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

PROGRAMME:

 

11h15 - 12h00

Maryam Khaksar Ghalati

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

Azizeh Nozad

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

Muhammad Ali Khan

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.

 


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.

 

Azizeh Nozad is a Postdoctoral fellow at Faculty of Science, University of Lisbon.

 

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.

 

Place:   Room 2.4, DMat of University of Coimbra
URL:  https://sites.google.com/view/rspucupmath20162017/p%C3%A1gina-inicial
Start Date:   2017-06-01
Start Time:   11:15
See more:   <Main>   <Research Seminar Program - UC|UP MATH PhD Program>  
 
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