This meeting aims to gather all of the CMUP members, including new members, who will informally present their current research. The talks will be non-technical and of interest to mathematicians from all fields of mathematics and its applications. **The participation of CMUP members is strongly encouraged.**

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**Coffee-Break (free of charge):**

Please use the following link (and fill in the corresponding form) if you intend to go to the coffee-break:

https://docs.google.com/forms/d/1NThXBVWtpWsFMeoZ5yBvA2NIDEO1CLwj8-NJ-KqL3Qc/edit

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

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**Titles and abstracts**

**14:00 - Helena Reis (**CMUP**) - **Opening session

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**14:05 ****- Victor Vargas **(Dynamical Systems)

**Title: **Thermodynamic formalism for discrete time linear dynamical systems

**Abstract: **

In this talk will be presented some results about existence of σ-additive Borel probability measures with non-trivial support and invariant by the action of a weighted shift L : X → X, where X is either the Banach space c^0(R) or l^p(R) (1 ≤ p < ∞). In order to do that, we adapt well known techniques of the classical thermodynamic formalism to the setting of the so called weighted shifts defining a transfer operator depending of a potential A : X → R satisfying suitable conditions and we prove that any fixed point μA of the dual of the transfer operator results in an invariant probability measure with full support. Furthermore, we are able to show that any μ_A obtained through this technique satisfies a variational principle of the pressure and, thus, it is an equilibrium state. In addition, we also show existence of accumulation points at zero temperature of the family (μ_tA)_t>1 in the weak* topology.

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**14:50 - Adjaratou Arame Diaw** (Geometry)

#### Title: Resolution of the singularities of a pair of foliations in dimension 2 and classification

**Abstract:**

According to the well-known theorem of A.Seidenberg, it is possible to reduce the complexity of a singularity of a holomorphic foliation defined on a complex surface to obtain a foliation with a very simple singularity in the sense of Seidenberg.

In this talk, we propose to generalize Seidenberg’s theorem to the pair of foliations. Indeed, given a pair of singular holomorphic foliations on a compact complex surface:

• What is the simplest and most accessible local model of the pair of foliations after simultaneous reduction of its singularities?

• Is it possible to know the analytic type of the pair of foliations after solving its singularities?

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**15.35 -** Coffee break

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**16:00 - Claude Marion **(Algebra)

**Title: F**ree groups and finite groups: a few problems

**Abstract: **

Given a free group F of finite rank and a pseudovariety V of finite groups (a class closed under taking subgroups, homomorphic images and finite direct products), one can endow F with the pro-V topology. Examples of pseudovarieties include the sets of all finite groups, all groups of odd order, all nilpotent groups and all solvable groups. Given a finitely generated subgroup H of F, a natural question arises:

(i) Are some pro-V topological properties, for example denseness or closedness, decidable for H?

On another area, following the classification of finite simple groups in the 1980s, it was established that every finite simple group can be generated by two elements. Some natural questions arise:

(ii) Can we impose some restrictions on a pair of generators?

(iii) What are the graph-theoretical properties of the generating graph of a finite simple group?

(iv) Given a subgroup H of a finite simple group, what can be said about the minimal number of generators for H?

We will discuss some of these problems.

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**16:45 - Alfonso Tortorella **(Geometry)

**Title:** Advances in Jacobi geometry through the boundary of Poisson and Contact Geometry

**Abstract: **

Over the last decades both Poisson geometry and contact geometry have developed into full-fledged research areas of pure Mathematics with plenty of notable applications at the crossroads of Geometry and Physics. Contact, locally conformal symplectic (lcs) and Poisson geometries share a common extension, namely Jacobi geometry. Since it unifies several important geometric theories, Jacobi geometry has a built-in potential to highlight the rich interplay and powerful connection between Poisson and contact geometry. In the last years, I have been interested in unlocking this potential by addressing fundamental questions about Jacobi, Poisson, and related geometries using techniques from deformation theory, functional analysis, Lie theory and the geometry/algebra of PDEs. In this talk, I will try to briefly tell you more about my activity in this direction.

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**17:30: Closing session**

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