Vector Distributions and Related Geometries


List of abstracts:


Speaker: Ian Anderson

Title: Non-rigid parabolic geometries of Monge type. Part I.

Abstract: In this talk, I will discuss recent work with Zhaohu Nie and Pawel Nurowski, which began with a desire to generalize Cartan's solution of the equivalence problem for the (2, 3, 5) geometry to other parabolic geometries. The first part of the talk will describe some computer experiments which led us to the notion of parabolic geometries of Monge type. The classification of non-rigid parabolic Monge geometries will be given in the second talk by Nie. For each classical simple Lie algebra, there is a particularly nice 3-gradation which defines a non-rigid parabolic geometry of Monge type. The flat models for these geometries are realized by particularly simple under-determined ODE systems. The direct calculations of the symmetries of these systems will be discussed and related to invariant subspaces of quadratic forms. The structure equations for these 3-gradations and their harmonic curvatures will be explicitly given.


Speaker: Gil Bor

Title: The dancing metric, rolling of the projective plane on its dual and G2 symmetry.

Abstract: The dancing metric is a pseudo-riemannian metric of signature (2,2) on the space of  non-incident pairs  (point, line) in the real projective plane. The null curves are given by the "dancing condition": at each moment, the point is required to move towards a point on the line, about which the line is turning. This metric establishes  a nice dictionary between classical projective geometry (incidence, cross ratio,. . .) and pseudo-riemannian geometry (null curves and geodesics, parallel transport, curvature. . . ). There is also an unexpected bonus: the recent twistor construction of An-Nurowski reveals a G2-symmetry, hidden deep in classical projective geometry. To uncover this symmetry, one needs to refine the dancing condition by a finer (2nd order) condition, analogous to the "no-twist" condition which is added to the "no-slip" condition of rolling riemannian surfaces.
This is joint work with Pawel Nurowski and Luis Hernandez Lamoneda.


Speaker: Robert Bryant

Title (conference talk): Existence and non-existence of canonical connections.

Abstract: Elie Cartan developed his `method of equivalence' in order to study the invariants of G-structures on manifolds. In many of the classical cases, his method amounts to defining a canonical connection (possibly of higher order) for such G-structures that are of finite type, i.e., when G satisfies the condition that the infinitesimal automorphisms of the flat G-structure constitute a finite-dimensional Lie algebra. However, this is not always possible, and one must sometimes resort to Cartan's original (and more general) notion of reduction to an e-structure on a higher dimensional manifold.  In this talk, I will discuss some examples of this phenomenon and illustrate how one can still work with these so-called `canonical pseudo-connections' in the more general case.

Title (colloquium): The geometry of periodic equi-areal sequences.

Abstract: A sequence of functions f_i on a surface S, with integer indices i, is said to be equi-areal (or sometimes, equi-Poisson) if it satisfies the relations df_(i-1) ^ df_i = df_i ^ df_(i+1)  (not = 0) for all i.  In other words, each successive pair (f_i,f_(i+1)) are local coordinates on S that induce the same area form on S, independently of i.

One says that f is n-periodic if f_i = f_(i+n) for all i. The n-periodic equi-areal sequences for low values of n turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras.

In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such periodic sequences and their surprising relationships with differential geometry, cluster algebras, and the theory of over determined differential equations.


Speaker: David Calderbank

Title: Higher codimension CR structures, Levi-Kaehler reduction and toric geometry

Abstract: CR structures in codimension one play an increasingly important role in differential geometry, deeply intertwined with Kaehler geometry. In this talk, based on joint work in progress with V. Apostolov, P. Gauduchon and E. Legendre, I discuss the relation between CR structures in higher codimension and Kaehler geometry, through a process called "Levi-Kaehler reduction". I focus in particular on the toric case, where Levi-Kaehler reduction provides a new way to construct distinguished metrics on toric varieties. When the Delzant polytope is a product of simplices, explicit quotients of products of spheres are obtained, generalizing Bryant's Bochner-flat metrics on weighted projective spaces.


Speaker: Emma Carberry

Title: Toroidal Soap Bubbles: Constant Mean Curvature Tori in S^3 and R^3.

Abstract: Constant mean curvature (CMC) tori in S^3, R^3 or H^3 are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes and whether tori can be deformed. I will address these questions for the spherical and Euclidean cases, using Whitham deformations.


Speaker: Boris Doubrov

Title: Subalgebras of semisimple Lie algebras: techniques and geometric applications.

