dimension greater than three, an extended version of the quadric ansatz method will be presented as an attempt to find solutions of the equation.

May 29, 2024

Tymon Frelik (University of Warsaw): The geometry of planar robots and the three-edge snake

ABSTRACT: One of the primary objectives of the GRIEG research project SCREAM: Symmetry, Curvature Reduction, and EquivAlence Methods was to investigate interesting geometric structures (Cartan, contact, (para-)CR, etc.) originating from simple mechanical systems. The simplest of these are the so-called "planar robots." Around a decade ago, Paweł Nurowski and Gil Bor established some general results and formulated the key questions related to these systems. In this talk, I will provide an overview of their geometry and will discuss in further detail one of the more interesting examples: the three-edge snake robot. This talk is based on an ongoing joint work with Paweł Nurowski.

May 15, 2024

Wojciech Kryński (IM PAN): Lewy curves in para-CR geometry

ABSTRACT: I'll introduce a class of curves called Lewy curves in para-CR geometry, following H.Lewy's original definition in CR geometry. I'll show that in dimension 3 the curves are always solutions to a 2nd order system of ODEs, meaning geometrically, they define a path geometry on a manifold. This path geometry uniquely determines a para-CR structure, allowing one to study para-CR structures in terms of naturally associated ordinary differential equations. In higher dimensions, the Lewy curves define a path geometry if and only if the para-CR structure is flat. In general they are described by a system of ODEs of higher order. Finally, I'll present a characterization of path geometries of Lewy curves in the class of general path geometries; I'll also discuss relations between the Lewy curves and chains — another class of curves canonically associated to para-CR and CR structures.

The talk is based on a joint work with O.Makhmali.

April 24, 2024

Michal Jóźwikowski (University of Warsaw): Can we derive the gravitational constant? Two surprising ideas regarding gravity

ABSTRACT: Theories of gravity, first Newtonian one, and later General Relativity, are definitely cornerstones of modern physics. After triumphant verification of General Relativity it may seem pointless to further question our understanding of gravity. However, there are some fundamental issues that we still lack to understand. These include the origin of inertia and the value of the gravitational constant, which was pointed, among the others, by Schroedinger and Dirac. In the talk I will present two preliminary attempts to fill in the gaps in our understanding of gravity by Sciama and by Dicke (but actually originated by Mach and Einstein).

April 10, 2024

Robert Wolak (Jagiellonian University, Kraków): Sasakian manifolds

ABSTRACT: Sasakian manifolds are considered by many as odd-dimensional counterparts of Kähler manifolds. We start with basic definitions and then continue with the fundamental geometric and topological properties of Sasakian manifolds. Some of these properties are obstructions to the existence of a Sasakian structure on a contact or more general odd-dimensional manifold. These obstructions are the fundamental tools in the proofs of the existence or non-existence results for given classes of odd-dimensional manifolds.

Sasakian manifolds can be also investigated as foliated manifolds. Some of the well-known results are, in fact, true for a larger class of foliated manifolds, i.e., transversely Kähler isometric flows. Finally, we will present some applications of Sasakian manifolds.

March 27, 2024

Arman Taghavi-Chabert (Łódź University of Technology): Cauchy-Riemann geometry and Einstein Lorentzian metrics

ABSTRACT: This talk is concerned with two aspects of the interaction between Cauchy-Riemann geometry and Lorentzian conformal geometry. On the one hand, it was realised, notably through the work of Sir Roger Penrose and his associates, and that of the Warsaw group led by Andrzej Trautman, that CR three-manifolds underlie Einstein Lorentzian four-manifolds that admit non-shearing congruences of null geodesics. These foliations play a fundamental rôle in mathematical relativity, and constitute one of the original ingredients in the formulation of twistor theory.

On the other hand, motivated by his investigation of CR chains, Charles Fefferman in 1976 constructed, in a canonical way, a Lorentzian conformal structure on a circle bundle over a given strictly pseudoconvex Cauchy-Riemann (CR) manifolds of hypersurface type.

After reviewing these two independent developments, I will show how these can be related to each other, by presenting modifications of Fefferman’s original construction, where the conformal structure is "perturbed" by some semi-basic one-form, which encodes additional data on the CR three-manifold. Our setup allows us to reinterpret previous works by Lewandowski, Nurowski, Tafel, Hill, and independently, by Mason. Metrics in such a perturbed Fefferman conformal class whose Ricci tensor satisfies certain degeneracy conditions, are only defined off sections of the Fefferman bundle, which may be viewed as "conformal infinity". The prescriptions on the Ricci tensor can then be reduced to differential constraints on the CR three-manifold in terms of a "complex density" and the CR data of the perturbation one-form. One such constraint turns out to arise from a non-linear, or gauged, analogue of a second-order differential operator on densities. A solution thereof provides a criterion for the existence of a CR function and, under certain conditions, for CR embeddability. This talk is partly based on arxiv:2303.07328.

