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:**

May 17th, 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://forms.gle/nww7XeEJYzgvbWadA

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):**

- Carlos Rito (Geometry)
- Alexandre Rodrigues (Dynamical Systems)
- Ivan Kaygorodov (Algebra)
- Mário Magalhães (Dynamical Systems)
- Jordi Delgado (Algebra / Semigroups, Automata and Languages)

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**Schedule: **

**14:30 - 15:05 **Carlos Rito (Geometry)* - Algebraic Surfaces and Computer Algebra*

**15:05 - 15:40 **Alexandre Rodrigues (Dynamical Systems)* - Dynamics near a periodically forced heteroclinic cycle*

**15:40 - 16:15 **Ivan Kaygorodov (Algebra)* - Classifications of small dimensional algebras*

**16:15 - 16:50 **Coffee break

**16:50 - 17:25 **Mário Magalhães (Dynamical Systems) *- Point processes of exceedances*

**17:25 - 18:00 **Jordi Delgado (Algebra / Semigroups, Automata and Languages) *- From algebra to geometry (and back)*

**Titles and abstracts:**

**Carlos Rito (CMUP)**

**Title:** Algebraic Surfaces and Computer Algebra

**Abstract:** The study of surfaces is classic in Algebraic Geometry, but these are not yet classified. Often the construction of a single example in a family is a challenge, even for the lowest values of the topological invariants. In this presentation I will review construction methods and show how the junction of classical Algebraic Geometry with recent methods of Computer Algebra has been useful in solving some problems.

**Alexandre Rodrigues (CMUP)**

**Title:** Dynamics near a periodically forced heteroclinic cycle

**Abstract:** There are few explicit examples in the literature of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. In this talk, we present a careful analysis of the rich dynamics generated by a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere. We derive the first return map near the “ghost" of the heteroclinic cycle and we reduce the non-autonomous system to a two-dimensional map. According to the magnitude of two small parameters, the associated flow may behave periodically, quasi-periodically or chaotically, depending on the specific character of the perturbation. We also provide a bifurcation study of the periodic solutions that emerge and coalesce near the robust cycle when the parameters vary. We reduce the study of the dynamics to other configurations where the dynamics is known. This is a joint work with Isabel Labouriau (University of Porto).

**Ivan Kaygorodov (CMUP)**

**Title:** Classifications of small dimensional algebras

**Abstract:** The talk is about two types of classifications of algebras: algebraic and geometric ways. We will talk about classifications of 2-dimensional algebras, and some small dimensional algebras from nilpotent non-associative varieties.

**Mário Magalhães (CMUP)**

**Title:** Point processes of exceedances

**Abstract:** The study of extreme events is useful for risk assessment, advanced planning and other areas where the estimation of extreme behaviour is of crucial importance. In this talk, I will analyse the convergence of point processes of exceedances arising from a dynamical system by evaluating a given potential along the orbit of a chosen point, which will take into account the effect of clustering when we consider a repelling periodic point. This is a joint work with A. C. M. Freitas and J. M. Freitas.

**Jordi Delgado (CMUP)**

**Title:** From algebra to geometry (and back)

**Abstract:** Geometric group theory aims to use geometric objects to understand groups, and eventually solve algorithmic problems on them. I shall briefly recall the idea of algorithmic problem, and the basics of two of the most natural such geometric constructions; namely Cayley graphs and Stallings automata.

Then, I shall present the subgroup intersection problem, and consider it in a natural context where the previous two geometric concepts turn out to appear surprisingly intertwined. Finally, I will show how geometry transparently provides insightful algebraic information, and in particular, allows us to solve the intersection problem in this context.

This is joint work with Enric Ventura.