Room M005  Department of Mathematics (FCUP) PROGRAMME: MORNING SESSION  Algebra, Combinatorics and Number Theory  11:00h Antonio Macchia  Title Proper divisibility as a partially ordered set (*)  Abstract We define the order relation given by the proper divisibility of monomials, inspired by the definition of the Buchberger graph of a monomial ideal. From this order relation we obtain a new class of posets. Surprisingly, the order complexes of these posets are homologically nontrivial. We prove that these posets are dual CLshellable, we completely describe their homology (with integer coefficients) and we compute their Euler characteristic. Moreover this order relation gives the first example of a dual CLshellable poset that is not CLshellable. (*) joint work with Davide Bolognini, Emanuele Ventura and Volkmar Welker 
12:00h Alberto José Hernandez Alvarado  Title The Quotient Module, Coring Depth and Factorisation Algebras  Abstract In this conference, I will be reviewing the main aspects of my thesis dissertation. I will introduce the notion of depth of a ring extension $B\subseteqA$ and give several examples as well as important results of recent years. I will then consider a finite dimensional Hopf algebra extension $R \subseteq H$ and its quotient module $Q := H/R^+H$ and show that the depth of such an extension is intrinsically connected to the representation ring of $H, A(H)$. In particular, we will see that finite depth of the extension is equivalent to the quotient module $Q$ being algebraic in $A(H)$. Next, I will introduce entwining structures and use them to show that a certain extension of crossed product algebras is a Galois coring and use that to give a theoretical explanation for a result of S. Danz (2011). Finally, I will discuss factorisation algebras and their roll in depth, in particular a result on the depth of a Hopf algebra $H$ in its generalised factorised smash product with $Q^{*op}$.  ***LUNCH BREAK***  AFTERNOON SESSION  Algebra, Logic and Topology  14:30h Pier Giorgio Basile  Title A lax version of the EilenbergMoore adjunction (*)  Abstract In Category Theory there is a well developed theory of monads, proved to be very useful for 1dimensional universal algebra and beyond. The relation between adjunctions and monads was first noticed by Huber (Homotopy Theory in General Categories): every adjunction gives rise to a monad. Then, Eilenberg, Moore and Kleisli realized that every monad comes from an adjunction. In particular, Eilenberg and Moore (Adjoint Functors and Triples) realized that, for every monad T, there is a terminal adjunction (called EilenbergMoore adjunction) which gives rise to T. Category Theory can be also developed in a 2dimensional case, that is, considering not only morphisms between objects but also morphisms (usually called 2cells) between morphisms themselves. Thereby, one can study lax versions of the theory of monads. In the pseudo version, that is when we replace commutative diagrams by coherent invertible 2cells, the relation between biadjunctions and pseudomonads has been investigated by F. Lucatelli Nunes in the paper On Biadjoint Triangles as a consequence of the coherent approach to pseudomonads of S. Lack. The next step consists of studying the lax notion of monads, in which the associativity and identity works only up to coherent (not necessarily invertible) 2cells. In this talk we present a work in progress where we try to generalize to the laxcontext the classical result of EilenbergMoore. For this purpose, having in mind the notion of lax extension of monads introduced and studied in the context of Monoidal Topology (Metric, topology and multicategory: a common approach  M.M. Clementino and W. Tholen), we use a generalization of Gray's laxadjunction (see the monograph Formal Category Theory). Then, we show some steps of the construction leading to the positive answer. (*) joint work with Fernando Lucatelli Nunes 
15:30h Fernando Lucatelli Nunes  Title Kan construction of adjunctions  Abstract I will talk about a basic procedure of constructing adjunctions, sometimes called Kan construction/adjunction. In the first part of the talk, I will construct abstractly such adjunctions via colimits. In the second part, we give some elementary examples: fundamental groupoid, sheaves, etc. We assume elementary knowledge of basic category theory (definition of categories, colimits and Yoneda embedding). 
ABOUT THE SPEAKERS  1. Antonio Macchia is a PosDoc researcher working at the University of Coimbra in the area of "Algebra and Combinatorics". 2. Alberto José Hernandez Alvarado is a former PhD student of the Joint PhD Program UCUP working at the University of Porto in the area of "Algebra, Combinatorics and Number Theory" under the supervision of professor Lars Kadison. 3. Pier Giorgio Basile is a PhD student of the Joint PhD Program UCUP working at the University of Coimbra in the area of "Algebra, Logic and Topology" under the supervision of professor Maria Manuel Clementino. 4. Fernando Lucatelli Nunes is a PhD student of the Joint PhD Program UCUP working at the University of Coimbra in the area of "Algebra, Logic and Topology" under the supervision of professor Maria Manuel Clementino.
