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Wednesday, April 29, 2015

**Abstract:** For much of the 20th century group theory was viewed as a part of algebra and groups were treated primarily as algebraic objects. Geometric group theory, which developed into a distinct area of mathematics in early 1990s, aims to view finitely generated groups as geometric objects and to explore the relationship between intrinsic geometry of such a group on one hand and algebraic properties of this group on ther other hand. I will discuss, in broad terms, some of the key directions and ideas in modern geometric group theory, as well as examples of geometric properties and invariants of finitely generated groups. Time permitting, I will say something about recent applications of geometric group theory in low-dimensional topology, particularly the work of Agol and Wise solving the Virtual Haken Conjecture.