Workgroup Seminars on holomorphic foliations and resurgence 2019./20., 2020./21.

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

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

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7. Tuesday, 26/01/2021, at 16.15, Goran Radunović, PMF-MO, will give a talk in Dynamical Systems Seminar in Zagreb under the title: Fractional integro-derivative of fractal zeta functions and Log-Minkowski measurability

a. Abstract: We introduce Logarithmic gauge Minkowski content which arises naturally from the theory of complex dimensions. The complex dimensions of a given set are usually defined as poles of the corresponding fractal zeta function and they generalize the notion of Minkowski dimension. In the most simple case the fractal zeta function has a simple pole at D where D is the Minkowski dimension of the given set, whereas the residue equals to the Minkowski content (modulo a multiplicative constant). Here we show that in case of poles of higher order one has to introduce a generalization of Minkowski content to obtain an analogue connection. Furthermore, in the most general case, one has to consider more complicated singularities of the fractal zeta function, including a combination of poles, zeroes and branch points. The general case can be explained in the context of an appropriate fractional integro-derivative of the fractal zeta function. The talk will be held via Zoom platform: Topic: Seminar DYNSYS https://us02web.zoom.us/j/88154578830 Meeting ID: 881 5457 8830

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8. Tuesday, 02/02/2021, at 16.15, Maja Resman, PMF-MO, University of Zagreb, will give a talk in Dynamical Systems Seminar in Zagreb under the title: Zeta functions and complex dimensions of orbits of dynamical systems

Abstract: The standard zeta function can be generalized to a zeta function of the so-called fractal string. Various (almost equivalent) definitions of zeta functions of a general bounded set that generalize this definition have been introduced by Lapidus, Frankenhuijsen, Radunovic, Zubrinic. It was shown that the poles and their principal parts of the meromorphic extension, if it exists, of a zeta function of a set reveal the intrinsic geometry of the set. In particular, the set is considered fractal if there are non-real complex dimensions. We prove the existence of the meromorphic extensions of fractal zeta functions of orbits of dynamical systems generated by parabolic germs of analytic diffeomorphisms on the real line and describe their complex dimensions. We relate them to some intrinsic properties of the generating function and to the geometry of its orbits. This is a joint work with G. Radunovic, University of Zagreb, and P. Mardesic, University of Burgundy. The talk will be held via Zoom platform: Topic: Seminar DYNSYS https://us02web.zoom.us/j/88154578830 Meetng ID: 881 5457 8830

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9. Tuesday, 12/01/2021, at 16.15, Martin Klimeš will give a talk in Dynamical Systems Seminar under the title: Dynamics of meromorphic vector fields on the Riemann sphere

Abstract: The talk will offer a brief overview on the dynamics of real-time flow of rational vector fields in one complex variable on CP^1, of their bifurcations, and of the structure the moduli space. The real phase portrait is also known as the horizontal foliation of the dual meromorphic differential form, and a great deal of the theory has a natural generalization also for horizonal foliations of meromorphic quadratic differentials.

The talk will be held via Zoom platform https://us02web.zoom.us/j/84365265465, Meeting ID: 843 6526 5465

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10. Workgroup: Wednesday, February 10, February 17, March 10, 2021, 10h, Faculty of Electrical Engineering and Computing, workgroup, G. Radunović: Introduction to resurgent functions (work seminars, based on D. Sauzin: Resurgent functions and splitting problems, Candelpergher: Resurgent functions)

(participants: Vesna Županović, M. Resman, P. Mardešić, M. Klimeš, G. Radunović)

11. Workgroup: Wednesday, March 17, April 14, 2021, 15h, Faculty of science, Dept of Mathematics, M. Klimeš: Strange derivatives (work seminar, participants: Resman, Županović, Radunović, Klimeš, Peran via Zoom)