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Mini-Workshop "Nijenhuis Geometry"

Department of Mathematical Sciences, Institute of Advanced Studies (IAS), Loughborough University

  • Date: Tuesday 18 February 2020
  • Time: 14:00-17:00
  • Venue: SCH.0.13 (Schofield Building)

The aim of the Workshop is to discuss new developments in Nijenhuis Geometry. Similar to Riemannian, Poisson or symplectic geometry, the Nijenhuis geometric structure is defined by means of a tensor of order 2 (i.e., by a matrix) but, in contrast to the above examples, this tensor is not a bilinear form but linear operator. The additional geometric condition imposed on this operator is that its Nijenhuis torsion identically vanishes. Open problems in this area include studying singular points of Nijenhuis operators, local normal forms, global properties and topological obstructions to the existence of such operators on closed manifolds.

Schedule

14:00-14:40 A. Bolsinov (Loughborough University) Introduction to Nijenhuis Geometry
14:45-15:30 V Matveev (Jena University, Germany)

Applications of Nijenhuis geometry: projectively equivalent metrics and Poisson-Nijenhuis
structures

15:30-16:00 Coffee break
16:00-16:45 A. Konyaev (Moscow State University, Russia) Left-symmetric algebras, singularities of Nijenhuis operators and linearisation problem

See below for abstracts

Abstract: A. Bolsivov - Introduction to Nijenhuis Geometry

The talk will be devoted to basic facts, definitions and constructions in Nijenhuis Geometry such as Nijenhuis torsion, splitting theorem, singular points, stability and local normal forms.  I will try to explain why Nijenhuis Geometry should be understood in much wider context than before by refocusing the research agenda from local description of Nijenhuis operators at generic points to singularities and global issues.  I will also discuss some open problems in the area.

Abstract: V. Matveev - Applications of Nijenhuis geometry: projectively equivalent metrics and Poisson-Nijenhuis structures

Nijenhuis operators naturally appear in  many geometric structures. In this talk I consider two of such structures: pairs of projectively equivalent metrics  and pairs of compatible Poisson structures (I will recall all necessary definitions). I concentrate on singular points of these structures and show how recent results in Nijenhuis geometry provide a local description near such points.

Abstract: A. Konyaev - Left-symmetric algebras, singularities of Nijenhuis operators and linearisation problem

A singular point of a Nijenhuis operator L is said to be of scalar type if all of its eigenvalues  collide at this point and L  becomes proportional to the identity operator. The tangent space at such a point  is naturally equipped with a structure of the so-called left-symmetric algebra. These algebras, also known as Vinberg, Koszul or pre-Lie algebras, naturally appear in differential geometry, algebra, infinite dimensional integrable systems and quantum mechanics. In the context of Nijenhuis Geometry, they carry  information about Nijenhuis operators near singular points  (this situation is strikingly similar to that in Poisson geometry, where the cotangent space at a singular point of a Poisson structure carries a natural Lie algebra structure).

In this talk we explore the duality between linear Nijenhuis operators and left-symmetric algebras, introduce the notion of a non-degenerate left-symmetric algebra and provide complete classification of left-symmetric algebras in terms of non-degeneracy in dimension two (note that unlike Lie algebras, the list of two-dimensional left-symmetric  algebras is quite long and contains infinite series with continuous parameters!). This classification turns out to be a non-trivial task which makes use of notorious results by A.Bruno and J.-C.Yoccoz. At the end of the talk we will discuss some open problems.