Abstract: In my lecture I will present joint work with Misha Kapovich and Bernhard Leeb. I will begin by describing the GENERALIZED TRIANGLE INEQUALITIES. A geodesic segment in a symmetric space of noncompact type and rank m is determined up to isometry by m real numbers (not just the usual length). The generalized triangle inequalities are a system of homogeneous linear inequalities that give conditions on 3 m-tuples of real numbers that are necessary and sufficient in order that one can assemble three geodesic segments with these parameters into a triangle. It is a remarkable fact that the triangle inequalities play a fundamental role in some important problems from algebra. Indeed, given a triple of dominant weights a,b,c (now we have 3 m-tuples of decreasing INTEGERS), in order that the corresponding triple tensor product of finite dimensional irreducible representations contain the trivial representation it is necessary that the triple a,b,c satisfies the triangle inequalities. It is a well-known recent theorem of Knutson and Tao (the Saturation Conjecture for GL(m)) that the triangle inequalities are also SUFFICIENT for the group GL(m). However for other groups they are no longer sufficient. I will present a conjecture about what happens for other groups. The main evidence for the conjecture is that there is also a "Saturation Conjecture" for the (spherical) Hecke algebra which I will prove (I will explain what this algebra is) . There is a very close connection between the two conjectures and in fact they are equivalent for GL(m). From this equivalence I will deduce a new proof of the theorem of Knutson and Tao.