[1] Varchenko defended his Ph.D. thesis Theorems on Topological Equisingularity of Families of Algebraic Sets and Maps in 1974 and Doctor of Science thesis Asymptotics of Integrals and Algebro-Geometric Invariants of Critical Points of Functions in 1982.

Using the semicontinuity, Varchenko gave an estimate from above for the number of singular points of a projective hypersurface of given degree and dimension.

[7] Varchenko introduced the asymptotic mixed Hodge structure on the cohomology, vanishing at a critical point of a function, by studying asymptotics of integrals of holomorphic differential forms over families of vanishing cycles.

[8] The second part of the 16th Hilbert problem is to decide if there exists an upper bound for the number of limit cycles in polynomial vector fields of given degree.

The infinitesimal 16th Hilbert problem, formulated by V. I. Arnold, is to decide if there exists an upper bound for the number of zeros of an integral of a polynomial differential form over a family of level curves of a polynomial Hamiltonian in terms of the degrees of the coefficients of the differential form and the degree of the Hamiltonian.

This construction gave a geometric proof of the Kohno-Drinfeld theorem [11][12] on the monodromy of the KZ equations.

[13][14] The weight functions appearing in multidimensional hypergeometric solutions were later identified with stable envelopes in Andrei Okounkov's equivariant enumerative geometry.

[15][16] In the second half of 90s Felder, Pavel Etingof, and Varchenko developed the theory of dynamical quantum groups.

It is classically known that the intersection index of the Schubert varieties in the Grassmannian of N-dimensional planes coincides with the dimension of the space of invariants in a suitable tensor product of representations of the general linear group

Varchenko was an invited speaker at the International Congress of Mathematicians in 1974 in Vancouver (section of algebraic geometry) and in 1990 in Kyoto (a plenary address).