Geometry and Differential Equations Seminar

ABSTRACT: This is a joint work with Marcin Zubilewicz. We focus on local curvature invariants associated with bi-Lagrangian structures. We establish several geometric conditions that determine when the canonical connection is flat, building on our previous findings regarding divergence-free webs (Domitrz and Zubilewicz 2023 Anal. Math. Phys. 13 4). Addressing questions raised by Tabachnikov (1993 Differ. Geom. Appl. 3 265–84), we provide complete solutions to two problems: the existence of flat bi-Lagrangian structures within the space of rays induced by a pair of hypersurfaces, and the existence of flat bi-Lagrangian structures induced by tangents to Lagrangian curves in the symplectic plane.

ABSTRACT: Goursat flags (or: \(1\)-flags) became popular in the second half of the 1990s. Moduli of their local classification were found towards the end of 1997. (The first of them appears in length \(8\).) In 1999 Kumpera and Rubin had a 60+ pages long draft advertising [special] MULTI-flags (published in a much shortened form only in 2002). The local classification problem for such flags kept being one of the hottest. In 2010 a finite classification in length \(4\) and a modulus in length \(7\) were produced. In 2020 the infinitesimal symmetries were (if only recursively) described. Formally only for special \(2\)-flags, but easily generalizable to multi-flags. In the winter 2023/24 A. Weber implemented those recurrencies in Mathematica, offering an explicit hold on the infinitesimal symmetries of the special \(2\)-flags. That greatly narrowed the field of search for moduli in length \(5\) and \(6\). Eventually in July 2024 moduli were found in length \(5\), in three singularity classes out of \(41 = (1/2)(3^{5-1} + 1)\) existing in that length \(5\).

ABSTRACT: Contact dynamics has gained popularity as a geometric formalism for modelling certain dynamical systems with dissipation. The category of contact manifolds is equivalent to the category of \(\mathbb{R}^\times\)-principal bundles endowed with \(1\)-homogeneous symplectic forms. Moreover, contact Hamiltonian vector fields are in one-to-one correspondence with symplectic Hamiltonian vector fields of \(1\)-homogeneous functions.

The celebrated Liouville-Arnol’d theorem states that if on a \(2n\)-dimensional symplectic manifold \((M, \omega)\) there are \(n\) independent functions \(f_i,\, i\in \{1, \ldots, n\}\) in involution (i.e., their Hamiltonian vector fields commute with each other), then \(M\) is foliated by Lagrangian submanifolds which are left invariant by the Hamiltonian flows of \(f_i\), and there are action-angle coordinates such that i) they are canonical coordinates for \(\omega\); ii) action coordinates specify the Lagrangian submanifold; iii) angle coordinates have ``constant speed'' with respect to the Hamiltonian dynamics of each \(f_i\), while action coordinates are constants of the motion. Several attempts (by Boyer, Jovanović, and other authors) have been made to generalise this result and extend the theory of integrable systems to the realm of contact manifolds. However, there assumptions are so restrictive that one cannot consider Hamiltonian functions leading to dissipation of energy. Instead, we have proven a Liouville-Arnol’d theorem for homogeneous functions on homogenous symplectic manifolds, and deduced from it a counterpart for contact manifolds.

[1] Leonardo Colombo, Manuel de León, Manuel Lainz, and Asier López-Gordón. Liouville-Arnold theorem for contact Hamiltonian systems. 2024. arXiv:2302.12061 [math.SG].
[2] Asier López-Gordón. The geometry of dissipation. PhD thesis, Universidad Autónoma de Madrid, 2024. arXiv:2409.11947 [math-ph].

ABSTRACT: We will discuss the regularity of stationary harmonic functions in the plane. We will alway assume that their Laplacian is a Radon measure nad the main point of this seminar will be to discuss the geometry of the support. We will show that the support is a union of smooth curves. We will also discuss the stability of such functions.

ABSTRACT: In this talk, I will give a brief overview of geometric integrators - numerical methods based on the idea of preserving some geometric structures in discretized versions of differential equations. I will focus on some recent results in that context, related to Poisson and Dirac geometry. We will also discuss Dirac structures (not) appearing within the framework of port-Hamiltonian systems and eventually of the systems with constraints. From the mathematical point of view, Dirac structures generalize simultaneously symplectic and Poisson structures. Time permitting, I will also mention an approach to the variational formulation of dynamics on Dirac structures.

The talk is based on the following papers:
[1] V.Salnikov, A.Hamdouni, From modelling of systems with constraints to generalized geometry and back to numerics, Z Angew Math Mech., Vol. 99, Issue 6, 2019.
[2] V.Salnikov, A.Hamdouni, D.Loziienko, Generalized and graded geometry for mechanics: a comprehensive introduction, Mathematics and Mechanics of Complex Systems, Vol. 9, No. 1, 2021.
[3] O. Cosserat, C. Laurent-Gengoux, A. Kotov, L. Ryvkin, V. Salnikov, On Dirac structures admitting a variational approach, Mathematics and Mechanics of Complex Systems, 2023.
[4] V. Salnikov, Port-Hamiltonian systems: structure recognition and applications, Programming and Computer Software, Volume 50, 2, 2024.
[5] V.Salnikov, A.Falaize, D.Loziienko. Learning port-Hamiltonian systems - algorithms, Computational Mathematics and Mathematical Physics, 2023.
[6] O.Cosserat, V.Salnikov, C.Laurent-Gengoux, Numerical Methods in Poisson Geometry and their Application to Mechanics, to appear in Mathematics and Mechanics of Solids, 2024.

ABSTRACT: Einstein's equations are not conformally invariant. However, the solutions exhibit surprisingly nice conformal properties (at least in even dimensions and for positive cosmological constant). There is a geometric explanation of this fact using the Fefferman-Graham obstruction tensor and the ambient metric construction. The talk will be devoted to this phenomena and some applications from joint work with A. Bac, M. Broda and J. Lewandowski.