Abstract: We review the theory of subalgebras of semisimple complex and real Lie algebras. This  includes various classes of reductive subalgebras (such as regular, irreducible, maximal) as well as non-reductive ones. As applications we outline the classification of all subalgebras in the complex simple Lie algebra G_2 and give the description of conformal structures with symmetry algebra of submaximal dimension in Riemannian and Lorentzian signatures, where the answers differ from other signatures.


Speaker: Maciej Dunajski

Title: On the quadratic invariant of binary sextics.

Abstract: Classical invariant theory was formulated and developed by Cayley, Salmon, Sylvester and others in the second half of the 19th century. Some of the problems left over from that time still remain open. One class of such problems has to do with finding the interpretation of the vanishing of invariants and covariants.
In this talk I shall provide a geometric characterisation of binary sextics with vanishing quadratic invariant. The problem is motivated by twistor approach to G_2 geometry, but its solution appears to be interesting in its own right. This is joint work with Roger Penrose.


Speaker: Mike Eastwood

Title: Differential complexes for the Grushin distributions.

Abstract: The Grushin distributions are degenerate in the sense of not having constant rank. Nevertheless, one can still ask about the geometry they define and for associated differential complexes replacing the de Rham complex. The first and second Grushin distributions are symmetry reductions of certain parabolic geometries (joint work with Der-Chen Chang and Ovidiu Calin). Joint work with Jan Slovak and Vladimir Soucek realises the higher Grushin distributions as symmetry reductions of certain non-parabolic geometries (recently introduced by Boris Doubrov and Alexandr Medvedev). This talk will introduce and discuss the Grushin distributions from this viewpoint.


Speaker: Marek Grochowski

Title: Basic facts about sub-Lorentzian geometry.

Abstract: A sub-Lorentzian structure (or a metric) on a manifold is a couple (H,g) made up of a smooth bracket generating distribution H and a smooth Lorentzian metric g on H. By a sub-Lorentzian manifold we mean a smooth manifold equipped with a sub-Lorentzian metric. I will present basic notions and facts concerning the geometry of sub-Lorentzian manifolds.


Speaker: Matthias Hammerl

Title: A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors.

Abstract: I will discuss a particular Fefferman-type construction which is based on the inclusion $SL(n+1)->Spin(n+1,n+1)$ and associates a split-signature $(n,n)$ conformal structure to a given $n$-dimensional projective structure. While for $n>2$ the induced Cartan connection is only normal in the flat case, it is shown that the induced normal conformal connection still admits a parallel pure tractor spinor, and in particular defines a canonical pure twistor spinor. This is an alternative approach to a construction by Dunajski-Tod. The talk is based on joint work with Josef Silhan, Vojtech Zadnik and Arman Taghavi-Chabert  from Masaryk University Brno and Katja Sagerschnig from ANU Canberra.


Speaker: Boris Kruglikov

Title: Symmetry bounds for filtered geometric structures.

Abstract: A filtered geometric structure is related to a nonholonomic distribution on a manifold and a possible reduction of the generalized frame bundle. We first discuss the bounds on the symmetry dimensions and when they are achieved. More interesting are the submaximal symmetry dimensions, and we consider the gaps between them and the maximal possible dimensions. We illustrate the phenomenon by numerous examples from parabolic geometries (joint work with Dennis The) and also from some non-parabolic geometries (including the joint work with Henrik Winther).


Speaker: Benjamin McKay

Title: Complex analytic vector distributions on compact Kaehler manifolds.

Abstract: Some observations on curvature of vector distributions due to Jean-Pierre Demailly, connected to recent work of Campana and Peternell in algebraic geometry.


Speaker: Thomas Mettler

Title: Conformal connections on projective surfaces

Abstract: I shall discuss the problem of finding a conformal connection defining a prescribed projective structure on a surface. It turns out that locally many such connections exist. This is in contrast to the global situation, where one can show that on a compact oriented surface whose genus exceeds one, there is at most one conformal connection in a given projective class.


Speaker: Felipe Monroy-Perez

Title: Hyperelliptic signals as natural controls for nonholonomic motion planning

Abstract: We address the general problem of approximating optimally non-admissible motions of a kinematic system with nonholonomic constraints, by means of admissible trajectories. Since the problem falls into the general subriemannian geometric setting, it is natural to consider optimality in the sense of approximating by means of subriemannian geodesics.

We consider systems modeled by a subriemannian Goursat structure, a particular case being the well known system of a car with trailers, along with the associated parallel parking problem.

In contrast with the "typical" trigonometric signals, we show that the more natural optimal motions are related with closed hyperelliptic plane curves for which the number of loops is determined by the highest possible order of the Lie brackets.


