The ubiquity of  Wrońskians

Time: 11-18 October, 2011

Venue: Banach Center, Warsaw





REPORT


Organizers: Piotr Pragacz (chair) and Grzegorz Kapustka.


Invited speakers for the mathematical session:

Teresa Crespo (Barcelona),
Letterio Gatto (Torino),
Zbigniew Hajto (Cracow),
Maxim Kazarian (Moscow),
Inna Scherbak (Tel Aviv).


Invited speakers for the historical session:

Jacek Bartyzel (Toruń),
Mariusz Bochenek (Toruń),
Grzegorz Karwasz (Toruń).


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SUMMARY of the MATHEMATICAL SESSION:


We studied different incarnations of Wrońskians:

- those of linear systems in algebraic geometry;
- those in ordinary differential equations;
- those in singularity theory.

Letterio Gatto gave 4 talks on:
"From Wrońskians of linear systems to Wroński-Schubert Calculus".

Cf. the preprint: L. Gatto, I. Scherbak: On Generalized Wronskians
See also Gatto's recent paper: "The Wronskian and its derivatives" in: http://www.actapeloritana.it/


Maxim E. Kazarian gave 2 talks on:
"Wrońskian preserving deformations."

Abstract: We consider tuples of functions in one variable depending on additional parameters. We study the partition of the parameter space according to the degeneracy of their Wronskians. The strata are labelled by collections of Schubert symbols (Young diagrams). We formulate conditions assuring triviality of the partition along the strata (which does not always hold!). The talk is based on Kazarian's PhD thesis (1993) and its accompanying
papers. It would be interesting to relate these result to the modern study of Wronskians appearing in the context of integrable systems, Bethe Ansatz of the Gaudin model, etc.

Cf. the article: M.E. Kazarian, Singularities of the boundary of fundamental systems, flat points of projective curves, and Schubert cells, J. Soviet Math. 52 (1990), 3338 - 3349.


Inna Scherbak and Maxim Kazarian gave jointly 3 talks on:
"Schubert calculus in a Grassmannian, Wroński map and generalized Wrońskians."

Abstract: There is a natural isomorphism between the representation ring for gl and the cohomology ring of the Grassmannian: both are governed by the Littlewood-Richardson coefficients and the Schur functions.The correspondence is classically known but it is
usually treated formally without any geometric explanation. Nowadays there is a direct geometric construction provided by the Gaudin model: it produces explicitly a basis in
the singular subspace of a given weight in the tensor product of irreducible gl-modules starting from the intersection points of the corresponding Schubert cycles on the Grassmannian. Properties of the Wronskians play the key role in this construction.
The correspondence works perfectly provided some transversality condition is satisfied.
This condition usually holds but unfortunately there are still cases when the
non-transversality is inevitable.

Cf. the article: E. Mukhin, V. Tarasov, A. Varchenko, Schubert calculus and representations of general linear group,  arXiv:0711.4079


Teresa Crespo and Zbigniew Hajto gave jointly 4 talks on:
"Picard-Vessiot extensions for real differential fields."

Cf. T. Crespo, Z. Hajto, E. Sowa, Picard-Vessiot theory for real fields


                                               
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In the HISTORICAL SESSION, we had 3 talks:


Jacek Bartyzel:
"Wprowadzenie do filozofii achrematycznej i metapolityki mesjanicznej Hoene-Wrońskiego."

Grzegorz Karwasz:
"Lectures in Physics by Hoene-Wroński."

Mirosław Bochenek:
"Ekonomiczne dzieło Józefa Marii Hoene-Wrońskiego."

                                                   

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Letterio Gatto prepared:    LIST OF OPEN  PROBLEMS



(Last modified on 4.12.2011.)