ABSTRACT: In 1963 Stanisław Łojasiewicz proved that for an arbitrary real analytic function \( f \) defined in a neighbourhood of \( 0 \in \mathbb{R}^n \) such that \( f(0) = 0 \) and \( f(x) \geq 0 \), there exists \( \theta \in (0,1/2] \) such that \[ |f(x)|^\theta \leq C \|\nabla f(x)\| \] for \( x \) close to \( 0 \), with \( \theta \) depending on the singularity of \( f \) at \( 0 \) and \( C > 0 \). He used the inequality to prove that any trajectory of the gradient flow \( \dot{x} = -\nabla f(x) \) starting in a neighbourhood of \( 0 \) converges to a point \( x_\infty \) such that \( f(x_\infty) = 0 \) and \( \nabla f(x_\infty) = 0 \). We will briefly overview how in the next 60 years the inequality, and its variant proved by L. Simon, turned out to be powerful tools in studying the asymptotic behavior of solutions to PDEs, problems in geometric analysis involving singularities of solutions and asymptotic cones of Riemannian metrics, studying Ricci flow, heat Yang-Mills flow, and others. Another branch of applications includes convergence analysis of gradient-like algorithms used in various optimization problems met in applications, including stochastic gradient descent in machine learning and (gradient descent) deep learning by back-propagating errors, as pioneered by Geoffrey E. Hinton (Nobel Prize in Physics 2024).

ABSTRACT: We classify the Ricci flat Lorentzian space times with shearfree congruences of null geodesics lifted as \(\mathbb{R}^2\)-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 Kahler 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.

See https://arxiv.org/abs/2405.14760

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.

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.

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.

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).

ABSTRACT: Sasakian manifolds are considered by many as odd-dimensional counterparts of Kahler 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 Kahler isometric flows. Finally, we will present some applications of Sasakian manifolds.

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 r le 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.

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.

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.

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.

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.

ABSTRACT: A general sublaplacian is an operator of the form \(div_H(M \nabla_H f)\) where \(div_H\) is a horizontal divergence, \(M\) is a symmetric positive definite matrix acting on the horizontal bundle, and \(\nabla_H\) 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 \(div(\nabla f)\) 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.

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.

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.

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.

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.

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.

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.

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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.

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.

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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.

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.

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.

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.

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.

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).

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.

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.

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

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 \(X(\lambda),Y(\lambda)\) 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.

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.

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 \(\mathbb{Z}\)-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 \(\mathbb{Z}\)-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).

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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.

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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.

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.

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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.

ABSTRACT: Supermanifolds, as first proposed by F. A. Berezin, D. A. Leites (1975), are 'manifold-like' objects in which the coordinates are \(\mathbb{Z}_2\) 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.

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 (Mobius 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 Poincar group as asymptotic symmetries".

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.

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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.

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.

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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".

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 \(\mathbb{R}^n\) 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.

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

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-Carath odory distances and discuss some results on their length-minimizing curves.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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 \(P(TM) \to M\). 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.

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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.

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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.

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ABSTRACT: Holomorphically (locally) homogeneous CR real hypersurfaces \(M^3\) in \(\mathbb{C}^2\) 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 \(\mathbb{C}^3\) (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.

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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.

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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.

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ABSTRACT: What's so great about the Archimedean screw? Well, for one thing, it's affine homogeneous as a surface in \(\mathbb{R}^3\). The Cayley surface is another classical example. Using a Lie algebraic approach, the affine homogeneous surfaces in \(\mathbb{R}^3\) 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 \(\mathbb{R}^3\) and some further classifications in \(\mathbb{R}^4\) and \(\mathbb{C}^4\).

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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.

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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 Pleba ski 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.

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.

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.

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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.

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.

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.

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.

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.

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.

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.)

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.

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ABSTRACT: In this talk I will explain the notion of the variety of minimal rational tangents (VMRT). VMRT is a fundamental tool in the program of studying the varieties that are covered by rational curves. The latter may be thought of as the closest analogoues to the notion of a line in the familiar Euclidean geometry, playing a similar role as geodesics in Riemannian geometry.

I will focus on the case when the underlying variety is a (complex) contact manifold. More precisely, when the contact manifold is homogeneous with respect to a Lie group G. In this case, the VMRT takes a particularly simple form, known as the sub-adjoint variety of G.

Finally, I will show how to use the sub-adjoint variety of G to obtain G-invariant second-order PDEs.

The review part of this talk is based on the paper "Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems" by J. Buczy ski and the speaker, to appear on Banach Centre Publications. The result about G-invariant PDEs is contained in the paper "Lowest degree invariant second-order PDEs over rational homogeneous contact manifolds" by D. Alekseevky, J. Gutt, G. Manno and the speaker, recently accepted by Communications in Contemporary Mathematics.

ABSTRACT: The puncture repair theorem says that if M_1 and M_2 are compact Riemannian or conformal manifolds of the same dimension, and there exists a conformal map f of a punctured domain U-{p} in M_1 into M_2, then f extends conformally to U. The talk will outline how this theorem can be generalised in the context of quasiconformal mappings in metric measure spaces, bringing to the fore the significance of Loewner conditions. There are also more general results by Balogh and Koskela concerning porous sets which I will outline.

ABSTRACT: Given a system of equations F(x,y)=0, we will prove a local version of IFT on existence of a solution y=\psi(x), without assuming that the rank of D_yF(0,0) is maximal, thus allowing singularities of F. We will also provide conditions which guarantee existence of a global implicit function y=\psi(x), for x and y in compact manifolds.

ABSTRACT: I will discuss a curious projection from a projective three--space to projective plane which takes lines to conics. The range of this map is characterised by Calyey's description of pairs poristic conics inscribed and circumscribed in a triangle. This is an example of a more general twistor construction, when the twistor space fibers holomorphicaly over a projective plane. The resulting twistor correspondence provides a solution to a system of nonlinear equations for an anti-self-dual conformal structure.

ABSTRACT: There is a classical theorem proved by D.B. Malament stating that the class of continuous timelike curves determines the topology of spacetime. The aim of my talk is to generalize this result to a certain class of sub-Lorentzian manifolds, as well as to some control systems and differential inclusions.

ABSTRACT: Using generalized Feix-Kaledin constructuion of quaternionic manifolds we will discuss a relation between quaternionic symmetries of manifolds arising by the construction from c-projective submanifold $S$, and c-projective symmetries of $S$. We will see that any submaximally symmetric quaternionic manifold arises by the construction and that the standard submaximally symmetric quaternionic model arises from the (unique) submaximally symmetric c-projective model. This suggests that the submaximally symmetric quaternionic structure should be also unique. Finally we will discuss the dimension of quaternionic symmetries of the Calabi metric showing that the dimension of the algebra of quaternionic symmetries is not fully determined by the dimension of algebra of c-projective symmetries of the submanifold.

