# Mathematics Colloquia and Seminars

\no This is a survey of my joint work with Bernhard Leeb and John Millson. \no Everybody knows how to construct triangles with the prescribed side-lengths $\al_1$, $\al_2$, $\al_3$ in the Euclidean plane: The necessary and sufficient conditions for this are the usual triangle inequalities $\al_i \le \al_j+\al_k$. In this talk I will explain how to solve (in a unified fashion) the analogous problem for other geometries $X$: nonpositively curved symmetric spaces (and their infinitesimal analogues)and Euclidean buildings. The notion of side-length'' in this generality becomes more subtle: {\em side-lengths} are elements of the appropriate Weyl chamber $\Delta$. One of the suprising results is that the generalized triangle inequalities'' for $X$ determine a polyhedral cone $D_3(X)\subset \Delta^3$, which depends on $X$ and on the type of geometry only weakly: $D_3(X)$ is completely determined by the finite Coxeter group corresponding to $X$. The linear inequalities describing $D_3(X)$ are determined by the Schubert calculus'' (computing the integer cohomology ring) in the associated generalized flag varieties. Our techniques for proving these results about $D_3(X)$ are mostly geometric (with a bit of dynamics): By relating triangles with weighted configurations at infinity'', the idea which goes back to Gauss.