Research projects

Overview

My research includes graph theory, algorithms, complexity theory, algebra, linear algebra and scientific education. I am working on the following research projects:

- Geometric Representations of Graphs. Existence of many geometric representations implies highly restrictive structure of graphs. We study the partial represenation extension problem, other restricted representation problems and structural implications of these representations.
- Symmetries of Graphs and The Graph Isomorphism Problem. Symmetries are described by automorphism groups. We have proved Jordan-like inductive characterization of automorphism groups for several graph classes of geometrically represented graphs including interval graphs and planar graphs. We also study the graph isomorphism problem restricted by lists.
- Algorithms and Complexity Theory. We study complexity of computational problems related to these topics and decide whether they are polynomial-time solvable or NP-hard.
- Energy-aware Computing and Numerical Linear Algebra. One of important problem for supercomputing is to make computations as efficient with respect to energy consumption as possible. The main difference is that some operations (e.g. comunication) consume much more energy than other operations (e.g. local computation). Our study compared different methods for solving linear systems Ax=b from the point of energy consumption.
- Scientific Education and Understanding of Human Mind. We build a mathematical theory and computer tools to write down, understand and share structures in our mind. We are testing this approach in education of linear algebra at Charles University. In years 2015 to 2017, we have been running an examination experiment testing understanding of Big Picture in linear algebra of freshmen students.