ABSTRACT: It is well-known that the mechanical system represented by the rolling of two Riemannian surfaces without slipping or twisting can be formulated in terms of a rank 2 distribution on a 5-dimensional manifold. In this talk, the notion of rolling is extended to a pair of Finsler surfaces which gives rise to a rank 2 distribution on a 6-dimensional manifold. Special classes of rolling Finsler surfaces will be discussed. Finally, a geometric formulation of rolling for a pair of Cartan geometries of the same type on manifolds of equal dimensions will be given.

ABSTRACT: In the talk I will discuss the results of [Lessinness and Goriely, Nonlinearity 30 (2017)]. To determine if an extremal of a given variational problem is indeed minimal, one needs to study the definiteness of the second variation. In general this is a difficult problem. However, for one-dimensional problems of mechanical type a clever use of the Sturm-Liouville theory allows to prove or exclude minimality from very simple global geometric properties of the extremal.

ABSTRACT: I plan to discuss geometric conditions for integrand f to define lower semicontinuous functional I_f(u). Of our particular interest is tetrahedral convexity condition introduced by Agnieszka Ka amajska in 2003, which is the variant of maximum principle expressed on tetrahedrons, and the new condition which we call tetrahedral polyconvexity. Those problems are strongly connected with open rank-one conjecture posed by Morrey in 1952, known in the multidimensional calculus of variations. The discussion will be based on joint work with Agnieszka Ka amajska.

ABSTRACT: It is well-known that the mechanical system represented by the rolling of two Riemannian surfaces without slipping or twisting can be formulated in terms of a rank 2 distribution on a 5-dimensional manifold. In this talk, the notion of rolling is extended to a pair of Finsler surfaces which gives rise to a rank 2 distribution on a 6-dimensional manifold. Special classes of rolling Finsler surfaces will be discussed. Finally, a geometric formulation of rolling for a pair of Cartan geometries of the same type on manifolds of equal dimensions will be given.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ABSTRACT: I will recall the construction of bihamiltonian structures of the equations mentioned in the title, belonging to B. Khesin and G. Misio ek (2002). This construction consists in the argument shift method on the Virasoro Lie algebra, in frames of which the three equations are distinguished by the choice of the shift point. If time permits, I will discuss also perspectives of generalizing to this case of some results from the finite-dimensional argument shift method.

ABSTRACT: One of many properties of harmonic functions, proved by Gauss in 1840, is the mean value property (MVP). It results from the converse theorem by Koebe, that functions having the MVP at every point with all admissable radii are harmonic. Further results by Volterra and Kelogg, known as One-Radius Theorem, and by Hansen and Nadirashvili showed that in case of a bounded domain it is enough to assume MVP with one radius at each point to assert harmonicity. I will present examples showing that none of the assumptions of these theorems can be dropped. As it occurs from Two-Radius Theorem by Delsarte, there is a significant difference when function has MVP with two radii r_1 and r_2, asserting its harmonicity whenever certain relation between r_1 and r_2 is fulfilled. The Delsarte conjectured (and proved in dimension 3) that in fact this relation is always true, so that MVP on any pair of distinct radii is sufficient. I will present a recent proof of the Delsarte Conjecture in all dimensions and present a counterpart of the conjecture on harmonic manifolds.
The talk is based on joint work with T. Adamowicz.

ABSTRACT: Wykład będzie dotyczył metody minimax na przykładzie klasycznego twierdzenia Ambrozettiego - Rabinowitza. Metody typu minimax wykorzystuje się do badania istnienia punkt w krytycznych funkcjonałów zdefiniowanych na odpowiednich przestrzeniach funkcyjnych. W typowej sytuacji punkty krytyczne odpowiadają rozwiązaniom równań róniczkowych posiadających naturę wariacyjną.

ABSTRACT: Multipeakons are special solutions to the Camassa-Holm equation that are described by a very interesting integrable system. We exploit a Riemannian metric that is associated to the system and construct dissipative prolongations of multipeakons near the singular points of the underlying Hamiltonian system.

ABSTRACT: Hypersurfaces in a Lagrangian Grassmannian give a geometric representation of a class of second order PDE (sometimes called 'Hirota type'). Hence it is worthwhile to study the natural geometric structures they carry, and the associated invariants. This approach had been used by A. D. Smith to classify non-degenerate hydrodynamically integrable hyperbolic Hirota-type PDE in 3 independent variables. I will present some early results of a joint work in progress with G. Manno, G. Moreno and A. D. Smith, extending the underlying geometry to higher dimensions.

Dmitri Alekseevsky (University of Hull): Neurogeometry of vision and conformal geometry of sphere

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13:30 - 14:30

Andre Belotto (Universite de Toulouse III): The Sard conjecture on Martinet surfaces

ABSTRACT: Given a totally nonholonomic distribution of rank two on a three-dimensional manifold M, it is natural to investigate the size of the set of points S(x) that can be reached by singular horizontal paths starting from a same point x in M. In this setting, the Sard conjecture states that S(x) should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. In this seminar, I will present a recent work in collaboration with Ludovic Rifford where we show that the conjecture holds whenever the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.

ABSTRACT: Solutions to a system of PDE's may be viewed as maps between manifolds (instead of real valued) minimizing an energy. Adopting this viewpoint allows one to formulate existence of minimizers in much more generality, even when a PDE approach is not available.
I will focus on minimizing a p-energy among homotopy classes of maps between certain metric types of spaces. In particular I will discuss the notion of homotopy in the (possibly discontinuous) case of Sobolev maps, and the proof of existence of a minimizer in this generality.

ABSTRACT: Effective dynamics of open systems can be described using an anticommutator matrix differential equation. In this talk we present the relations between Lie and Jordan algebras and give the conditions, for which such an equation is reduced to a problem in the corresponding Jordan algebra. The example model we study is an effective model of a three-level atom interacting with an electric field.

ABSTRACT: Multipeakons are important and interesting class of solutions to the Camassa-Holm equation. They correspond to solitons, solutions of the Korteweg-de Vries equations. Multipeakons obey the system of Hamiltonian ODEs. However, derivative of the Hamiltonian posseses discontinuity. I will discuss several aspects related to the dynamics of multipeakons.

ABSTRACT: We look for nontrivial solitary wave solutions of nonlinear Sch dinger equations. Our problem is motivated by nonlinear optics and Bose-Einstein condensates. For instance, in nonlinear optics, a nonlinearity is responsible for nonlinear polarization in a medium and by means of the slowly varying envelope approximation we can study the (approximated) propagation of the electromagnetic field in the medium. In this talk we present recent results and variational methods which allow to find ground and bound states of nonlinear Sch dinger equations. Moreover we discuss how to find the exact propagation of electromagnetic fields in nonlinear media.

