Geometry and Differential Equations Seminar

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

Organizers: Bronisław Jakubczyk, Janusz Grabowski, Paweł Nurowski, Wojciech Kryński


December 19, 2018

Janusz Grabowski (IM PAN): Remarks on contact geometry

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.

December 12, 2018

Konrad Lompert (Politechnika Warszawska): Invariant Nijenhuis tensors and integrable geodesic flows on homogeneous spaces

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.)

December 05, 2018

Paweł Goldstein (MIM UW): Topologically nontrivial counterexamples to Sard's theorem and approximation of $C^1$ mappings

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 $k$.

November 21, 2018

Paweł Nurowski (CFT PAN): Hopf fibration 7 times in physics


November 14, 2018

Wojciech Kryński (IM PAN): Invariants and isometries of contact sub-Riemannian structures

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.

November 07, 2018

Ben Warhurst (University of Warsaw): A canonical connection in Subriemannian contact geometry


October 31, 2018

Omid Makhmali (IM PAN): Causal structures from a microlocal viewpoint

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.

October 24, 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.

October 17, 2018

Jun-Muk Hwang (Korea Institute for Advanced Study): Cone structures arising from varieties of minimal rational tangents

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.

October 03, 2018

Paweł Nurowski (CFT PAN): Kerr's theorem revisited

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.