Numerical Linear Algebra and Discrete Structures

Numerical linear algebra is a modern area of mathematics which is extensively evolving due to the need to adapt and develop numerical algorithms for implementation in the modern information technology. Algorithm is considered efficient if it is selected in accordance with the specific structure of the given problem. Therefore, the first step in the successful implementation of numerical algorithms, is detailed analysis of the benefits caused by the specific structure of the observed problems. In the case of linear problems, investigation of matrix properties based on some of its special structures will be the subject of research for this project. In the case of discrete structures, the study will focus on positional games on graphs, investigations of various aspects of the algebraic theory of semigroups, and of combinatorial, algebraic and model-theoretical properties of homogeneous discrete structures, characterized by a high level of symmetry. The results pertaining to winning strategies in positional games are expected. And new insights into the structure of varieties of semigroups and related algebraic structures are expected, too, with the objective to characterize homomorphism-homogeneity, the Bergman property and other similar properties in various classes of discrete structures.