International Conference on Symmetry and Geometric Structures

12.11.2017 - 18.11.2017 | Warsaw


A path geometry in 2D (Gil Bor)

I will describe several interrelated models of a certain SL(2,R)-invariant  path geometry in the plane, already appearing in Arthur Tresse's 1896  list of locally homogenous path geometries. In one model, the paths are Hook's ellipses (sharing a common center) of fixed area. In
another, the paths are Kepler ellipses (sharing a common focus) with fixed length of their minor axis. In a third model the paths are circles
tangent to and contained in a fixed circle. These three models are related by duality and a quadratic transformation. But my favourite is the Greek model: conic sections by planes tangent to a fixed hyperboloid of revolution. I will try to replace formulas with  pictures as much as
possible. This is joint work with Maciej Dunajski.


On (systems of) ODEs of C-class (Andreas Cap)

My talk reports on recent joint work with B. Dourbrov and D.The on the geometric theory of scalar ODEs up to contact transformations and systems of ODEs up to point transformations. In particular, I will discuss equivalent descriptionsin terms of canonical Cartan geometries and the question when that geometry descendsto the space of solutions. This is closely related to Cartan's concept of C-class equations. The methods we use are inspired by the use of BGG sequences to obtain restrictions on the curvature of a parabolic geometry from restrictions on its harmonic curvature.


Rational curves in flag varieties and vector distributions with large symmetry algebras (Boris Doubrov)

We study flag structures on contact manifolds which naturally appear in the geometry of bracket-generating vector distributions. We introduce the notion of a symbol for a given flag structure, which is just a homogeneous element of negative degree in the symplectic Lie algebra equipped with some parabolic grading. Such symbols also define rational curves in the isotropic flag varieties. Similar to Tanaka theory, each such symbol determines a unique flat model, whose symmetry algebra can be computed by a two-step prolongation procedure. This leads to interesting examples of vector distributions, which are also flat from the Tanaka point of view and have finite-dimensional Tanaka prolongations with positive part of much larger dimension than its negative part.


Conics and Twistors (Maciej Dunajski)

I shall describe the range of the Radon transform on the space of conics in CP2, and show that for any function F in this range, the zero
locus of F is a four–manifold admitting an anti-self-dual metric which can be constructed explicitly. This is a joint work with Paul Tod.


Some results on split  G_2^*-structures (Anna Fino)

I will present a classification of the holonomy algebras of manifolds admitting an indecomposable torsion free G_2^*-structure, i.e. for which the holonomy representation does not leave invariant any proper non-degenerate  subspace. Some of these Lie algebras  can be realized as holonomy algebras of left-invariant metrics on Lie groups. Moreover, using Cartan's theory of exterior differential systems I will show that all the Lie algebras from this list for which the socle of its holonomy representation is one-dimensional can indeed be realised as the holonomy of a local metric. All these Lie algebras are contained in the maximal parabolic subalgebra that stabilises one isotropic line of R^{4,3}. The talk is based on joint work with Ines Kath.


Slice theorem for CR structures and its applications (Kengo Hirachi)

Following the works of Bland-Duchamp, I formulate a slice theorem for integrable CR structures near the standard sphere and apply it to the analysis of the Q-prime curvature and the Bergman kernel.


On the symmetry algebras of 5-dimensional CR-manifolds (Boris Kruglikov)

We will demonstrate that the symmetry algebra of a 5Dholomorphically non-degenerate CR manifold has dimension at most 15 andthe next possible symmetry dimension is 11. All lower dimensions are alsorealised. We will present a countable set of pairwise non-equivalentmodels realising the sub-maximal bound. All of them are Levi-degeneratealong a submanifold, in the complement to which they are spherical. Thedimension gaps for Levi-nondegenerate CR structures are bigger and dependson the signature as I will also explain. The talk is based on thefollowing join work with A.Isaev (ANU): arXiv:1607.06072.2.


