Proposed by
Michal Křížek 
Title: Geometry of Simplicial Finite Element Meshes
Description :
Ph.D. topic description
Description / Comments:
This Ph.D. topic focuses on geometric aspects of simplicial finite element methods and aims at several open problems such as regularity of the bisection algorithm in higher dimensions, construction of nonobtuse simplicial meshes in an arbitrary tetrahedron, decomposition of an arbitrary simplex into path simplicies, face-to-face simplicial tiling of the four dimensional space and finding the sufficient and necessary geometric condition for the convergence of the finite element method. See the attached description for more information.
Proposed by
Marcello Ortaggio
Title: Aspects of gravity and electromagnetism in higher-dimensional spacetimes
URL Proposal: not available
Description / Comments:
In recent years, there has been a significant increase in interest in the properties of gravity in more than four spacetime dimensions, motivated, e.g., by fundamental theories such as string theory, along with the idea of extra dimensions and braneworld models of TeV gravity. The study of higher-dimensional Lorentzian spaces is of interest also from a purely geometrical viewpoint.
Several higher-dimensional solutions of classical General Relativity have been known for some time, and various techniques to study exact solutions have been extended to an arbitrary number n of dimension. There are, however, general qualitatively new features when n>4. These concern, for example, the Goldberg-Sachs theorem, peeling properties of gravitational and electromagnetic fields, and the “uniqueness” of certain black holes solution within algebraically special spacetimes.
In relation to the above points, topics of interest for a PhD thesis include the asymptotic behaviour of the gravitational and electromagnetic field, and applications of classification schemes and general formalisms (such as Newman-Penrose, Geroch-Held-Penrose) to the study of exact solutions in higher dimensions.
Selected references:
A. Coley, R. Milson, V. Pravda, A. Pravdova, Classification of the Weyl tensor in higher dimensions, Class.Quant.Grav. 21 (2004) L35-L42
M. Ortaggio, V. Pravda and A. Pravdová, Algebraic classification of higher dimensional spacetimes based on null alignment, 2013 Class. Quantum Grav. 30 013001
G. Bernardi de Freitas, M. Godazgar, H. S. Reall, Uniqueness of the Kerr–de Sitter Spacetime as an Algebraically Special Solution in Five Dimensions, Commun. Math. Phys. 340 (2015) 291-323
M. Ortaggio, A. Pravdová, Asymptotic behaviour of the Weyl tensor in higher dimensions, Phys. Rev. D90 (2014) 104011
M. Ortaggio, V. Pravda, Electromagnetic fields with vanishing scalar invariants, Class. Quant. Grav. 33 (2016) 115010
