Workgroup Seminars on holomorphic foliations and resurgence 2019./20.

University of Zagreb, Department of Mathematics

Organizers: Goran Radunović, Maja Resman

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1.       Tuesday, February 18, 2020, 16.15 h  

(Department of Mathematics, room 104)

L. Teyssier, Université de Strasbourg: Non-algebraizable planar saddle-nodes

 

Abstract:

    We call "planar foliation" the local structure of the integral curves of a holomorphic vector field in the complex 2-plane. Far from stationary points the foliation is regular and can be (locally) straightened onto a product of discs. A generic singular foliation also admits a simple local structure, since it is linearizable by Poincaré's theorem.

  Geometrically speaking these properties are tantamount to saying that the typical foliation is the expression in a local chart of a global foliated

compact complex surface. Is that a general fact? If not, how can one build examples of germs of a foliation which cannot arise as the localization of a global foliation?

   By considering saddle-node foliations (non-linear "irregular singular points") we are able to answer these questions. More specifically, we study in detail the process that brings a polynomial saddle-node foliation into its Loray normal form, and how doing so enlarges the field of definition of the foliation in a controlled way. As a result, normal forms whose field of definition has infinite transcendence degree over the rationals cannot be

locally conjugate to an algebraic foliation. 

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2.       Thursday, February 27, 2020, 16.15 h

(Department of Mathematics, Room 104)

M. Resman: Introduction to theory of divergent series

Abstract: In this talk we will introduce the notion of divergent Gevrey summable series and the Borel-Laplace summation method for such series, that recovers sectorially analytic functions. We illustrate the method on some well-known examples in differential equations, for example that of Euler series.

The seminar is first in the series of seminars based on the following literature:

1. F. Loray, Analyse des séries divergentes, dans Quelques aspects des mathématiques actuelles, Mathématiques 2eme cycle, Ellipses (1998), 111–173

2. D. Sauzin: Resurgent functions and splitting problems, https://arxiv.org/pdf/0706.0137.pdf (2006)

3. D. Sauzin, Introduction to1-summability and resurgence, https://arxiv.org/pdf/1405.0356.pdf (2014)

4. C. Mitschi, D. Sauzin, Divergent series, Summability and resurgence. I. Monodromy and resurgence. Lecture notes in Mathematics, Springer (2016)

5. C. Rousseau: Divergent series, past, present and future. https://dms.umontreal.ca/~rousseac/divergent.pdf (2013) https://dms.umontreal.ca/~rousseac/Rousseau_divergent_series.pdf

7. J. P. Ramis, Séries divergentes et théories asymptotiques, Paris : Société mathématique de France, Series: Panoramas et synthèses (1993)

8. B. Candelpergher, Une introduction à la résurgence, Gaz. Math.42(1989), 36–64.

9. M. Loday-Richaud, Divergent Series, Summability and Resurgence, II Simple and Multiple Summability, Springer (2016)

10. W. Balser, From divergent power series to analytic functions, Theory and application ofmultisummable power series, Lecture Notes in Mathematics, Springer-Verlag, Berlin (1994)

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3.       Tuesday, March 3, 2020, 15.30 h

(Department of Mathematics, Room 104)

M. Resman: Algebra of formal series and Borel-Laplace solutions of difference equations


Abstract: In this seminar we familiarize with operations on formal series. We state main properties of Borel transform as an isomorphism between differential algebras of Gevrey-1 formal series and of analytic germs. The goal: solving linear (and later non-linear) difference equations by Borel-Laplace method. Based on
D. Sauzin, Introduction to 1-summability and resurgence, https://arxiv.org/pdf/1405.0356.pdf (2014).

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4.       Tuesday, March 10, 2020, 15.30 h

(Department of Mathematics, Room 104)

M. Resman: Algebra of formal series and Borel-Laplace solutions of difference equations – part II


Abstract: In this seminar we familiarize with operations on formal series. We state main properties of Borel transform as an isomorphism between differential algebras of Gevrey-1 formal series and of analytic germs. The goal: solving linear (and later non-linear) difference equations by Borel-Laplace method. Based on
D. Sauzin, Introduction to 1-summability and resurgence, https://arxiv.org/pdf/1405.0356.pdf (2014).

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5.     Tuesday, May 12, 2020, 16.15h (on-line)

 

M. Resman: Borel-Laplace solutions of nonlinear difference equations and resurgent functions

 

Abstract: We solve some non-linear difference equations by Borel-Laplace method. The Borel transform of the solution, unlike for linear difference equations, does not necessarily possess meromorphic extension to whole C. Nevertheless, the singularities are ‘simple’ – it belongs to the algebra of so-called resurgent functions. Based on D. Sauzin, Introduction to 1-summability and resurgence, https://arxiv.org/pdf/1405.0356.pdf (2014).

 

Predavanje ce se odrzati putem platforme Zoom na linku:

https://us02web.zoom.us/j/6198436007

 

 

6.       Tuesday, May 19, 2020, 17h (on-line)

M. Resman: Rjesavanje nelinearnih jednadzbi Borel-Laplaceovom metodom i resurgentne funkcije, nastavak.

Sažetak: Rješavamo neke specijalne nelinearne diferencijske jednadzbe Borel-Laplaceovom metodom. Borelova transformacija rjesenja, za razliku od linearnih jednadzbi, nema nuzno meromorfno proširenje na čitavu kompleksnu ravninu. Ipak, singulariteti su u nekom smislu 'jednostavni' za analizu- pripadaju tzv. algebri resurgentnih funkcija.

Bazirano na preprintu  David Sauzin: 1-summability and resurgence, https://arxiv.org/pdf/1405.0356.pdf (2014)

Predavanje ce se odrzati putem platforme Zoom na linku:

https://us02web.zoom.us/j/6198436007?pwd=WjdQNkF4MGs3WEtaK2F1MGpDTUgzUT09
Meeting ID: 619 843 6007, Password: seminar