This meeting aims to gather all CMUP's members, especially its post-docs, so that they can briefly present their current research.

Date, time and place:

27 of April, from 14:30 to 18:00.

Room FC1.031.

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/E9mGvOmpWvfgl3hH2

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Speakers (and their group / line membership):

• Marco Martins Afonso (Analysis / Mathematical Models and Applications)
• Sylvain Lavau (Geometry)
• Claude Marion (Algebra)
• Sebastián Pérez (Geometry / Dynamical Systems)
• Alexander Zakharov (Algebra / Semigroups, Automata and Languages)

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Schedule:

14:30-14:35 -- Opening session (Jorge Freitas)

14:35-15:10 -- Claude Marion (Algebra)

15:10-15:45 -- Marco Martins Afonso (Analysis / Mathematical Models and Applications)

15:45-16:20 -- Sebastián Pérez (Geometry / Dynamical Systems)

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

16:50-17:25 -- Sylvain Lavau (Geometry)

17.25-18:00 -- Alexander Zakharov (Algebra / Semigroups, Automata and Languages)

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Titles and abstracts:

Claude Marion -- On generation of finite simple groups

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

(i) Can we impose some restrictions on a pair of generators? For example, given a triple (a,b,c) of positive integers, what are the finite simple groups that can be generated by two elements of respective orders dividing a and b, and having product of order dividing c? In other words, what are the finite simple groups that are a quotient of the triangle group T =Ta,b,c =⟨x,y,z: x^a =y^b =z^c =xyz=1⟩?

(ii) Given a finite group G, the generating graph Γ(G) of G has as set of vertices the non- trivial elements of G and two vertices of Γ(G) are joined by an edge if and only if they are distinct and generate G as group elements. What are the graph-theoretical properties of the generating graph of a finite simple group?

(iii) 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 questions.

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Marco Martins Afonso -- Applications of Mathematics in Fluid Dynamics

We show some results obtained analytically in fluid dynamics and turbulent flows, with special emphasis on the mathematical tools and concepts employed. While a part of them is of common usage in statistical physics, a few others are met less frequently: multiple-scale expansions, Furutsu-Novikov-Donsker theorem in functional analysis, multivariate Hermite polynomials with second quantization, Cayley-Hamilton theorem for time-ordered matrix exponentials, multi-fractals, Heun equation and continued fractions, et cetera. In particular we focus on the issue of particle transport by a velocity field known deterministically or stochastically, and we study the complex interplay of the several control parameters of the problem.

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Sebastián Pérez -- A travel from the hyperbolic to the non-hyperbolic world

A fundamental problem in differentiable dynamical systems is to describe the dynamics of open sets in the space of all dynamical systems, interesting either in terms of their own rich mathematical structure or relevance to other areas of sciences. In this sense, the hyperbolic theory has been extremely
successful: hyperbolicity is a geometric, topological and statistical approach of dynamical system and they is in core of the chaotic dynamics, one of the main paradigms of current science. For these reasons is that understand the "scope" of the hyperbolic systems among dynamical systems is one of the most important problems in differentiable dynamics. On the other hand, there exist two main prototypical configurations of non-hyperbolic dynamics in open sets,  namely, heterodimensional cycles and homoclinic tangencies.
In each case one begin with a special hyperbolic diffeomorphism and carefully modified (perturbations, homotopy, etc ..) it to move into the complement of the closure of the hyperbolic diffeomorphisms set. In this talk we will explain this passage from the hyperbolic to the non-hyperbolic world through an example involving cycles.

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Sylvain Lavau -- Zoology of Lie infinity-algebras

A Lie infinity-algebra is a homotopical generalization of a Lie algebra. This notion is an important tool in geometry, algebraic geometry and mathematical physics. We will present the main notions of graded algebra underlying this concept, and give some examples of Lie infinity algebras that are found very naturally.

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Alexander Zakharov -- On finitely generated submonoids of free groups

Free groups are some of the most basic and important examples of groups. All subgroups of free groups are free, by a theorem of Nielsen and Schreier, and finitely generated ones have nice algorithmic properties, since they can be represented by Stallings automata. However, the structure of finitely generated submonoids of free groups is much more complicated. In particular, these include all finitely generated submonoids of free monoids, which can be even not finitely presented, let alone free. The isomorphism problem is one of the most natural algorithmic questions about groups or monoids. We solve the isomorphism problem for a big class of submonoids of free groups, which can be described in a few different ways. This is joint work with Pedro Silva.

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Organization: CMUP Direction Board (Alexandre Rodrigues, Ana Moura, and André Oliveira).