ABSTRACT: The name "hydrodynamic integrability" refers to a property which identifies a nontrivial class of (nonlinear) PDEs. A PDE fulfills this property if it possesses "sufficiently many" hydrodynamic reductions. Hydrodynamic reductions are special solutions which can be obtained in a formally analogous way as B. Riemann did in his classical paper "The Propagation of Planar Air Waves of Finite Amplitude" (1860). Since that pioneering work, there has been a plethora of spin-offs, where the method of hydrodynamic reductions has been studied, generalized and successfully used in many applications. However, the geometry behind hydrodynamic integrability has been a mystery until 2010, when there appeared the back-to-back papers "Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian" by E. Ferapontov et al. (IMRN, arXiv) and "Integrable GL(2) geometry and hydrodynamic partial differential equations", by A. D. Smith (Comm. Anal. Geom., arXiv). In this seminar I will review the milestones set by the aforementioned papers, framing them against an appropriate geometric background. Even though I will not announce any new result, I will duly stress a conjecture formulated in Ferapontov paper, which is currently under study by A. D. Smith, G. Manno, J. Gutt and myself. More details about the progress of our work will be given by J. Gutt in a forthcoming seminar within this series.

ABSTRACT: B. Feix and D. Kaledin independently showed that there exists a hyperk hler metric on a neighbourhood of the zero section of the cotangent bundle of a real analytic Kahler manifold. Moreover, they generalized this construction for hypercomplex manifolds, where the hypercomplex structure is constructed on a neighbourhood of the zero section of the tangent bundle of a complex manifold with a real analytic connection with curvature of type (1,1).

In this talk we will discuss a generalization of this construction to quaternionic geometry. Using twistor methods, starting from a 2n-manifold M equipped with a c-projective structure with c-projective curvature of type (1,1) and a line bundle L with a connection with curvature of type (1,1), we will construct a quaternionic structure on a neighbourhood of the zero section of TM\otimes\mathcal{L}, where \mathcal{L} is some unitary line bundle obtained from L. The obtained quaternionic manifolds abmit a compatible S^1 action and we will prove that the quotient in the lowest dimensional case is an asymptotically hyperbolic Einstein- Weyl space with a distingushed gauge. Finally, we will mention some further directions concerning twisted Armstrong cones and Swann bundles.

The presented results are a joint work with D. Calderbank (University of Bath).

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ABSTRACT: I will focus on example of Bianchi surfaces to explain what we understand by integrable discretization of class of surfaces, the class which is described by nonlinear integrable system of differential equations.

ABSTRACT: We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a $2n$-dimensional manifold. In the $\mathbb C$-analytic category this set consists of the Martinet hypersurface $\Sigma _2$, the restriction of the singular symplectic form $\omega$ to $T\Sigma_2$ and the kernel of $\omega^{n-1}$ at the point $p\in \Sigma_2$. In the $\mathbb R$-analytic and smooth categories this set contains one more invariant: the canonical orientation of $\Sigma_2$. We find the conditions to determine the kernel of $\omega^{n-1}$ at $p$ by the other invariants. In dimension $4$ we find sufficient conditions to determine the equivalence class of a singular symplectic form-germ with the structurally smooth Martinet hypersurface by the Martinet hypersurface and the restriction of the singular symplectic form to it. We also study the singular symplectic forms with singular Martinet hypersurfaces. We prove that the equivalence class of such singular symplectic form-germ is determined by the Martinet hypersurface, the canonical orientation of its regular part and the restriction of the singular symplectic form to its regular part if the Martinet hypersurface is a quasi-homogeneous hypersurface with an isolated singularity.

ABSTRACT: I will recall the group of reduced path (in Chen-Humbly-Lyons sense) and its connection with the signature of the path. Then I will discuss a recent article (2016) about a (infinite dimensional) Lie structure of a character group of a graded connected Hopf algebra. Finally, I will show how the reduced path group is embeded in the character Lie group of the shuffle Hopf algebra and discuss some of its propoerties.

ABSTRACT: We study flatness of multi-input control-affine systems. We give a geometric characterization of systems that become static feedback linearizable after an invertible one-fold prolongation of a suitably chosen control. They form a particular class of flat systems. Namely, they are of differential weight $ n + m+1$, where $n$ is the dimension of the state-space and $m$ is the number of controls. Using the notion of Ellie Cartan, they are absolutely equivalent to a trivial system under 1-dimensional prolongation. We propose conditions (verifiable by differentiation and algebraic operations) describing that class and provide a system of PDE's giving all minimal flat outputs.
The talk is based on a joint work with Florentina Nicolau.

ABSTRACT: We will present selected nonholonomic systems appearing in the theory of mobile robots and their control using transverse function method. Appearance of nonholonomic constraints reduces the kinematic freedom in the configuration space so that the number of free for moving dimensions (number of controls) is smaller then the number of configuration variables, even if all configurations are reachable. The transverse function method proposed by Morin and Samson, to be presented at the seminar, anables one to move, approximately, in the forbiden directions. For general systems on Lie groups it proposes a method of control which moves the system, approximately, in the direction of Lie brackets of the vector fields of the system. The method can be interpreted as decoupling control using smooth dynamic fedback.

ABSTRACT: A classical feature of any Riemannian manifold M is that each point admits a neighbourhood over which exists polar-like coordinates, namely normal coordinates. Assuming given a subset X of M which is not submanifold, we can nevertheless equip its smooth part with the restriction of the ambient Riemmannian structure and try to understand the behaviour of geodesics nearby any non smooth point. The most expected occurrence of such situation is when M is an affine or projective space (real or complex) and X is an affine or projective variety with non-empty singular locus.

In a joint work with D. Grieser (Univ. Oldenburg, Germany) we discuss the problem of an exponential-like map at the singular point of a class of isolated surface singularities of an Euclidean space, called cuspidal surface. I will state the trichotomy of this class of surface regarding the existence and the injectivity of an exponential-like mapping at the singular point of this class of surface... and explain a bit if times allows.

ABSTRACT: Various important structure over integrable partial differential equation, such that Lax pair, Lie algebra-valued zero-curvature representations and Gardner's deformations, can be view in the set-up of system of nonlocal variables (or differential coverings). We will discuss interrelations of these structure on example of the Korteweg-de Vries equation.

