His research interests include the n-body problem, the Borda count voting system, and application of mathematics to the social sciences.
[4] For instance, as he has pointed out, plurality voting can lead to situations where the election outcome would remain unchanged if all voters' preferences were reversed; this cannot happen with the Borda count.
[4][6] Saari also applies similar methods to a different problem in political science, the apportionment of seats to electoral districts in proportion to their populations.
[SS][S85][S95b] In celestial mechanics, Saari's work on the n-body problem "revived the singularity theory" of Henri Poincaré and Paul Painlevé, and proved Littlewood's conjecture that the initial conditions leading to collisions have measure zero.
In his view, Arrow's impossibility theorem in voting theory, the failure of simple pricing mechanisms, and the failure of previous analysis to explain the speeds of galactic rotation stem from the same cause: a reductionist approach that divides a complex problem (a multi-candidate election, a market, or a rotating galaxy) into multiple simpler subproblems (two-candidate elections for the Condorcet criterion, two-commodity markets, or the interactions between individual stars and the aggregate mass of the rest of the galaxy) but, in the process, loses information about the initial problem making it impossible to combine the subproblem solutions into an accurate solution to the whole problem.
[11] Saari grew up in a Finnish American copper mining community in the Upper Peninsula of Michigan, the son of two labor organizers there.
Frequently in trouble for talking in his classes, he spent his detention time in private mathematics lessons with a local algebra teacher, Bill Brotherton.