10:15 - 10:30
André Carvalho and Beatriz Santos
10:30 - 11:20
Rui Prezado
Abstract: Let f : X → Y be a continuous map between compact Hausdorff spaces. We say f is monotone, respectively light, if for all y ∈ Y , the fiber f −1(y) is connected, respectively totally disconnected. These maps were studied by [Eil34] (for metric spaces) and [Why50], where they show that every continuous map f betweencompacy Hausdorff spaces admits a factorization f = g ◦ h where g is light and h is monotone, unique up to a unique isomorphism. This is the so-called monotone-light factorization of compact Hausdorff spaces, which may be described as the pullback-stabilization/localization of the factorization system induced by the reflection CHaus → Stn. In [CJKP97], this notion was explored for other reflections, with connections to Galois theory and Grothendieck descent theory.Break 11:20 - 11:30
11:30 - 12:20
Carlos Fitas
Abstract: I have been studying, together with my advisor, submonads of the presheaf monad and their algebras. After recalling a few fundamental notions about V-Cat, V-Rel and V-Dist, where V is a quantale, we introduced two new characterizations of the submonads of the presheaf monad: one in terms of a special class of V-distributors; and another as those monads which are fully (BC)*, lax idempotent and satisfy certain fully faithful conditions. By (BC)* we mean a new Beck-Chevalley type condition which gives us an interaction between V-Cat and V-Dist, analogous to the interaction between Set and Rel given by the usual Beck-Chevalley condition on Set. The algebras for these submonads are the V-categories satisfying a condition involving their multiplication. In the special case of the formal ball monad, the algebras can also be seen as the V-categories with a particular class of weighted colimits.Lunch 12:20 - 14:00
14:00 - 14:50
Raquel Bernardes
Abstract: Motivated by the fact that σ-locales generalize measurable spaces, and seeking to overcome some limitations of measure theory, Alex Simpson [1] proposed a new way of dealing with the problem of measuring subsets, via an approach to measure theory in the framework of point-free topology. This talk will be a general review of Simpson’s work, emphasizing his motivation, the idea behind his approach, as well as its advantages.Break 14:50 - 15:00
15:00 - 15:50
Pedro Ribeiro
Abstract: Perhaps since Pythagoras or Diophantus, the study of the representation of a given integer as a sum of squares has been a long standing mathematical interest. Due to its physical resonance, the sum of three squares, or the number of lattice points in a sphere, seems to deserve special attention.Break 15:50 - 16:00
16:00 - 16:50
Mathematical model to reconstruct the mechanical properties of an elastic medium
Rafael Henriques
Abstract: In this talk, a mathematical model to reconstruct the mechanical properties of an elastic medium is presented, in view of contributing to the resolution of the problem of optical coherence elastography [1].Coffee Break