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In this paper, we describe a connection that exists among (a) the number ofsingular points along the trajectory of Toda flow, (b) the cohomology of acompact subgroup $K$, and (c) the number of points of a Chevalley group$K({\mathbb F}_q)$ related to $K$ over a finite field ${\mathbb F}_q$. The Todalattice is defined for a real split semisimple Lie algebra $\mathfrak g$, and$K$ is a maximal compact Lie subgroup of $G$ associated to $\mathfrak g$.Relations are also obtained between the singularities of the Toda flow and theintegral cohomology of the real flag manifold $G/B$ with $B$ the Borel subgroupof $G$ (here we have $G/B=K/T$ with a finite group $T$). We also compute themaximal number of singularities of the Toda flow for any real split semisimplealgebra, and find that this number gives the multiplicity of the singularity atthe intersection of the varieties defined by the zero set of Schur polynomials.Comment: 28 pages, 5 figure

Toda lattice, cohomology of compact Lie groups and finite Chevalley groups