March 13, 2024

Tomasz Cieślak (IM PAN): Triangular systems of wave equations occurring in AFG equations

ABSTRACT: I will review a starting stage of a common project with Wojtek Kamiński from Faculty of Physics of the University of Warsaw related to his approach to Anderson-Fefferman-Graham equation. One of the first steps of our interest is existence and propagation of regularity of the solutions to triangular hyperbolic systems of PDEs appearing in AFG equations.

February 28, 2024

Jarosław Buczyński (IM PAN): Three stories of Riemannian and holomorphic manifolds: actions of several copies of the group of invertible complex numbers, holonomy groups, and distributions

ABSTRACT: As requested by the organisers, the talk is an advanced (more differential geometry oriented) version of the talk I gave as a colloquium in November, but I assume that some of the people have not been there, and will talk from scratch, and the abstract is essentially the same (but people who attended the colloquium will also hear new stuff).

On Wednesday morning you are going to hear a bunch of stories about manifolds, focusing on two main characters: a compact holomorphic manifold and a Riemannian manifold. The talk consists of three seemingly independent parts. In the first part, the main character is going to be a compact holomorphic manifold, and as in every story, there will be some action going on. This time we act with the group of invertible complex numbers, or even better, with several copies of those. The spirit of late Andrzej Białynicki-Birula until this day helps us to understand what is going on. The second part is a tale of holonomies, it begins with "a long time ago,..." and concludes with "... and the last missing piece of this mystery is undiscovered til this day". The main character here is a quaternion-Kahler manifold, but the legacy of Marcel Berger is in the background all the time. In the third part we meet legendary distributions, which are subbundles in the tangent bundle of one of our main characters. Among others, distributions can be foliations, or contact distributions, which like yin and yang live on the opposite sides of the world, yet they strongly interact with one another. Ferdinand Georg Frobenius is supervising this third part. Finally, in the epilogue, all the threads and characters so far connect in an exquisite theorem on classification of low dimensional complex contact manifolds. In any dimension the analogous classification is conjectured by Claude LeBrun and Simon Salamon, while in low dimensions it is proved by Jarosław Wiśniewski, Andrzej Weber, in a joint work with the narrator.

January 10, 2024

Aleksandra Borówka (Jagiellonian University, Kraków): Quaternionic manifolds with rotating circle action

ABSTRACT: B. Feix (and D. Kaledin independently) showed that there exists a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of any real-analytic Kahler manifold. B. Feix provided an explicit construction of its twistor space and showed that any hyperkahler manifold admitting a rotating circle action near its maximal fixed point set arises locally in this way. The construction have been further generalized to hypercomplex manifolds quaternionic manifolds and quaternion-Kahler manifolds. In this talk we will discuss the cases of the construction. Then we will show how to apply it, to obtain a local classification result for quaternionic manifolds with rotating circle action near maximal fixed point set. Finally we will mention connections with c-map.

December 13, 2023

Maciej Dunajski (University of Cambridge): Quasi Einstein Metrics on Surfaces

ABSTRACT: We prove that the intrinsic Riemannian geometry of compact cross-sections of any Einstein extremal horizon must admit a Killing vector field. This extremal horizon is a special case of a quasi-Einstein structure. We shall discuss another global example of such structures corresponding to projective metrizability.

December 06, 2023

Benjamin Warhurst (University of Warsaw): Moduli for sublaplacians on the second Heisenberg group

ABSTRACT: A general sublaplacian is an operator of the form div_H(M grad_H f) where div_H is a horizontal divergence, M is a symmetric positive definite matrix acting on the horizontal bundle, and grad_H is a horizontal gradient. In the Euclidean setting one can always find a change of coordinates that brings such an operator into the standard form div(grad f) using the symmetric square root C of M, however this is not always possible on a stratified group since C must also extend to an automorphism of the Lie algebra of the group. If the group is free then extending C to an automorphism is not a problem and the symmetric square root works. The second Heisenberg group is perhaps the simplest nonfree stratified group. In this case we employ a recently developed theory of horizontal jets to reveal that the classes of contact equivalent sublaplacians are uniquely determined by a positive real parameter.

November 22, 2023

Rouzbeh Mohseni (IM PAN): Twistors spaces of foliated manifolds.

ABSTRACT: Let M be an even-dimensional Riemannian manifold, the twistor space Z(M) is the parametrizing space for compatible almost complex structures on M. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit formulations and proofs of foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation, we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings.

