**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 to**1-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.

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

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