# Geometry and Differential Equations Seminar

Time and place: IMPAN, room 106, Wednesday, 12:15-13:45

Organizers: Bronis³aw Jakubczyk, Pawe³ Nurowski, Wojciech Kryñski

### Talks:

May 30, 2018

Javier de Lucas Araujo (University of Warsaw): Poisson-Hopf algebra deformations of a class of Hamiltonian systems

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.

May 23, 2018

Giovanni Moreno (University of Warsaw): Varieties of minimal rational tangents and second-order PDEs

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. Buczyñski 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.

May 17, 2018 (Thursday, 10:15 AM, room 408)

Ben Warhurst (MIM UW): Puncture repair in metric measure spaces

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.

May 09, 2018

Bronis³aw Jakubczyk (IM PAN): A Global Implicit Function Theorem

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.

April 25, 2018

Maciej Dunajski (University of Cambridge): From Poncelet Porism to Twistor Theory

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.

April 18, 2018

Marek Grochowski (UKSW, Warsaw): Causality in the sub-Lorentzian geometry

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.

April 11, 2018

Aleksandra Borówka (Jagiellonian University, Kraków): C-projective symmetries of submanifolds in quaternionic geometry

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.

March 21, 2018

Omid Makhmali (IM PAN): Geometries arising from rolling bodies (part II)

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.

March 14, 2018

Micha³ JóŸwikowski (IM PAN): Minimality in one-dimensional variational problems from global geometric properties of the extremals

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.

March 07, 2018

Piotr Kozarzewski (MIM UW): On the condition of tetrahedral polyconvexity

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 Ka³amajska 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 Ka³amajska.

February 28, 2018

Omid Makhmali (IM PAN): Geometries arising from rolling bodies

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.

February 21, 2018

Helene Frankowska (CNRS and Sorbonne Université): Integral and Pointwise Second-order Necessary Conditions in Deterministic Control Problems

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.

January 24, 2018

Bronis³aw Jakubczyk (IM PAN): Division of differential forms: Koszul complex, Saito's theorem and Cartan's lemma with singularities

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.

January 10, 2018

Pawe³ Nurowski (CFT PAN): Kerr's Theorem

ABSTRACT: -

December 13, 2017

Wojciech Kryñski (IM PAN): (3,5,6)-distributions, bi-Hamiltonian systems and contact structures on 5-dimensional manifolds.

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.

December 06, 2017

Pawe³ Nurowski (CFT PAN): On optical structures in space-time physics

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.

November 08, 2017

Henrik Winther (University of Tromso): Submaximally Symmetric Quaternionic Structures

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.

November 03, 2017

Omid Makhmali (McGill University): Local aspects of causal structures and related geometries

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.

October 25, 2017

Lenka Zalabova (University of South Bohemia, Èeské Budìjovice): On automorphisms with natural tangent action for Cartan geometries

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)

October 18, 2017

Aleksandra Borówka (Jagiellonian University, Kraków): Armstrong cones and generalized Feix--Kaledin construction

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.

October 11, 2017

Arman Taghavi-Chabert (University of Turin): Twistor geometry of null foliations

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.

October 4, 2017

Travis Willse (University of Vienna): Curved orbit decompositions and the ambient metric construction

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.

Septembet 27, 2017

Christoph Harrach (University of Vienna): Poisson transforms for differential forms adapted to homogeneous parabolic geometries

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.

Septembet 20, 2017

Shin-Young Kim (Masaryk University, Brno): Geometric structures modeled on some horospherical varieties

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.

Septembet 13, 2017

Sean Curry (University of California, San Diego): Compact CR 3-manifolds, and obstruction flatness

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.

2016/2017

2015/2016

2014/2015

2013/2014