This meeting aims to gather all CMUP's members, especially its post-docs, so that they can briefly present their current research. The talks will be non-technical in order to be reachable for all areas. The participation of everyone is strongly encouraged.

**Date, time and place:**

30 of November from 14:30 to 18:00.

Room FC1.031.

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**Coffee-Break:**

Please use the following link (and fill in the corresponding form) if you intend to go to the coffee-break (it is free of charge): https://goo.gl/forms/FhbqQOMo0hugFI6E2

Please notice that it is important to fill in the link in order to have an estimate of the number of participants of the Meeting.

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**Speakers (and their group / line membership):**

- Carla Dias (Dynamical Systems)
- Elisa Sovrano (Dynamical Systems)
- Jaime Silva (Geometry)
- Martin Klimes (Geometry / Dynamical Systems)
- Theo Zapata (Algebra)

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**Schedule: **

**14:30 - 14:35 **Opening session

**14:35 - 15:10 **Martin Klimes (Geometry / Dynamical Systems) - *Dynamics of polynomial vector fields in one complex variable*

**15:10 - 15:45 **Carla Dias (Dynamical Systems) - *Gibbs-Markov-Young structures and stochastic stability*

**15:45 - 16:20 **Jaime Silva (Geometry) - *Hodge Structures of complex varieties*

**16:20 - 16:50 **Coffee break

**16:50 - 17:25 **Theo Zapata (Algebra) - *The congruence kernel of some weakly branch groups*

**17:25 - 18:00 **Elisa Sovrano (Dynamical Systems) - *Chaotic dynamics in the twisted horseshoe map*

**Titles and abstracts:**

**Martin Klimes (CMUP)**

**Title:** Dynamics of polynomial vector fields in one complex variable

**Abstract:** This is probably one of the simplest case of holomorphic dynamics onecan encounter: the real time trajectories of a polynomial vector field in the complex plane. Yet its far from completely trivial. I will talk about some older results of Douady and Sentenac on the geometry of the phase portrait, and about some current work.

**Carla Dias (CMUP) **

**Title:** Gibbs-Markov-Young structures and stochastic stability

**Abstract: **In the 1960's, Sinai and Bowen showed that all smooth uniformly hyperbolic dynamical systems admit a nite Markov partition.Sinai, Ruelle and Bowen then used this remarkable geometric structure, and the associated symbolic coding of the system, to study ergodic properties such as the rate of decay of correlations. Some years ago, L.-S. Young proposed an alternative geometric structure, which we call Gibbs-Markov-Young (GMY) structure, as a way of studying the ergodic properties of certain dynamical systems. In this talk, we discuss the relation between GMY structure, Lyapunov exponents and stochastic stability in the setting of random perturbations. This is a joint work with J.F. Alves and H. Vilarinho.

**Jaime Silva (CMUP)**

**Title:** Hodge Structures of complex varieties

**Abstract: **My purpose in this talk is to introduce and motivate one of my topics of research - Hodge structures on complex varieties - through some simple considerations on cohomology theory. I will start by motivating the usual cohomology theory of a manifold through the classification of spaces up to homeomorphism and the more commonly known Euler characteristic. Then, by referring more subtle problems such as classification up to biholomorphism, I will introduce Hodge structures as an invariant that takes into account the algebraic structure. I will finish by presenting some of my own results and, if time allows, say how they can be used to obtain some information on other fields.

**Theo Zapata (Universidade de Brasília, CMUP)**

**Title:** The congruence kernel of some weakly branch groups

**Abstract: **Focusing on the Brunner-Sidki-Vieira group and the Basilica group, we shall explain how to use profinite groups to solve the congruence subgroup problem for certain self-similar groups generated by automata; a detailed description of the congruence kernel will be given.

This is a joint work with F. Zapata (UnB).

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**Elisa Sovrano (CMUP) **

**Title: **Chaotic dynamics in the twisted horseshoe map

**Abstract: **The mathematical phenomenon of chaos is of interest in different scientific fields (e.g., astronomy, meteorology, ecology, and economics). However, there is no satisfactory single mathematical definition of chaos.

In the field of Dynamical Systems, a beautiful example encapsulating chaos is given by the geometric structure of the Smale’s horseshoe [S. Smale, Math. Intelligencer, 1998]. In this case, chaotic dynamics are related to the possibility of realizing a coin-tossing experiment.

In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction, such as the “topological horse- shoes” by [J. Kennedy, J. A. Yorke, Trans. Amer. Math. Soc., 2001]. In this framework, we exploit the topological tool developed in [A. Medio, M. Pireddu, F. Zanolin, IJBC, 2009] to show the presence of chaos in the twisted horse- shoe planar map [E. S, IJBC, 2016] and in some higher dimensional discrete dynamical systems.