**Gordan Savin**

University of Utah, Salt Lake, USA

December 31, 2009

**Geometry of binary hermitian forms**:
In a remarkable booklet `The sensual (quadratic) form', J. H. Conway introduces an elegant approach to integral binary quadratic forms. The main tool is a 3 valence tree which is in fact an SL_2(Z)-equivariant retract of the 2-dimensional hyperbolic space.
Let A be the ring of integers of a quadratic imaginary extension
of the field of rational numbers. The 3-dimensional hyperbolic
space has a 2-dimensional, GL_2(A)-invariant retract. The retract is a CAT(0) complex. In this lecture I will explain how the retract can be used to derive some results on hermitian binary forms over the ring A.

This is a joint work with Mladen Bestvina.