ABSTRACT: In the talk I will formulate geometric characterization of extremals for the sub-Riemannian geodesic problem. The conditions we get are equivalent to the ones derived by means of the Pontryagin Maximum Principle, yet the derivation is much simpler. The idea is based on a natural concept of homotopy between sub-Riemannian trajectories. If time allows I will speak about second-order optimality conditions.

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ABSTRACT: In previous talks we discussed mean value type properties implying sub-Laplacian-harmonicity. In this talk we consider the converse.

ABSTRACT: I will report on recent progress in classification of complex contact manifolds focusing on the case of dimension 7. This is related to the classification of 12-dimensional quaternion-Kaehler manifolds. The tools we use include representation theory and actions of (complex) reductive groups, minimal rational curves, symplectic geometry.
This is a joint work with Jaros aw Wi niewski.

ABSTRACT: Living tissues, such as stems, leaves and flowers in plants and bones in animals, grow into a great variety of shapes. In some cases, Nature has found ways to control this growth with remarkable accuracy. In this talk I shall discuss some free boundary problems modeling controlled growth, namely
(I) Growth of 1-dimensional curves in \(\mathbb{R}^3\) (plant stems), where stabilization in the vertical direction is achieved by a feedback response to gravity. The presence of obstacles (rocks, branches of other plants) yields additional unilateral constraints. In this case, the evolution can be modeled by a differential inclusion in an infinite dimensional space.
(II) Growth of 2- or 3-dimensional domains, controlled by the concentration of a morphogen, coupled with the minimization of an elastic deformation energy.
Some very recent existence, uniqueness, and stability results will be presented, together with numerical simulations. Several research directions will be discussed.

ABSTRACT: The talk is based on my recent results with T. Mettler and on a paper by B. Kruglikov and E. Ferapontov (arxiv:1607.01966). I will present connections between the \(GL(2)\)-structures, complex geometry and integrable systems.

ABSTRACT: The recent detections of gravitational waves are impressive achievements of experimental physics and another success of the theory of General Relativity. The detections confirm the existence of black holes, show that they may collide and that during the merging process gravitational waves are produced. The existence of gravitational waves was predicted by Albert Einstein in 1916 after linearizing his equations. However, later he changed his mind finding arguments against the existence of nonlinear gravitational waves, which virtually stopped development of the subject until the mid 1950s. The theme was picked up again and studied vigorously by various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson and Andrzej Trautman, where the theoretical obstacles concerning gravitational wave existence were successfully overcome, thus giving the Green Light for experimentalists to start designing detectors, culminating in the recent LIGO/VIRGO discovery. We will tell the story of this theoretical breakthrough.

ABSTRACT: This talk will continue to discuss some elementary aspects of harmonic analysis and sub-elliptic pde on Carnot groups.

ABSTRACT: I will present the construction of families of Lie--Poisson brackets depending on finite or infinite number of parameters and investigate when those brackets are compatible. In this way I will obtain classes of bi-Hamiltonian systems which can be in a natural way interpreted as a deformation of systems known before in rigid body mechanics or continuum mechanics. One of the interesting problems is to find the answer to the questions if these systems are integrable, what is the Lax form of describing equations and how they behave when we contract some parameters.

ABSTRACT: Bellaiche was making his desingularization on the nilpotent approximations' level, but settled for a homogeneous space [of a Lie group] of reasonable dimension. Chitour-Jean-Long have been making their desingularization purely Lie-algebraically, in enormously many dimensions, with mainly applications to the Motion Planning Problem in view. Now their construction is being used by Hakavuori & Le Donne in their tackling of isolated corners in the SR geometry. In the talk some key points in both these approaches will be indicated and discussed.

ABSTRACT: We will discuss a recent paper by Hakavuori and Le Donne on a long-standing problem of regurality of sub-Riemannian geodesics. We will concentrate on a reduction procedure of the general case to the problem on the Carnot groups. Next lecture on this topic will be given by P. Mormul (University of Warsaw) on May 18th.

ABSTRACT: I will discuss a class of dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian \(Gr(3, 5)\), revealing a remarkable correspondence with Einstein-Weyl geometry and the theory of \(GL(2, \mathbb{R})\) structures. Generalisations to higher dimensions will also be discussed. The talk is based on joint work with B Doubrov, B Kruglikov and V Novikov.

ABSTRACT: Consider some class of delay differential equations of neutral type and study asymptotic behaviour of its solutions. The main approach is to interpret delay equations as linear ordinary differential equations in a Hilbert space setting. There will be presented some growth estimations of the norm of the corresponding semigroup of operators and of individual solutions. Moreover the notion of polynomial stability will be presented and some criterion of polynomial stability of neutral systems will be derived in terms of location of the spectrum of the semigroup generator. As an application a control problem will be also considered, that is the asymptotic behaviour of diameters of reachable sets is shown.

ABSTRACT: This talk will discuss some elementary aspects of harmonic analysis and sub-elliptic pde on Carnot groups.

ABSTRACT: The basic model of open quantum system with time-periodic Hamiltonian will be presented. The general structure of appropriate quantum dynamical map and master equation derived in the Markovian regime and based on the Floquet theory will be shown. Finally, some examples of open systems with external periodic driving will be given, including the recently developed theory of Markovian Dynamical Decoupling.

ABSTRACT: We consider a \(GL(2,\mathbb{R})\)-structure on a manifold \(M\) of even dimension and present a construction of a canonical almost-complex structure on a bundle over \(M\). The integrability of the almost-complex structure is characterised in terms of the torsion of a connection on M. We present some applications of the construction to the geometry of \(GL(2,\mathbb{R})\)-structures and the associated twistor spaces.

ABSTRACT: It is well-known that normal extremals in sub-Riemannian geometry are curves which locally minimize the energy functional. However, the known proofs of this fact are computational and the relation of the local optimality with the geometry of the problem remains unclear. In the talk I will provide a new proof of this result, which gives insight into the geometric reasons of local optimality. Also the the relation of the regularity of normal extremals with their optimality become apparent in our approach. The talk is based on a joint work with professor Witold Respondek.

ABSTRACT: I will discuss most important results from the articles: U Boscain, M Caponigro, T Chambrion, M Sigalotti, "A Weak Spectral Condition for the Controllability of the Bilinear Schr dinger Equation with Application to the Control of a Rotating Planar Molecule", and T Chambrion, P Mason, M Sigalotti, U Boscain, "Controllability of the discrete-spectrum Schr dinger equation driven by an external field".

