Since the institution at Novosibirsk had not yet been fully credentialed, he had defended his Ph.D. thesis at Moscow State University in 1958, on a subject in Riemann spaces.

In 1961, Toponogov became a professor at a newly created Institute of Mathematics and Computing in Novosibirsk affiliated with the state university.

Toponogov's scientific interests were influenced by his advisor Abram Fet, who taught at Tomsk and later at Novosibirsk.

In 1995 Toponogov made the conjecture:[2] On a complete convex surface S homeomorphic to a plane the following equality holds: where

are the principal curvatures of S. In words, it states that every complete convex surface homeomorphic to a plane must have an umbilic point which may lie at infinity.

[3][4] In the same paper, Toponogov proved the conjecture under either of two assumptions: the integral of the Gauss curvature is less than