Metrics admitting projective symmetries (Gianni Manno)

Let M be a Riemannian or pseudo-Riemannian manifold. A projective symmetry is a vector eld on M whose local flow sends geodesics into geodesics (viewed as unparametrized curves). In 1882 Sophus Lie raised the problem of finding 2-dimensional metrics admitting at least a projective symmetry. In this talk we show how to find normal forms of such metrics and discuss some results in the 3-dimensional case.
The talk is based on joint papers with Robert L. Bryant, Vladimir Matveev and Andreas Vollmer.


Equivalences of 5-dimensional CR manifolds (Joël Merker)



The Hessian in projective differential geometry (Thomas Mettler)

The Hessian operator in projective differential geometry gives rise to a Monge-Ampere equation and Calabi observed, that solutions to this Monge-Ampere equation determine affine hyperspheres. In my talk I will explain that the relevant Monge-Ampere equation — when interpreted as a submanifold of some suitable jet bundle — fibres over several geometric spaces of interest. This allows to connect affine hyperspheres to other PDEs that arise naturally in projective differential geometry, notably extremal conformal structures and minimal Lagrangian connections.


Extrinsic geometries and differential equations of type sl3 (Tohru Morimoto)



Infinitesimal symmetries of special multi-flags reduce the local classification problem to linear algebra (Piotr Mormul)

Special m-flags, m>=2, constitute a natural follow-up to Goursat flags. The latter compactify (in certain precise sense) the contact Cartan distributions on the jet spaces J^r(1,1), while the former do the same with respect to the jet spaces J^r(1,m).

Sequences of Cartan prolongations of rank-(m + 1) distributions are the key players in producing (only locally) virtually all rank-(m + 1) distributions generating special m-flags. Immediately there emerges an immense tree of singularities of positive codimensions, all of them adjoining the unique open dense Cartan-like strata.

The local classification problem is well advanced for the Goursat flags, most notably after the work [2]. However, it is much less advanced for special multi-flags. The complete classification of them was given in [4]: in length r = 3 for all m>=2, and in length 4 for m=2 (the number of equivalence classes 34). After the year 2010 researchers were aiming at defining various invariant stratifications in the spaces of germs of special multi-flags. The actual state of the art is reflected in [1].

A new (2017) approach to the classification starts with the effective (recursive) computation of all infinitesimal symmetries of special multi-flags. The recursive patterns depend uniquely on the so-called singularity classes of special m-flags defined still in [3]. Those classes are coarser (hence fewer) than the RVT classes recapitulated in [1].

Polynomial visualisations of objects in the singularity classes are called EKR's (Extended Kumpera-Ruiz). They `only' feature finite families of real parameters. Then the classification problem is rephrased as a search for ulti- mate normalizations among such parameter families. Having an explicit hold of the infinitesimal symmetries at each prolongation step, the freedom in varying those parameters is being reduced to solvability questions of (huge) systems of linear equations.

In fact, that linear algebra involves only partial derivatives, at the reference point, of the first m + 1 components of a given infinitesimal symmetry (which initially, by the classical Backlund theorem, are completely free functions of m + 1 variables). Keeping the preceding part of a [germ of a] flag in question frozen imposes a sizeable set of linear conditions upon those derivatives up to certain order. Then some other linear combinations of them appear, or not, to be free - just in function of the local geometry of the prolonged distribution. This, in short, determines the scope of possible normalizations in the new (emerging from prolongation) part of EKR's.


[1] A. Castro, S. J. Colley, G. Kennedy, C. Shanbrom; A coarse stratification of the monster tower; arXiv: 1606.07931v2 [math.AG] (12 Jan 2017).

[2] R. Montgomery, M. Zhitomirskii; Points and curves in the monster tower. Memoirs AMS 956 (2010).

[3] P. Mormul, Geometric singularity classes for special k-flags, k>=2, of arbitrary length (2003). (https://www.mimuw.edu.pl/~mormul/Mor03.ps)

[4] P. Mormul, F. Pelletier; Special 2-flags in lengths not exceeding four: a study in strong nilpotency of distributions; arXiv: 1011.1763 [math.DG] (2010).


