2016-2017 Sessions
1st of June, 2017 - Room 2.4, DMat of University of Coimbra
11h15 - 12h00
Speaker: Maryam Khaksar Ghalati¹
Title: Two mathematical approaches in ophthalmology
Abstract: In this talk, I will present two mathematical approaches in ophthalmology. The first approach is in the light scattering effect in retina. The discontinuous Galerkin solutions of Maxwell's equations are introduced as an efficient tool in order to simulate the full complexity of the retina. I will present the numerical results that shows the stability and convergency of the numerical methods. The efficiency of our method is examined by simulating the light scattering in a 2D domain which tries to represent a single nucleus of the outer nuclear layer of the retina. The second approach is in the visual optics and using the special functions in ophthalmic surface modelling. The ocular aberrations are commonly described in terms of a series of Zernike polynomials that offer distinct advances due to their normalization on a circular pupil. However, in certain cases with slow convergence they may not be the most appropriate choice. I will show numerical results comparing the fitting accuracy of Zernik polynomials with Bessel circular functions for different surfaces. This comparison will extend to other class of functions and among them Chebyshev polynomials seems applicable in modelling ophthalmic surfaces.
Lunch Break
14h00 - 14h50
Speaker: Azizeh Nozad²
Title: Reducibility of nilpotent cone for G-Higgs bundle moduli space
Abstract: Let G be a non-compact reductive Lie group with a choice of a maximal compact subgroup H ⊂ G and let X be a compact oriented surface of genus g ≥ 2. A G-Higgs bundle is a pair of a holomorphic vector bundle together with a Higgs field which is a section of a special vector bundle. We study the obstructions to a deformation retraction from the moduli spaces of G-Higgs bundles to the moduli space of semistable H-principal bundles over X. The main idea is to use the C∗-action on the moduli space of G-Higgs bundles, which is given by multiplication on the Higgs field, and to find a fixed point with unstable underlying principal bundle. This also allows the study of the reducibility of the nilpotent cone of the moduli spaces of G-Higgs bundles.
14h50 - 15h40
Speaker: Muhammad Ali Khan³
Title: Statistical instability for the contracting Lorenz flow
Abstract: We consider a one parameter family of one-dimensional maps, introduced by Rovella, obtained through modifying the eigenvalues λ2< λ3< 0 < λ1 of the geometric Lorenz attractor, replacing the expanding condition λ3+ λ1> 0 by a contracting one λ3+ λ1< 0. By referring the techniques of Benedicks-Carleson, Rovella proved that there exists a positive Lebesgue measure set of parameters, the so called Rovella parameters, for which the corresponding map has a positive Lyapunov exponent in the critical value. Later on, Metzger proved the existence of a unique absolutely continuous (with respect to Lebesgue) invariant probability measures (SRB) for those maps. Recently, Alves and Soufi showed that on the set of Rovella parameters those maps are strongly statistically stable, i.e. the function which maps each Rovella parameter to the density of the SRB measure is continuous (in the L1-norm). In our work we show that if we add parameters with an attracting periodic orbits to Rovella parameters, then we do not have statistical stability in this extended set of parameters.
¹ Maryam Khaksar Ghalati is a former PhD student of the Joint PhD Program UC|UP, in the area “Numerical Analysis and Optimization” under the supervision of professors Adérito Araújo and Sílvia Barbeiro.
² Azizeh Nozad is a Postdoctoral fellow at Faculty of Science, University of Lisbon.
³ Muhammad Ali Khan is a PhD student of the Joint PhD Program UC|UP, working at the University of Porto, in the area "Dynamical Systems" under the supervision of professor José Ferreira Alves.
25th of November, 2016 - Room M005, DMat of University of Porto
10h00 - 11h00
Speaker: Antonio Macchia¹
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 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.
11h00 - 12h00
Speaker: Alberto José Hernandez Alvarado²
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
14h00 - 15h00
Speaker: Pier Giorgio Basile³
Title: A lax version of the Eilenberg-Moore adjunction**
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.
15h00 - 16h00
Speaker: Fernando Lucatelli Nunes¹¹
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).
¹ Antonio Macchia is a Postdoctoral researcher at University of Coimbra in the area Combinatorial Commutative Algebra.
(*) joint work with Davide Bolognini, Emanuele Ventura and Volkmar Welker.
² Alberto José Hernandez Alvarado is a former student of the UC|UP Joint PhD Program in Mathematics, he worked under the supervision of Prof. Lars Kadison.
³ Pier Giorgio Basile is a student of the UC|UP Joint PhD Program in Mathematics, working on Category Theory under the supervision of Prof. Maria Manuel Clementino.
(**) joint work with Fernando Lucatelli Nunes.
¹¹ Fernando Lucatelli Nunes is a student of the UC|UP Joint PhD Program in Mathematics, working on Category Theory under the supervision of Prof. Maria Manuel Clementino.