November 8, 2023

Fabio Di Cosmo (Universidad Carlos III, Madrid): On the Categorical Foundations of Information Geometry

ABSTRACT: In this talk, I will review the categorical approach to Information Geometry started in the 70's by Chentsov. Information geometry is a method of exploring the world of information by differential geometry, mainly Riemannian geometry. In this setting, the notion of a statistical model is the departure one and many properties of statistical inference can be interpreted as geometrical properties of the associated manifolds. In particular, a distinguished role in this theory is played by the Fisher-Rao metric tensor, which ubiquitously appears in estimation theory. Chentsov interpreted this metric tensor using a categorical approach: The Fisher-Rao metric tensor is the unique invariant tensor under a family of transformations forming the morphisms of a category. This approach to information theory was also extended to the quantum setting. In this case, however, the Riemannian metric tensors which are monotone with respect to completely positive trace-preserving maps are characterized by an operator-monotone function, and many different metric tensors have been employed to address different quantum problems. In the last part of the talk, I will present a different category, which is called the NCP category, where one can deal at the same time with classical and quantum systems. In this setting, one can consider a generalized version of a statistical model, which is provided by Lie categories embedded into the NCP one. As a first consequence, one can derive an analogous Cramer-Rao bound for estimators of these models in terms of a symmetric form on the algebroid associated with the Lie category.

October 25, 2023

Giovanni Manno (Politecnico di Torino): 2-dimensional metrics admitting infinitesimal projective symmetries

ABSTRACT: A projective symmetry is a vector field whose local flow preserves unparametrized geodesics. We shall give an overview of some methods for classifying and obtaining normal forms of 2-dimensional metrics admitting a projective symmetry. Of such metrics, we shall discuss the integrability of their geodesic flow.

May 24, 2023

Serhii Koval (Memorial University of Newfoundland, Canada): 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.

May 17, 2023

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.

April 05, 2023

Irina Yegorchenko (IM PAN 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.

March 22, 2023

Maciej Dunajski (University of Cambridge): Legacy of Eisenhart


March 21, 2023

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.

March 01, 2023

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.

February 22, 2023

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.

February 08, 2023

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.

January 25, 2023

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.

January 11, 2023

Benjamin Warhurst (University of Warsaw): 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. 

December 14, 2022

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.

November 30, 2022

Witold Respondek (Lódź University of Technology and INSA de Rouen Normandie): 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).

November 09, 2022

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.

October 26, 2022

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.

October 12, 2022

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, Haifa): On singular (3,5)-distributions

June 08, 2022

Andriy Panasyuk (University of Warmia and Mazury, Olsztyn): 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.

May 11, 2022

Prim Plansangkate (Prince of Songkla University, Thailand): 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.

May 04, 2022

Mikołaj Rotkiewicz (University of Warsaw): 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).

April 13, 2022

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


April 06, 2022

Ivan Beschastnyi (CIDMA, Aveiro): 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.

March 23, 2022

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

pdf video

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.

March 09, 2022

Michał Jóźwikowski (University of Warsaw): 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.

February 16, 2022

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.

February 02, 2022

Andrew J. Bruce (Swansea University): 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.

January 19, 2022

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".

January 05, 2022

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.

December 08, 2021

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.

December 01, 2021

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.

November 24, 2021

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".

November 17, 2021

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.

November 10, 2021

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.

October 27, 2021

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.

October 20, 2021

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.

April 07, 2021

Andrei Agrachev (SISSA, Trieste): 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.

March 24, 2021

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.

March 10, 2021

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.

February 24, 2021

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.

February 10, 2021

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



January 13, 2021

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.

December 16, 2020

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.

December 02, 2020

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

pdf video

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.

November 18, 2020

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.

November 04, 2020,

Sergei Tabachnikov (Pennsylvania State University): Flavors of bicycle mathematics

pdf video

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.

October 21, 2020

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

pdf misc. video


October 14, 2020

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

pdf video maple

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.

June 10, 2020

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.

June 03, 2020

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

pdf video

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.

May 27, 2020

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

pdf notes video


May 20, 2020

Michael Eastwood (University of Adelaide): Homogeneous hypersurfaces

pdf references video

screw and shells note

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.

May 13, 2020

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

pdf video

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.

May 06, 2020

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

pdf video

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.

March 11, 2020

Mikołaj Rotkiewicz (University of Warsaw): 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.

January 22, 2020

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.

January 15, 2020

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


January 08, 2020

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.

December 18, 2019

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.

December 11, 2019

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.

December 04, 2019

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.

November 27, 2019

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.

November 13, 2019

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.

November 06, 2019

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.)

October 30, 2019

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.

October 23, 2019

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.

October 09, 2019

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.

October 02, 2019

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.

August 28, 2019

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.