Voss surfaces in conformal Lorentzian geometry (Emilio Musso)



On local bisymplectic realizations of compatible Poisson brackets (Andryi Panasyuk)

In a seminal paper "The local structure of Poisson manifold" (1983) A. Weinstein proved that for any Poisson manifold (M,P) there exists a
local symplectic realization, i.e. nondegenerate Poisson manifold (M',P') and a local surjective submersion f:M'->M with f_*P'=P. Global
aspects of this problem were afterwards intensively studied as they are related to the theory of symplectic and Poisson grupoids, to the
integration problem of Lie algebroids, and to different quantization schemes.

In this talk I will discuss a problem of local simultaneous realization of two compatible Poisson structures by means of two nondegenerate ones. Note the following essential difference between the two realization problems: there is only one local model of the nondegenerate Poisson bivector P' given by the Darboux theorem and there are many local models of bisymplectic bihamiltonian structures. So besides the problem of existence it is important to understand how many nonequivalent realizations there are in the second case.


Differential flatness of rank 2 distributions and symmetries (Witold Respondek)

Integral curves of a rank m distribution on X can be interpreted as solutions of an underdetermined system of ordinary differential equations. This system (or the corresponding distribution) will be called differentially flat if there exit m functions on TX that, together with their time-derivatives up to a certain order, parameterize all solutions. If those functions are actually functions on X only, the system will be called x-flat, if they are defined on TX, then the system is called (x,x')-flat. It is known since the work of Cartan that, on an open and dense subset of X, a rank 2 distribution defines a flat system if and only if it is x-flat and locally equivalent to the Goursat normal form. Two natural questions arise. First: are there x-flat distributions other than the Goursat normal form? Second: can a rank 2 distribution be flat without being x-flat, i.e, are there (x,x')-flat rank 2 distribution?

We will answer both questions. In particular, the answer to the second question is positive, and we will describe (x,x')-flat rank 2 distributions and discuss their geometry and symmetries of related affine distributions.


Holonomies of generalised configuration spaces (Nuno Romao)


Integrable (3+1)-dimensional systems related to contact geometry (Artur Sergyeyev)

In this talk we present a new class of (3+1)-dimensional dispersionless integrable systems possessing nonisospectral Lax pairs written in terms of contact vector elds. In particular, we show that to any pair of rational functions of variable spectral parameter in general position there corresponds a (3+1)-dimensional integrable system, thus addressing a long-standing prob lem of nding a systematic construction for integrable partial differential systems in four independent variables; please see arXiv:1401.2122 for details.


Jet-determination of symmetries of parabolic geometries (Dennis The)

For many geometric structures, their symmetry algebras are uniquely determined by a finite jet at a given point.  I will discuss recent work with Boris Kruglikov on the case of parabolic geometries.  In this context, all symmetries are everywhere 2-jet determined.  At points where harmonic curvature is nonzero, we can sharpen this to 1-jet determinacy, and I'll discuss a key new result on Tanaka prolongation that led to this fact.  The existence of a 2-jet (but not 1-jet) determined symmetry (at a point where harmonic curvature vanishes) is often a strong restriction on the geometry.  In a variety of cases, including torsion-free geometries, parabolic contact structures, and various other classes, this assumption leads to a rigidity result: namely, flatness on an open neighbourhood with the given point in its closure.


On geometry of 2-nondegenerate CR structures of hypersurface type via bigraded Tanaka prolongation (Igor Zelenko)

The talk is devoted to the local geometry of 2-nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such
structures was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension (dimM=5), and for dim M=7 in certain cases by C. Porter. We develop a bigraded analog of Tanaka's prolongation procedure to construct a canonical absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension, classify all bigraded Tanaka symbols in the case of one-dimensional Levi kernel, and find the bigraded Tanaka prolongation for symbols satisfying additional natural regularity assumptions. Most of the talk is based on the joint work with C. Porter, some of the results are obtained also in collaboration with D. Sykes.


How to construct exact normal forms in local differential geometry and what are they for? (Michail Zhitomirskii)

I will discuss these questions basing mainly on the classical problems of classification of Riemannian metrics, conformal structures, and (2,5) distributions.


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