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Proposed theses topics

Proposed by Alexander Zuevsky

Title: Vertex algebras and modular forms in geometric correspondence  

URL Proposal:  not available

Description / Comments:
We would like to clarify further the role of vertex algebra techniques in geometric correspondences. Applications in computation of higher genus characters for vertex operator algebras are also expected. Number theory consequences follow.

Proposed by Alexander Zuevsky

Title: Algebraic approach to exactly solvable models  

URL Proposal:  not available

Description / Comments:
Lie and vertex algebras background in constructing and solving models in Mathematical Physics.

Proposed by Alexander Zuevsky

Title: Indentities for automorphic forms originating from vertex algebras  

URL Proposal:  not available

Description / Comments:
We would like to study identities in Number theory originating from construction and computation of characters for vertex algebras on higher genus Riemann surfaces.

Proposed by Miroslav Rozložník   

Title: Analysis and implementation of iterative methods for solving saddle-point systems in applications  

Description :    PhD topic description

URL Proposal:  not available

Description / Comments:
This topic focuses on the analysis of the convergence and numerical behavior of iterative methods for solving large-scale saddle-point systems that arise in various applications such as discretizations of partial differential equations via mixed formulations of the finite element method or nonlinear programming with equality constraints. A large amount of work has been devoted to a wide selection of solution techniques for saddle-point systems varying from the fully direct approach through the use of iterative stationary methods up to the combination of direct and iterative techniques including preconditioned iterative schemes. For an excellent survey on applications, solution methods and results on algebraic solution of saddle-point systems, we refer to the paper by M. Benzi, G. H. Golub and J. Liesen, and numerous reference therein. Significantly less attention has been, however, paid to the numerical behavior of such schemes. In this work we would like to concentrate not only on the effects of rounding errors but also on the analysis of inexact Krylov subspace methods, where some computations in the solution schemes are due to savings intentionally approximated and performed with some degree of inexactness. This may have a substantial effect on the behavior of such solution techniques. We believe that careful analysis of these effects can contribute to better understanding of relaxation strategies that are successfullyb used in various applications.


References:

M. Benzi, G. H. Golub, J. Liesen. Numerical solution of saddle point problems. Acta Numerica, 2005, pp. 1– 137.

P. Jiránek, M. Rozložník, Maximum Attainable Accuracy of Inexact Saddle Point Solvers, SIAM Journal on Matrix Analysis and Applications 29 (4), 2008, 1297-1321.

M. Rozložník: Saddle point problems, iterative solution and preconditioning: a short overview, Proceedings of the XV-th Summer School Software and Algorithms of Numerical Mathematics, I. Marek ed., University of West Bohemia, Pilsen, 97-108 (2003).

M. Rozložník, V. Simoncini, Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning, SIAM Journal on Matrix Analysis and Applications 24 (2), 2002, 368-391.

Proposed by Tomas Masopust   

Title: Computational and Descriptional Complexity Questions in Discrete-Event Systems  

URL Proposal:  not available

Description / Comments:
The aim of this PhD thesis is to investigate descriptional and computational complexity questions of problems arising in discrete-event systems, such as supervisory control, detectability, opacity, etc.

Please do not hesitate to contact me for more details or with requests for diploma theses.

Proposed by Tomáš Vejchodský   

Title: Deterministic and stochastic modeling in molecular and cell biology  

Description :    Diploma thesis (in Czech)

URL Proposal:  not available

Description / Comments:
This diploma thesis will concern the deterministic and stochastic modeling of chemical systems in terms of ordinary differential equations (ODE) and stochastic simulation algorithms (SSA). The motivation comes from the biology, from the mathematical modeling of gene regulatory networks, cell cycle, circadian rhythms, etc.
Goals of the thesis will specified according to the individual interests of the student. It is possible to concentrate on the numerical methods for partial differential equations (PDE), mathematical modeling, analysis of PDE, or to the stochastic analysis.
The co-advisor of the thesis will be Dr Radek Erban (University of Oxford, UK) and possibly others. There is a possibility of a visit of the group of Dr Radek Erban in Oxford. http://people.maths.ox.ac.uk/erban/
For more information see the attached flier (in Czech).

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

Proposed by Michal Křížek   

Title: Geometry of Simplicial Finite Element Meshes  

Description :    Ph.D. topic description

URL Proposal:  not available

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.