The Schedule

Day 1

17th of February

10:30 - 10:45

Virtual Reception

André Carvalho

10:45 - 11:30

Lax factorisation systems and categories of partial maps

Leonardo Larizza

Abstract: Factorisation systems in category theory are structures describing morphisms in a category by factorising them into pairs of composable morphisms. These structures are strongly related to the orthogonality relation between morphisms, which entails the existence of some diagonal morphisms for certain squares. In this seminar we present the new notion of lax weak orthogonality between morphisms, which involves lax squares and the factorisation systems it generates. Then we will introduce lax versions of functorial and algebraic weak factorisation systems. These lax factorisation systems are discussed, keeping the theory of ordinary factorisation systems as a blueprint and providing useful properties. An overview of the examples of such lax factorisation systems is presented in the context of partial maps. We conclude with a discussion of general constructions of these examples and their description in the particular case of sets with partial maps.

Break 11:30 - 11:45

11:45 - 12:30

The frame of reals and a characterization of normal frames given by z-embedded sublocales

Ana Bélen Avilez García

Abstract: In classical Topology the reals with their usual topology and the continuous real-valued functions play an important role. In particular, they allow us to define zero and z-embedded sets, which give a characterization of normality: A topological space is normal if and only if every closed set is z-embedded. In this talk we will see the point-free version of this result. First we recall the category of frames and locales and present the frame of reals. This will allow us to define continuous real-valued functions in the point-free setting. We will also go through the notions of sublocale, normal frame and cozero element in order to arrive at the point-free characterization of normality given in terms of z-embedded sublocales.

Lunch 12:30 - 14:00

14:15 - 15:00

Tree sets and the Return Theorem

Herman Goulet-Ouellet

Abstract: In 2015, Berthé and her coauthors introduced the notion of extension graph. This proved to be a very useful device for linking several combinatorial and algebraic properties of uniformly recurrent languages, particularly so for the so-called tree sets. In this talk, we will present their approach, as well as one striking result it produced, dubbed the Return Theorem. If time permits, we will also discuss ongoing efforts to push this approach further.

15:00 - 15:45

Endomorphisms of the product of two free groups

André Carvalho

Abstract: In this talk we will describe the endomorphisms of the product of two free groups of finite rank and show how this description can be used to solve the Whitehead problems for endomorphisms, monomorphisms and automorphisms. The structure of the group of automorphisms for groups in this class will also be discussed and finiteness conditions on the fixed and periodic points subgroups will be given. Finally, we will briefly present some results on the dynamics of a continuous extension of an endomorphism to the completion of the group.

Break 15:45 - 16:00

16:00 - 16:30

Homological properties of Schur functors

Tiago Cruz

Abstract: In representation theory, Schur functors are a fundamental tool to relate representations of Schur algebras with representations of symmetric groups. In fact, many results of the representation theory of symmetric groups are obtained exploiting this connection.The aim of this talk is to discuss abstract frameworks of Schur--Weyl duality. These can be seen as algebraic analogues of resolution of singularities. At the centre of such frameworks are Schur functors. We will discuss some homological properties of Schur functors and their quality, including recent results for integral Schur algebras.

Day 2

24th of February

10:30 - 10:45

Virtual Reception

Carla Jesus

10:45 - 11:30

Internal and enriched category theory

Rui Prezado

Abstract: The main subject of an introductory course of category theory is usually the ordinary categories. However, in the realm of category theory, we find many more categorical structures of interest. This talk aims to introduce two very fundamental ones: the internal categories and the enriched categories. We also talk about an embedding result and sketch ideas towards the multi-categorical counterpart, which is a result recently proven in the context of my Ph.D. work on Grothendieck descent theory.

Break 11:30 - 11:45

11:45 - 12:30

Sublocale lattices and the T_D-duality

Igor Arrieta Torres

Abstract: A major difference between the category of locales (point-free spaces) and topological spaces is that subobject lattices in the latter are complete and atomic Boolean algebras, whereas in the former they are more complicated objects.In this talk we will see that given a locale, properties of (subobjects of) its sublocale lattice often characterize geometric properties of the locale itself. Using this approach, we will revisit some properties of the important T_D axiom of Aull and Thron.
References:[1] I. Arrieta, On joins of complemented sublocales, DMUC preprint 20-23.[2] I. Arrieta, A.L. Suarez, The coframe of D-sublocales of a locale and the T_D-duality, to appear in Topology and its Applications.

Lunch 12:30 - 14:00

14:15 - 15:00

Regularity Theory for Nonlinear Elliptic PDE

David Jesus

Abstract: This talk will consist of an introduction to regularity theory for fully nonlinear elliptic partial differential equations. We will introduce the concepts of compactness of solutions, stability results, approximation theorems and geometric iterations which together are able to produce improved regularity. We will also introduce some basic but essential definitions, such as the notion of viscosity solutions which is the primary working frame when studying equations of the fully nonlinear form

F(D²u) = f.

We will define and characterize Hölder spaces C^k,^α which are usually the aimed regularity. Finally, if time allows it, we will discuss some recent research, where C²^,α, regularity was obtained, without assuming that the operator F is neither convex nor differentiable, provided a suiting approximation regime is in force.

15:00 - 15:45

The use of splines in fractional differential equations

Carla Jesus

Abstract: Fractional partial differential equations can be used to represent anomalous diffusion that describes a wide range of phenomena in different fields such as finances [1] or biology [2]. When dealing with a fractional derivative operator, we face challenges such as non-locality and singularity. Previous to the introduction of linear splines in the context of fractional differential equations [3], the majority of the numerical methods were first order accurate. In this talk, we will introduce anomalous diffusion (subdiffusive and superdiffusive processes), fractional partial differential equations resorting to the fractional Riemann-Liouville derivatives for 0<α<1 and 1<α<2 and show how to use not only linear splines but also fractional splines in order to approximate this type of derivatives.
References[1] A. Cartea, D. del-Castillo-Negrete, Fractional di↵usion models of option prices in markets with jumps. Physica A: Statistical Mechanics and its Applications, 374 (2) (2007), 749–763.[2] A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez, K. Burrage, Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization, J. R. Soc. Interface, 11 (97) (2014).[3] E. Sousa. Numerical approximations for fractional diffusion equations via splines, Comput. Math. Appl., 62 (2011), 938–944.

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