The Schedule
20th of May
20th of May
10:15 - 10:30
Reception
Reception
André Carvalho and Beatriz Santos
10:30 - 11:20
Lifting of monotone-light factorizations
Lifting of monotone-light factorizations
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.As part of a project aiming to study categorical Galois theory for categorical structures, we study liftings of monoidal reflections and respective factorization systems to the categories of enriched categories. Of particular interest is the monoidal reflection ∆_⊗ → [0, ∞]^op of the quantale of distribution functions into the complete real half-line, and its monotone-light factorization for a tensor ⊗ satisfying certain properties. This lifts to a reflection ∆_⊗-Cat → [0, ∞]^op-Cat of probabilistic metric spaces (see [HR13]) into Lawvere metric spaces, which gives us a monotone-light factorization system in this context.
These results are achieved by expressing the various notions of reflections [CHK85] in terms of bilimits, and verifying when these bilimits are preserved by 2-functors. This is part of on-going joint work with Maria Manuel Clementino and Fernando Lucatelli Nunes.
References:
[CHK85] C. Cassidy, M. Hébert, G. M. Kelly. Reflective subcategories, localizations and factorization systems. J. Austral. Math. Soc. (Series A), 38:287–329, 1985.[CJKP97] A. Carboni, G. Janelidze, G. M. Kelly and R. Paré. On Localization and Stabilization for Factorization Systems. Applied Categorical Structures, 5:1–58, 1997.[Eil34] S. Eilenberg. Sur les transformations continues d’espaces métriques compacts. Fund. Math., 22:292–296, 1934.[HR13] D. Hofmann, C. D. Reis. Probabilistic metric spaces as enriched categories. Fuzzy Sets and Systems, 210:1-21, 2013.[Why50] G. T. Whyburn. Open mappings on locally compact spaces. Mem. Amer. Math. Soc. 1, 1950.
Break 11:20 - 11:30
11:30 - 12:20
On presheaf submonads of quantale enriched categories
On presheaf submonads of quantale enriched categories
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
Point-free measures: a localic approach to measure theory
Point-free measures: a localic approach to measure theory
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.Firstly, we will revise some basic concepts of σ-locales, and talk about the adjunction between the category of measurable spaces and measurable maps and the category of Boolean σ-locales and σ-localic maps. Then, we will generalize the standard definition of measure to an arbitrary lattice with countable suprema, and given a measure on a σ-locale X, we will extend it to the coframe of all σ-sublocales of X, S(X). Finally, we will show how this approach overcomes some classical paradoxes of measure theory: in more detail, we will show how to construct a measure defined, in particular, on all subsets of a space, and propose a suitable candidate for a natural model of the phenomenon of randomness.
References
[1] A. Simpson, Measure, randomness and sublocales, Ann. Pure Appl. Logic 163 (2012), 1642–1659.
Break 14:50 - 15:00
15:00 - 15:50
The Zeta function attached to the sum of three squares
The Zeta function attached to the sum of three squares
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.In this talk we shall discuss the zeta funcion attached to the representation of a number as a sum of three squares, a concrete example of a three-dimensional Epstein zeta function. In particular, we will indicate two different methods to bound the distance between consecutive zeros that this zeta function possesses at its critical line. We shall also discuss how the remaining zeros tend to "cluster" around other lines in the critical strip. Finally, we will argue why this zeta function acts as a bridge connecting two drastically different classes of Epstein zeta functions.
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].The mathematical model for the mechanical deformations is based on the on time-harmonic equations of linear elasticity in multilayer domains. The numerical solution is obtained through finite elements in a three-dimensional domain. The performance of the method is illustrated with numerical examples.
The mathematical model for solving this direct problem is the computational basis for addressing the inverse problem of determining the set of parameters that characterize the mechanical properties of the medium, knowing the field of displacements for a given excitation. We consider different optimization methods to solve the inverse problem and discuss their performance. We report several computational results which illustrate their behavior in terms of accuracy and efficiency. We will give some lights for future work.
References
[1] R.Henriques, Mathematical model to reconstruct the mechanical properties of an elastic medium, master thesis, http://hdl.handle.net/10316/93596, 2020.
Coffee Break