Speaker: Tohru Morimoto

Title: Klein - Cartan programme for differential equations

Abstract: After the thoughts of Klein and Cartan in understanding various different geometries, we ask in turn what are "the atoms" of differential equations and how the differential equations are created from the atoms.

We propose a programme to study this question from the viewpoint of nilpotent geometry and analysis, surveying our joint works with Doubrov and Machida, and discussing several problems arising there.


Speaker: Zhaohu Nie

Title: Non-rigid parabolic geometries of Monge type. Part II classification.

Abstract: Parabolic geometries of Monge type are defined by special gradings of simple Lie algebras, namely, gradings with the property that their -1 component contains a nonzero co-dimension 1 abelian subspace whose bracket with its complement is non-degenerate. We completely classify the simple Lie algebras with such gradings in terms of elementary properties of the defining set of simple roots. We then characterize those parabolic geometries of Monge type which are non-rigid in the sense that, apart from the flat models, they have nonzero harmonic curvatures in positive weights, which will also be listed.


Speaker: Katharina Neusser

Title: Strongly essential Killing fields of |1|-graded parabolic geometries.

Abstract: We present some new techniques to study infinitesimal symmetries of parabolic geometries admitting a zero of higher order. We will see how these techniques can be used to obtain restrictions on the curvature of geometries that admit such symmetries. We demonstrate this in the case of almost Grassmannian structure of type (2,n). This talk is based on a joint work in progress with Karin Melnick.


Speaker: Antoni Pierzchalski

Tittle: Conjugate submersions

Abstract: Two submersions φ and Ψ of a Riemannian manifold are called to be (p,q)-conjugate, 1/p+1/q=1, if , at each point of M, the kernels of their differentials are orthogonal and the norms of the Jacobi matrices (with respect to orthonormal bases) satisfy: |Jφ|^p= |Jψ|^q when φ and Ψ are real functions and p=q=2 the submersions reduce to the conjugate functions investigated recently eg., by M. Eastwood and P. Nurowski. We are going to present a geometric interpretation of the (p,q)-conjugacy in the language the conformal modulus of the foliations defined by the horizontal (= orthogonal to the kernel) distributions defined by the submersions and to show that a pair of conjugate submersion is a critical point of the functional subordinating to any pair of foliations the product of their moduli. The results base on the joint work with M. Ciska-Niedziałomska.


Speaker: Jean-Baptiste Pomet

Titile: On dynamic equivalence of control systems and distributions

Abstract: Dynamic equivalence is the one where transformations between curves are not pointwise but allow the "new" position to depend of the original position, velocity and higher derivatives, and vice-versa. The transformation should be invertible on the set of curves in question (integral curves of a distribution, trajectories of a control system etc.). Dynamic equivalence is called "absolute equivalence" by E. Cartan and also "Lie-Bäcklund" equivalence, in the context of PDEs. Here we will only interested in differential systems with one independent variable.

It is especially difficult to prove that two distributions or control systems are NOT dynamically equivalent for very few invariants are known due to the lack of a bound on the number of derivatives involved.

This talk will present a survey of the few results available and some open questions.


Speaker: Matthew Randall

Title: Generalised Ricci solitons in dimension 2

Abstract: In this talk we describe a class of equations on (pseudo)-Riemannian manifolds that generalise equations like Einstein and the Ricci soliton equations. This is an overdetermined system of PDEs. We specialise to the case of 2 dimensions and provide explicit examples, inspired by the work of Jacek Jezierski. This is joint work in progress with Pawel Nurowski.


Speaker: Witold Respondek

Title: Control systems of minimal differential weight and Cartan absolute equivalence

Abstract: Pfaffian systems that are trivial, in the sense of Cartan absolute equivalence, are those whose corresponding UDE (underdetermined differential equation) can be integrated without integration (via differentiation of suitably choosen functions only). Equivalently, they are control systems that are linearizable via dynamic feedback. We will give a complete geometric characterization of the systems of minimal differential weight, that is, systems linearizable via one-dimensional dynamic feedback. Integrating the corresponding UDE requires thus one more differentiation than the total number of variables. This is a joint research with Florentina Nicolau.


Speaker: Colleen Robles

Title: Degeneration of Hodge structure.

Abstract: I will discuss how one can use Hodge theory, flag varieties and flag domains to address the questions: How can a smooth projective variety degenerate?  How do we access how ``severe'' one degeneration is relative to another?  To what extent can the degenerations be classified?


