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Research Seminar Program (RSP) - UC|UP Joint PhD Program in Mathematics
 
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Description:

Room M005 - Department of Mathematics (FCUP)

 

PROGRAMME:

 

MORNING SESSION - Algebra, Combinatorics and Number Theory

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11:00h Antonio Macchia

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Title Proper divisibility as a partially ordered set (*)
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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 non-trivial. We prove that these posets are dual

CL-shellable, 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 CL-shellable poset that is not CL-shellable.

(*) joint work with Davide Bolognini, Emanuele Ventura and Volkmar Welker


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12:00h Alberto José Hernandez Alvarado

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Title The Quotient Module, Coring Depth and Factorisation Algebras
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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

$Q^{*op}$.

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***LUNCH BREAK***

 

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AFTERNOON SESSION - Algebra, Logic and Topology

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14:30h Pier Giorgio Basile

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Title A lax version of the Eilenberg-Moore adjunction (*)
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Abstract In Category Theory there is a well developed theory of monads, proved to

be very useful for 1-dimensional 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 Eilenberg-Moore adjunction) which gives rise to T.

Category Theory can be also developed in a 2-dimensional case, that is, considering

not only morphisms between objects but also morphisms (usually called 2-cells)

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 2-cells, 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) 2-cells. In this talk we present a

work in progress where we try to generalize to the lax-context the classical result of

Eilenberg-Moore. 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 lax-adjunction (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


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15:30h Fernando Lucatelli Nunes

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Title Kan construction of adjunctions
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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).


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ABOUT THE SPEAKERS
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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 UC|UP 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

UC|UP 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 UC|UP working at the University of Coimbra in the area

of "Algebra, Logic and Topology" under the supervision of professor

Maria Manuel Clementino.

Place:   Room M005 - Department of Mathematics - FCUP
Start Date:   2016-11-25
See more:   <Main>   <Research Seminar Program - UC|UP MATH PhD Program>  
 
Attached Files
 
File Description
Poster 1 /event/events/RSPPoster.pdf
Poster 2 /event/events/PosterRSP2.pdf
Number of registers: 2.1
     
© 2012 Centre for Mathematics, University of Coimbra, funded by

Science and Technology Foundation

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