Geometric methods in group theory and group-based cryptography

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My research is in Pure Mathematics, more precisely, in Geometric Group Theory. This flourishing and rapidly developing field of study attracts many brilliant researchers. It is interdisciplinary in nature. It is rooted in the work of many mathematicians. It started getting its shape in the 1980s when Gromov clearly articulated the idea that infinite groups, usually regarded as algebraic objects, can be characterized by their geometric properties, and that tools from topology can be used in group theory. Since then, research in this area made its way also to harmonic analysis, logic, probability and computer science. A wealth of available techniques and potential applications ensures that the field will remain a mainstream in mathematics for a long while. There are two strands in geometric group theory: groups can be studied via their intrinsic geometry, or via their actions on appropriate spaces. In my research I am interested in both. The intrinsic geometry of a group reveals algebraic, algorithmic and asymptotic properties of the group. Group actions on spaces, such as real trees or tree-graded spaces, allow one to gain insight into the structure of the group. It is fascinating to see how quickly the development of new methods leads to striking applications, both in pure mathematics and in the real life. An example of the former is recent solution to the following two Tarski's problems from the 1940s: Are the 2-generated free group and the 3-generated free group elementarily equivalent? Is the elementary theory of a free group decidable? On the practical side, the interest in these methods is motivated by the needs of the group-based cryptography. These seemingly very different areas of research can be approached using the versatile theory of solving systems of equations and inequations over groups. A canonical description of the solution set is of importance to a logician, while a cryptographer would focus on decidability and complexity. My interest and passion lie in the study of the systems of equations over relatively hyperbolic groups, using a variety of available techniques from different areas, borrowing existing methods and developing new ones.