ABSTRACT: The classical isoperimetric inequality in the Euclidean plane \(\mathbb{R}^2\) states that for a simple closed curve \(M\) of the length \(L\), enclosing a region of the area \(A\), one gets \(L^2\geqslant 4\pi A\) and the equality holds if and only if \(M\) is a circle. We will show that if \(M\) is an oval, then \(L^2\geqslant 4\pi A+8\pi |A_{0.5}|\), where \(A_{0.5}\) is an oriented area of the Wigner caustic of \(M\), and the equality holds if and only if \(M\) is a curve of constant width.

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ABSTRACT: Wykład będzie okazją do dyskusji o ogłoszonym przed tygodniem wydarzeniu pierwszej rejestracji fal grawitacyjnych spowodowanym zderzeniem dwu czarnych dziur a także o wcześniejszych wynikach teoretycznych na ten temat.

ABSTRACT: The configuration space of a simple 3-segment snake-like robot carries a natural structure of a principal bundle with non-integrable connection. I will explain its origin, and its application to basic problems of control (after M. Ishikawa). Volunteers get a hands-on experience driving a virtual model of the robot. This will be an entertainment-oriented talk, accessible to undergraduates.

ABSTRACT: I will present a short introduction to the Tannaka theory of tensor categories, which turns out to be useful in computing a differential Galois group of the equations describing a Hamiltonian quantum system. This knowledge allows us to prove (non)integrability of the system.

ABSTRACT: A joint work with Tomasz Adamowicz.

ABSTRACT: It is known that a geometric structure of a Veronese web is described by the Hirota dispersionless nonlinear equation. Seen as Lorentzian hyper-CR Einstein Weyl space the same structure is given by the so-called hyper-CR equation. In this talk we propose a simple geometric procedure of generating different nonlinear PDEs describing Veronese webs and interpolating between two equations mentioned. Backlund transformations between different types will be also discussed. A joint work with Boris Kruglikov.

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ABSTRACT: I will discuss a classification of "ts-invariant" sub-Lorentzian structures on 3 dimensional contact Lie groups. The approach is based on invariants arising form the construction of a normal Cartan connection and the classification of 3 dimensional Lie groups due to Snobl and Winternitz. (Joint work with Alexandr Medvedev and Marek Grochowski.)

ABSTRACT: The BGG resolution, introduced in a seminal paper by Bernstein-Gelfand-Gelfand, and generalised by Lepowsky, is an important device in the representation theory of semi-simple Lie algebras. Its differential-geometric interpretation provides a supply of invariant differential operators between certain natural vector bundles on generalised flag manifolds. The work of Baston and Cap-Slovak-Soucek leads to analogues of these operators for parabolic Cartan geometries, with a very direct construction due to Calderbank-Diemer. I will give an example-based introduction to these ideas.

ABSTRACT: The BGG resolution, introduced in a seminal paper by Bernstein-Gelfand-Gelfand, and generalised by Lepowsky, is an important device in the representation theory of semi-simple Lie algebras. Its differential-geometric interpretation provides a supply of invariant differential operators between certain natural vector bundles on generalised flag manifolds. The work of Baston and Cap-Slovak-Soucek leads to analogues of these operators for parabolic Cartan geometries, with a very direct construction due to Calderbank-Diemer. I will give an example-based introduction to these ideas.

ABSTRACT: It was Gianna Stefani who first started to look for something simpler than canonical exponential coordinates of the 1st or 2nd kind - in her Bierutowice 1984 talk. Then she used that to locally simplify the control systems linear in controls - to define a prototype of the nilpotent approximation (NA in short) of the initial system. Agrachev & Sarychev joined in in 1987, Hermes & Kawski in 1991, Risler in 1992, Bellaiche in 1996. That last contibutor proposed an algorithm, of improving a given set of local coordinates to privileged (or: adapted) ones, that was purely polynomial, avoiding any exponentiation. In short (perhaps too short) Bellaiche successfully debugged Stefani's original approach of 1984. A little polished version of his procedure will be reproduced during the talk. The second part is aimed at showing that the Bellaiche proposal is hardly a fully blown algorithm as in the title above. Since it is general, it is also cumbersome and - potentially - extremely memory-thirsty. It also leads sometimes to illisible visualisations of NA's. (A given NA has, as a rule, a plethora of various visualisations.) In concrete classes of distributions dynamic modifications of `polynomial Bellaiche' are needed that would lead to much simpler visualisations. This is particularly important in the SR geometry when one reduces a local minimization problem to a simpler one showing up in the NA of an SR structure. Two instances of such `much simpler' visualisations will be given.

ABSTRACT: In this second part, I will provide a solid mathematical foundation to the statement that "the simplest nonlinear PDEs of order three (in two independent variables) are of Monge-Ampere type". Basically, I will mimic all the steps, explained in the first part, which allowed to "reconstruct" a classical (non-elliptic) Monge-Ampere equation out of its characteristics. As it will turn out, everything goes rather smoothly, except for the definition of the "third-order analog" of the Lagrangian Grassmannian, which I denote by \(L(2,5)\) and refer to as the "meta-symplectic Grassmannian". I will explain in detail how to define the four-dimensional space \(L(2,5)\), how to frame it in the jet-theoretic framework for nonlinear PDEs, and how to recognize in its hyperplane sections the natural third-order analogues of the Monge-Ampere equations. Finally, I will show how such a perspective on third-order Monge-Ampere equations can help in solving equivalence problems and in finding exact solutions.

ABSTRACT: Multidimensional Monge-Ampere equations are, in a sense, the simplest nonlinear PDEs of order two, and to explain this point of view, I will briefly outline the ideas and the results contained in the paper "Contact geometry of multidimensional Monge-Ampere equations: characteristics, intermediate integrals and solutions" by D. Alekseevsky et al. (Ann. Inst. Fourier, 2012). In particular, I will stress the role of characteristics in the description of these equations: a characteristic is a direction in the manifold of independent variables along which the Cauchy-Kowalevskaya theorem fails in uniqueness, and for (non-elliptic, two-dimensional) Monge-Ampere equations, the knowledge of all the characteristics corresponds to the knowledge of the equation itself. This easy feature, which is usually overlooked, plays a key role here, and it can properly formulated in terms of the geometry of the three-dimensional Lagrangian (or "symplectic") Grassmannian \(L(2,4)\).

