Geometry and Differential Equations Seminar

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

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 Hajłasz and Pekka Pankka, we answer the question *in the positive* for all $n>3$, by constructing an explicite example. We also give a much simpler, direct proof of the case $n=3$, using the ideas behind the proof of Freudenthal's theorem.

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

ABSTRACT: -

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.

ABSTRACT: -

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. Buczyński 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 Kałamajska 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 Kałamajska.

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.

ABSTRACT: -

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. Misiołek (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 punktów krytycznych funkcjonałów zdefiniowanych na odpowiednich przestrzeniach funkcyjnych. W typowej sytuacji punkty krytyczne odpowiadają rozwiązaniom równań różniczkowych 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

ABSTRACT: -

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 Schödinger 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 Schödinger 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 hyperkähler metric on a neighbourhood of the zero section of the cotangent bundle of a real analytic Kähler 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).

ABSTRACT: -

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.

ABSTRACT: -

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 Jarosław Wiśniewski.

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 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: A GL(2)-structure is a natural generalisation of a conformal metric on a manifold. The talk is based on our recent results with T. Mettler and on a paper by B. Kruglikov and E. Ferapontov (arxiv:1607.01966). We shall 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 $GL(2, R)$ 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,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,R)-structures and the associated twistor spaces. This is a report on a joint work with Thomas Mettler.

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 Schrödinger 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 Schrödinger 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.

ABSTRACT: -

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. Bäcklund transformations between different types will be also discussed. A joint work with Boris Kruglikov.

ABSTRACT: -

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 \v{C}ap-Slov\'ak-Sou\v{c}ek 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 \v{C}ap-Slov\'ak-Sou\v{c}ek 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.

ABSTRACT: -

ABSTRACT: -

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 Michał 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 B\"acklund 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.

ABSTRACT: -

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

ABSTRACT: -

ABSTRACT: -

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 R^n space can be represented as an element of the tensor algebra generated by 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 R^n space can be represented as an element of the tensor algebra generated by 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.

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

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.

ABSTRACT: -

ABSTRACT: In this talk I will introduce the concept of an N-manifold as refinement of the notion of 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.

ABSTRACT: -

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, Hajłasz 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, Hajłasz and Newtonian spaces will be presented and their connections to PDEs on metric measure spaces will be mentioned as well.

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

ABSTRACT: -

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 się przybliyć metodę konstrukcji symplektycznych przestrzeni k-symetrycznych typu niezwartego. W przypadku zwartym struktura symplektyczna jest indukowana poprzez niezmienniczą formą kaehlerowską, ktrej istnienie - przy grupie izotropii o nietrywialnym centrum - udowodnił A. Borel w latach 50' ubiegłego stulecia. Poprzez odpowiednią dualność między zwartymi i niezwartymi przestrzeniami k-symetrycznymi, pokażę jak uzyskać 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.