Geometric group theory, topology, mathematical logic and algorithms

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Recent developments in pure mathematics demonstrate the power of an interdisciplinary approach to problem solving. Borrowing ideas and techniques from different, sometimes fairly distant, areas of mathematics helped prove long-standing conjectures that seemed to be out of reach. Yet, even more importantly, when various mathematical traditions and ways of thinking are brought together, they influence the development of both new and previously existing branches of research, increase the power of mathematics as a whole and enhance its beauty. ***The modern study of infinite groups brings several areas of mathematics into contact with group theory. The theme of my research is the interplay between the theory of infinite discrete groups, mathematical logic, geometry and topology.***My proposed research program is aimed at the development of techniques to study the elementary theory of relatively hyperbolic groups and related questions. While groups were introduced as algebraic objects, mathematical logic studies groups from the point of view of their elementary theory; that is, the focus is on the set of all first order sentences in the language of group theory which are true in a given group. A natural question is to what extent the elementary theory allows one to distinguish algebraic properties of groups. Borrowing tools from geometry and topology proved extremely useful in answering questions of this kind.****My interest in this field of research has been motivated by several famous open questions asking about connections between group theory, geometry and logic. Research in this direction has a strong influence on the development of geometric group theory, mathematical logic and related areas of pure mathematics.**