ABSTRACT: I will present a deformation-theoretic approach to the problem of classifying multiply-transitive homogeneous models of parabolic geometries determined by distributions (equivalently, of strongly regular distributions whose symbol Tanaka-prolongs to a semisimple Lie algebra), developed recently in collaboration with Ian Anderson. \((2,3,5)\)-distributions will serve as a toy example: we'll reproduce the models from Cartan's 1910 paper, as well as the Doubrov-Govorov one.

ABSTRACT: I will present a deformation-theoretic approach to the problem of classifying multiply-transitive homogeneous models of parabolic geometries determined by distributions (equivalently, of strongly regular distributions whose symbol Tanaka-prolongs to a semisimple Lie algebra), developed recently in collaboration with Ian Anderson. \((2,3,5)\)-distributions will serve as a toy example: we'll reproduce the models from Cartan's 1910 paper, as well as the Doubrov-Govorov one.

ABSTRACT: In the talk I will present an interpretation of the Pontryagin Maximum Principle in the language of contact (instead of symplectic) geometry. I will show its applications to the study of abnormal geodesics in subriemannian geometry. The talk is based on a joint work with prof. Witold Respondek.

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ABSTRACT: We consider local geometry of sub-pseudo-Riemannian (in particular sub-Riemannian and sub-Lorentzian) structures on contact manifolds. We construct fundamental invariants of the structures and show that the structures give rise to Einstein-Weyl geometries in dimension 3, provided that certain additional conditions are satisfied.

ABSTRACT: I will present a geometric interpretation of the integration-by-parts formula. This will lead us to some geometrical constructions (a bundle of higher semiholonomic velocities) and canonical vector bundle morphisms used in our geometrical definition of force and momentum maps. Based on a joint paper with Micha Jóźwikowski.

ABSTRACT: The theory of elliptic systems with critical growth, i.e. such that the nonlinearity is a priori only in L^1, has been one of the most active areas in elliptic PDE's in the past 20 years.Examples of such systems include systems describing harmonic mappings between manifolds, surfaces with prescribed mean curvature and their higher-dimensional generalizations. Uhlenbeck-Riviere decomposition of antysymmetric matrices of differential forms, originating in the theory of Yang-Mills field theory and adapted to the more general PDE setting by Riviere is one of the most useful tools in that field.

ABSTRACT: The Wei-Norman algorithm used for solving a system of nonlinear time-varying ODEs bases on the Magnus expansion and Lie groups and Lie algebras properties. In this talk it will be presented, whether the analogous method may be formulated for Jordan algebras. At the end I will present some physical applications, for instance the Jordan-GNS construction.

ABSTRACT: The talk will discuss applications of Cartan's equivalence method to geometry of differential equations. Examples will describe the approach to the problems of finding zero-curvature representations, recursion operators and Backlund transformations for multi-dimensional PDEs based on Cartan's theory of Lie pseudo-groups.

ABSTRACT: Consider the minimum time optimal control problem for the linear system \(x'=Ax+bu\) with \(A\) and \(b\) satisfying the Kalman rank condition where for simplicity the control is assumed to be single-input, with \(u\in[-1,1]\). Results stating that
(i) the minimum time function \(T\) to reach the origi is Holder with exponent \(1/N\) in the reachable set (which contains a neighborhood of 0);
(ii) the optimal control is unique, bang-bang with an upper bound on the number of switchings;
(iii) the reachable sets at every time are strictly convex;
are classical from the early stages of control theory. Simple examples show that \(T\) is never everywhere differentiable, and even Lipschitz, in any neighborhood of the origin. The talk will be devoted to describing the following results
(1) \(T\) is differentiable in an open set with full measure;
(2) more precisely, \(T\) is analytic out of a closed set \(C\) which is countable union of Lipschitz graphs of \(N-1\) variables;
(3) some information on the exceptional set \(C\) can also be provided: in particular, the set where \(T\) is not Lipschitz is fully described.
Methods of nonsmooth analysis and of geometric measure theory are used.

ABSTRACT: Goursat Monster Tower seems to be an ideal environment for the search of possible non-smooth local SR minimizers. It features, from level 3 on, singular (critical) submanifolds of various codimensions, foliated by - smooth - abnormal curves of the Goursat field of planes living in a given level of the tower. At any moment one can spring out from such submanifold along a vertical curve, smooth again, retaining the abnormality of the concatenated curve. The jump point is an isolated corner on such an extremal. When it is tangential (or: entirely critical in the Montgomery-Zhitomirskii terminology), then the extremal, in the vicinity of that corner, is not an SR geodesic. This extends to all levels in the GMT a 1997 observation made in the level 3, when the corner point has been of the Giaro-Kumpera-Ruiz (1978) original type.

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ABSTRACT: We show that sub-Lorenzian structures of special type on the Heisenberg group can be extended to Einstein-Weyl structures.

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ABSTRACT: In this talk I shall recall classical results of several authors generalizing the Newlander-Nierenberg theorem on integrability of complex structures to \((1,1)\)-tensors with a more general Jordan decomposition. Also, I shall discuss some relations of these results to an approach of P. Nagy to canonical connections of \(3\)-webs.

ABSTRACT: In "Free Lie algebroids and the space of paths" M. Kapranov has introduced a formal algebraic model for the groupoid of paths on a manifold. The construction is related to K.T. Chen's iterated integrals and leads to an interesting interpretation of certain notions of differential geometry. I shall review the article in the context of G. Pietrzkowski's recent talks.

ABSTRACT: In "Free Lie algebroids and the space of paths" M. Kapranov has introduced a formal algebraic model for the groupoid of paths on a manifold. The construction is related to K.T. Chen's iterated integrals and leads to an interesting interpretation of certain notions of differential geometry. I shall review the article in the context of G. Pietrzkowski's recent talks.

ABSTRACT: An absolutely continuous path in \(\mathbb{R}^n\) space can be represented as an element of the tensor algebra generated by \(\mathbb{R}^n\) (equivalently, by a formal power series of \(n\) noncommuting variables). There is a remarkable subgroup of this tensor algebra, which has universal properties. We will discuss some of the problems concerning these topics considered by T. Lyons and coworkers, in particular recent results published in Ann. of Math.

ABSTRACT: An absolutely continuous path in \(\mathbb{R}^n\) space can be represented as an element of the tensor algebra generated by \(\mathbb{R}^n\) (equivalently, by a formal power series of \(n\) noncommuting variables). There is a remarkable subgroup of this tensor algebra, which has universal properties. We will discuss some of the problems concerning these topics considered by T. Lyons and coworkers, in particular recent results published in Ann. of Math.