Speaker: Yuri Sachkov

Title: Subriemannian geometry on rank 2 Carnot groups

Abstract: We study nilpotent left-invariant sub-Riemannian structures with the growth vectors (2,3,4), (2,3,5), and (2,3,5,8).

For the growth vector (2,3,4), i.e., for the left-invariant SR structure on the Engel group, we prove the cut time is equal to the first Maxwell time corresponding to discrete symmetries (reflections) of the exponential mapping. For the growth vector (2,3,5), i.e., for the left-invariant SR structure on the Cartan group, the same fact is a conjecture supported by mathematical and numerical evidence.

For the growth vector (2,3,5,8), we study integrability of the normal Hamiltonian vector field  H. We compute 10 independent integrals of H, of which only 7 are in involution.  After reduction by 4 Casimir functions, the vertical subsystem of H (on the dual to the Lie algebra of the 8-dimensional nilpotent Lie algebra) shows numerically a chaotic dynamics, which leads to a conjecture on non-integrability of H in the Liouville sense.


Speaker: Jan Slovak

Title: Conformally Fedosov Geometry

Abstract: The recent article with Michael Eastwood ( shows that one can combine projective differential geometry with the notion of a Fedosov manifold to obtain what we call conformally Fedosov manifolds. It is remarkable that these geometric structures allow for a tractor calculus very similar to that related to contact parabolic geometries and I will also comment the relation to the more recent article by Andreas Cap and Tomaas Salac on pushing down the Rumin complex from the contact geometries to their conformally symplectic quotients (

Speaker: Hector J. Sussmann

Title: Distributions, sub-Riemannian metrics and minimizers.

Abstract: The talk will present a general introduction to the theory ofminimizers for subriemannian metrics associated with smooth distributions,emphasizing the special phenomena that occur in the real-analytic case.


Speaker: Arman Taghavi-Chabert

Title: Kerr theorems in odd dimensions

Abstract: We show how certain classes of integrable totally null m-plane distributions on the complex conformal (2m+1)-sphere CS^(2m+1) can be realised as complex submanifolds of an auxilliary space, the twistor space of CS^(2m+1). (Joint work with Boris Doubrov)


Speaker: Denis The

Title: Homogeneous integrable Legendrian contact structures in dimension five

Abstract: Given a contact manifold, a splitting of the contact distribution into a direct sum of two Legendrian sub-distributions is called a Legendrian contact structure.  I will describe recent work on classifying (complex) homogeneous integrable such structures in dimension five.  Locally, these are equivalently described as certain compatible overdetermined systems of 2nd order PDE in the plane. In this realization, the fundamental (harmonic) curvature is easy to compute, and many homogeneous structures admit a simple representation. Classifying real forms of the algebraic models obtained yields corresponding lists for real Legendrian contact structures, and definite and indefinite CR structures (in the integrable case).  Many of the resulting CR structures can be realized as tubular hypersurfaces.  This talk is based on work with Sasha Medvedev and Boris Doubrov.


Speaker: Travis Willse

Title: Holography of (2, 3, 5)-distributions and nearly Kahler geometry

Abstract: Cartan's 1910 study of (2, 3, 5)-distributions, that is, 2-plane distributions (M^5, D) satisfying the genericity condition [D, [D, D]] = TM, revealed a intimate connection with the exceptional Lie group G_2. A recent resurgence of interest in this beautiful geometry, fueled partly by its realization as a (distinctive) example of parabolic geometry, has exploited this relationship to expose new features, including Nurowksi's construction of a canonical conformal structure that any D induces on its underlying manifold. Apprehending some of these results from the viewpoint of projective tractor geometry reveals new consequences of the role of G_2 in this setting: (1) (2, 3, 5)-distributions are the natural structures for compactifying strictly nearly Kahler structures (N, g, J) in dimension 6, and (2) (at least for real-analytic D) the compactification construction can be reversed, that is, each oriented (2, 3, 5)-distribution arises as the compactifying structure for a unique (N, g, J), enabling a holographic program for these distributions.

This is joint work with Rod Gover and Roberto Panai.



Speaker: Michail Zhitomirskii

Title: Homogeneous subsets of the tangent bundle.

Abstract: The talk is devoted to the classification with respect to the group of local diffeomorphisms of the symmetry algebras of all possible local homogeneous subsets of TR^n and T_h C^n with n=2,3. Such symmetry algebras is a certain part of all local transitive Lie algebras. They might be finite-dimensional or infinite-dimensional. I will explain why the classification is possible and the main tools. The classification allows to prove that beyond just few cases, which can be effectively described, all local homogeneous subset are induced in a natural way by a subset of an n-dimensional Lie algebra.