# A day in combinatorics and complex dynamical systems

We are pleased to announce the meeting **A day in combinatorics and complex dynamical
systems**
that will take place on zoom on **Tuesday March 16 of 2021.** It will feature 4 talks, two
in
the morning dedicated to complex dynamics, and two in the afternoon dedicated to combinatorics and graph
theory.

#### March 16, 2021, from 10.30 to 16.15, Chile time (Continental) (13:30 UTC) or (GMT-3)

- 10:30 - 11:20 Luna Lomonaco (IMPA, Brazil)
- 11:25 - 12:15 Mónica Moreno Rocha (CIMAT-Mexico)
- 14:30 - 15:20 Maycon Sambinelli (Universidade Federal do ABC, Brazil)
- 15:25 - 16:15 Andrea Jiménez – (CIMFAV-Instituto de ingeniería Matemática, Chile)

Feel free to distribute this site and e-mail to your colleagues or
anyone you think might be interested, and please encourage graduate students.

Zoom Meeting ID: 845 7859 5055

Passcode: 974895

https://reuna.zoom.us/j/84578595055?pwd=QWtFKzZXMEZCWmQrUnpmZFJvTVo4dz09

Please do not post Zoom meeting and passwords on the open web. We do not plan to record the meeting.

#### Mating quadratic maps with the modular group

Speaker: Luna Lomonaco (IMPA, Brazil)

10:30 - 11:20

Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere (implicit maps sending z to w). The iteration of such a multi-valued map generates a dynamical system on the Riemann sphere (dynamical system which generalises rational maps and finitely generated Kleinian groups). We consider a specific 1-(complex)parameter family of (2:2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every a in the connectedness locus M_{\Gamma}, this family is a mating between the modular group and rational maps in the family Per_1(1); we develop for this family a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials; and we show that M_{\Gamma} is homeomorphic to the parabolic Mandelbrot set M_1. This is joint work with S. Bullett (QMUL).

###### Unbounded and invariant Fatou components

Mónica Moreno Rocha (CIMAT-Mexico)

11:25 - 12:15

If f denotes a polynomial of degree at least 2, it is fairly easy to see the existence of a neighborhood of infinity where the iterates of f converge to infinity. Such neighborhood belongs to an unbounded Fatou component that is completely invariant under f and whose boundary coincides with the Julia set of f (thus, the residual Julia set is empty). In the 1980’s, Baker considered the case of replacing f by a transcendental entire function and explored the existence of unbounded and invariant Fatou components. Around the same time, Makienko proposed a conjecture for rational functions that involves a completely invariant Fatou component and the residual Julia set. In this talk we will briefly review Baker’s results and Makienko’s conjecture in the case of transcendental meromorphic functions. Then, we will concentrate on the particular case of elliptic functions and discuss results and open questions regarding unbounded, invariant Fatou components and the consequences for Makienko’s conjecture in this setting.

#### Towards Gallai's path decomposition conjecture

Maycon Sambinelli (Universidade Federal do ABC, São Paulo, Brazil)

14:30 - 15:20

Path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. In 1968, Gallai conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most the ceiling of n/2. In this lecture, we will discuss the state of the art of this conjecture.

###### Grid subdivisions, minors and treewidth in planar graphs.

Andrea Jiménez (Instituto de ingeniería Matemática CIMFAV-Universidad de Valparaíso, Chile)

15:25 - 16:15

Minors, subdivisions and treewidth are classical notions that appear for example in the seminal characterization of planar graphs by Kuratowski and in the celebrated Graph Minor Theorem by Robertson and Seymour. The importance of grid minors comes from one of the result of Robertson and Seymour, which roughly claims that a graph of large treewidth necessarily contains a large grid minor. The concept of treewidth is fundamental in algorithmic graph theory since many problems which are hard to solve in general, can be efficiently solved when restricted to classes of graphs with bounded treewidth. In this talk, we discuss the computational complexity of the Minor Problem, the Subdivision Problem and the Treewidth Problem with input: grids and planar graphs. Surprisingly, some of these problems are still open. We describe a reduction which proves that the respective Subdivision Problem is NP-complete and a possible plan to prove that the other two are NP-complete as well.

## Organizing committee

##### Gerardo Honorato

gerardo.honorato@uv.cl

CIMFAV-Instituto de ingeniería Matemática-UV, Chile

##### Andrea Jimenez

andrea.jimenez@uv.cl

CIMFAV-Instituto de ingeniería Matemática-UV, Chile

##### Francisco Valenzuela-Henriquez

francisco.valenzuela@pucv.cl

Instituto de Matemáticas-PUCV, Chile