ABSTRACT: I shall discuss the necessary and sufficient conditions for a Riemannian four-dimensional manifold \((M,g)\) with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel sections of a certain connection on a complex rank-four vector bundle over \(M\). They provide a natural generalisation of the Bach tensor which vanishes identically for anti-self-dual conformal structures. This is joint work with Paul Tod.

ABSTRACT: We will introduce an interesting group having all features of a Lie group (topology, differential structure) which can also be called a noncommutative vector space. It has several universal properties: all finite dimensional manifolds, Lie groups, symplectic manifolds can be constructed as homogeneous manifolds of this group. A realization problem in control theory will be solved using this group.

ABSTRACT: This talk will introduce the affine dimension of a set in Euclidean space. It is a quantity similar to Hausdorff dimension which arises in the theory of convolution operators.

ABSTRACT: We will discuss the classical convexity result for planar subharmonic functions and its generalizations to the setting of elliptic PDEs and systems of PDEs in Euclidean domains.

ABSTRACT: A well known Lusternik theorem says that if \(X,Y\) are Banach spaces, \(W\subset X\) and \(F:W\to Y\) is a \(C^1\) map such that its derivative \(dF(x_0)\) is submersive at an interior point \(x_0\in int(W)\), then \(F(0)\in int(F(W))\).
We will extend this theorem to the cases where:
(a) \(x_0\) is a boundary point of \(W\);
(b) \(dF(x_0)\) is not submersive.
In case (b) we will assume \(F\) of class \(C^2\) and use an additional condition on the Hessian of \(F\).

ABSTRACT: We exploit a correspondence between Kronecker webs and hyper-Hermitian metrics in split signature to derive Plebanski heavenly equation.

ABSTRACT: A Carnot group \(G\) is a nilpotent Lie group with a geometric structure; these arise in modelling sub-elliptic operators, nonholonomic systems, and sub-Riemannian geometry. A coordinate change, that is, a bijective map \(\phi: \Omega \to G\), where \(\Omega\) is an open subset of \(G\), may be described geometrically as contact, quasiconformal or conformal. We show that conformal mappings are affine, except in a few special cases, and that if the group \(G\) is rigid, that is, the space of contact mappings is finite-dimensional, then so are quasiconformal maps. This is joint work with Alessandro Ottazzi.

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ABSTRACT: First I will present a construction of normal forms for general sub-Riemannian metrics. Using these normal forms I will describe nilpotent approximation for step 2, corank 2 metrics. In this latter case I will compute the cut locus and prove that (in general) it does not coincide with the first conjugate locus.

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ABSTRACT: In this talk I will introduce the concept of an N-manifold as refinement of the notion o a supermanifold in which the structure sheaf carries an additional grading, called weight, that takes values in the natural numbers. I will provide several motivating examples which largely come for the theory of jets, before discussing some generalities.

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ABSTRACT: The purpose of this talk is to give a brief introduction to the first order Calculus on metric measure spaces. We discuss various approaches to define gradients and metric counterparts of Sobolev spaces. In particular, Haj asz and Newtonian spaces will be presented and their connections to PDEs on metric measure spaces will be mentioned as well.

ABSTRACT: The purpose of this talk is to give a brief introduction to the first order Calculus on metric measure spaces. We discuss various approaches to define gradients and metric counterparts of Sobolev spaces. In particular, Haj asz and Newtonian spaces will be presented and their connections to PDEs on metric measure spaces will be mentioned as well.

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ABSTRACT: I will give an overview of Cartan's approach to the study of conformal and projective structures on manifolds, and its extensions due to Kobayashi and Ochiai. The main motivation is to provide a gentle introduction to the recent theory of parabolic geometries (Cap, Slovak et al.).

ABSTRACT: I will give an overview of Cartan's approach to the study of conformal and projective structures on manifolds, and its extensions due to Kobayashi and Ochiai. The main motivation is to provide a gentle introduction to the recent theory of parabolic geometries (Cap, Slovak et al.).

ABSTRACT: I will give an overview of Cartan's approach to the study of conformal and projective structures on manifolds, and its extensions due to Kobayashi and Ochiai. The main motivation is to provide a gentle introduction to the recent theory of parabolic geometries (Cap, Slovak et al.).

ABSTRACT: W czasie seminarium postaram si przybliy metod konstrukcji symplektycznych przestrzeni k-symetrycznych typu niezwartego. W przypadku zwartym struktura symplektyczna jest indukowana poprzez niezmiennicz form kaehlerowsk , ktrej istnienie - przy grupie izotropii o nietrywialnym centrum - udowodni A. Borel w latach 50' ubieg ego stulecia. Poprzez odpowiedni dualno mi dzy zwartymi i niezwartymi przestrzeniami k-symetrycznymi, poka jak uzyska analogiczny rezultat dla form symplektycznych na niezwartej przestrzeni k-symetrycznej.

ABSTRACT: Rozwijana przez wspomnianych autorow teoria jest spektakularnym zastosowaniem struktur bihamiltonowskich (czyli par zgodnych struktur Poissona) do jakosciowej analizy ukladow calkowalnych w sensie Liouville'a. W pierwszym z dwoch wykladow postaram sie przedstawic zarys teorii osobliwosci ukladow calkowalnych, ktora zawiera m.in. kwestie polozen rownowagi, stabilnosci, i t.p. W drugim opowiem o tym, jak przeklada sie na te kwestie (w istocie algebraiczna) teoria osobliwosci struktur bihamiltonowskich.

ABSTRACT: Rozwijana przez wspomnianych autorow teoria jest spektakularnym zastosowaniem struktur bihamiltonowskich (czyli par zgodnych struktur Poissona) do jakosciowej analizy ukladow calkowalnych w sensie Liouville'a. W pierwszym z dwoch wykladow postaram sie przedstawic zarys teorii osobliwosci ukladow calkowalnych, ktora zawiera m.in. kwestie polozen rownowagi, stabilnosci, i t.p. W drugim opowiem o tym, jak przeklada sie na te kwestie (w istocie algebraiczna) teoria osobliwosci struktur bihamiltonowskich.

ABSTRACT: We construct point invariants of ordinary differential equations and generalise Cartan's invariants in the case of order two and three. If the invariants vanish then the solution space of an equation is equipped with a paraconformal structure, an adapted connection and two-parameter family of totally geodesic hypersurfaces.