Programme

15:00-16:00 Vladimir Matveev
Separation of variables for spaces of constant curvature.

I discuss orthogonal separation of variables for spaces of constant curvature, with the emphasis on pseudo-Riemannian metrics of constant curvature. I will give a local description of all possible separating coordinates, and write the transformation to flat coordinates. The problem was actively studied since at least hundred years and was considered to be solved in a series of papers and books of E. Kalnins with different coauthors, I will discuss this. The results are joint with A. Bolsinov and A. Konyaev. They appeared within the Nijenhuis Geometry project and are a by-product of our classification of compatible inhomogeneous geometric Poisson brackets of type 3+1.

16:00-16:30 Coffee Break

16:30-17:30 Claudia Chanu
Geometry of regular and not regular separation: the example of the bi-Helmoltz equation
Separation of variable is a standard usefull ansatz in order to determine some solutions of a PDE. We give the geometrical framework of two different types of separation, called by Kalnins and Miller ”regular” and ”non-regular” separation. The

geometric interpretation of non regular separation provides an effictive method to understand and characterize some known examples, as well as the so called fixed energy separation or the constrained separation for Schroedinger equation. Furthermore, we will see how to apply this tool for an exploration of multiplicatively separable solutions of the bi-Helmoltz equation \(\Delta^2f =Ef \).

This equation appears classically in the theory of sound because it is used as a model for the vibrations of a (thin) solid plate. Even if it will be shown that regular separation never occurs, however the equation naturally admits families of separated solutions (coinciding with separated solutions of Helmholtz equation). In some examples we geometrically characterize the coordinate systems where additional separated solutions are determined.

Morning: Free discussions

14:30-15:30 Jan Schumm
Local normal forms for Kähler metrics with essential c-projective vector fields
We will discuss normal forms for Kähler metrics that admit essential c-projective vector fields. J-planar curves are those whose acceleration vector lies in the span of the velocity vector and the complex structure J applied to the velocity vector.
An essential c-projective vector field sends J-planar curves to J-planar curves without preserving the Levi-Civita connection of the Kähler metric. A list of metrics for Kähler surfaces admitting such an essential c-projective vector field has been published in [1].
We will discuss their normal forms in the sense that some of the metrics in the list are diffeomorphic.

[1] A. V. Bolsinov, V. S. Matveev, T. Mettler, and S. Rosemann.
“Four-dimensional Kähler metrics admitting c-projective vector fields”. In: J. Math. Pures Appl. (9) 103.3 (2015), pp. 619–657.

15:30-16:00 Coffee Break

16:00-17:00 Filippo Salis
Kähler-Einstein metrics induced by complex projective spaces

Even though the existence of holomorphic and isometric immersions of a given Kähler manifold into a complex space form can be considered as a classical topic in complex differential geometry, many interesting questions are still unanswered (cfr. e.g. [1, 2]). In this talk, we will deal in particular with the open problem of the classification of Kähler-Einstein manifolds that can be Kähler immersed into a complex projective space endowed with the Fubini-Study metric. A special focus will be specifically paid to the case of Kähler metrics admitting local symmetries of rotational type, that we have been studying jointly with G. Manno (see [3]).

References:

[1] Calabi E.: Isometric imbeddings of complex manifolds, Ann. of Math. (2), 58, 1-23 (1953).

[2] Loi A., Zedda M.: Kähler immersions of Kähler manifolds into complex space forms. Lectures notes of the Unione Matematica Italiana, 23 Springer, Bologna (2018).

[3] Manno G., Salis F.: 2-dimensional Kähler-Einstein metrics induced by finite dimensional complex projective spaces, New York J. Math., 28, 420-432 (2022).

09:30-10:30 Konrad Schöbel
A Route to Classifying Superintegrable Systems in Arbitrary Dimension

We show that the classification space for irreducible non-degenerate second order superintegrable systems is naturally endowed with the structure of a quasi-projective variety and a linear isometry action. On constant curvature manifolds this leads to a simple algebraic equation defining a variety which classifies those superintegrable Hamiltonians that satisfy all integrability conditions generically. This includes all examples known to date. We outline how to recover the known classification in dimension three and discuss further applications of this approach.

10:30-11:00 Coffee Break

11:00-12:00 Giorgio Gubbiotti
Computing the growth of degrees of a discrete Euler top
With techniques from algebraic geometry we prove that the Kahan-Hirota-Kimura discretisation of the Euler top possesses quadratic growth of the iterates, making it integrable in the sense of the algebraic entropy. More specifically, we rewrite the corresponding birational map as the composition of a standard Cremona transformation in dimension three with a projectivity, and we construct the space of initial conditions in the sense of Okamoto and Sakai. We also find a covariant net of quadrics which yields two functionally independent invariants, thus proving integrability also in the naive sense. If time permits, we will present a generalisations to a wider class of maps sharing the key properties of the KHK discretisation of the Euler top.

12:00-12:30 Sergio Benenti
Expanding Universe and expanding confusion.

Per le particelle si adotta il Modello Standard, per l'Universo il Modello Lambda-Cold-Dark-Matter.

Ma nessuno ha mai saputo dirmi come e dove sono stati definiti. Per quel che ne so entrambi fanno cilecca.

Così me ne sono fatto uno tutto mio (alla Razzi) basato su una sequenza di postulati ben definiti, bene ordinati e condivisibili. Per funzionare necessita di soli quattro dati cosmologici (tra cui la 'costante' di Hubble). Ha risposto bene sia strutturalmente che numericamente. Rispetto a quanto riportato nella Nobel Lecture di Riess (2011) dice qualcosa in più. Finale a sorpresa