Alexander Varchenko

He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to singularity theory, real algebraic geometry, and the theory of quantum integrable systems".

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. Varchenko proved the existence of the bound in the infinitesimal 16th Hilbert problem.

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). In 1973 he received the Moscow Mathematical Society Award.

In 1971, Varchenko proved that a family of complex quasi-projective algebraic sets with an irreducible base forms a topologically locally trivial bundle over a Zariski open subset of the base. This statement, conjectured by Oscar Zariski, had filled up a gap in the proof of Zariski's theorem on the fundamental group of the complement to a complex algebraic hypersurface published in 1937. In 1973, Varchenko proved René Thom's conjecture that a germ of a generic smooth map is topologically equivalent to a germ of a polynomial map and has a finite dimensional polynomial topological versal deformation, while the non-generic maps form a subset of infinite codimension in the space of all germs.

In 1969 Varchenko identified the monodromy group of a critical point of type A n {\displaystyle A_{n}} of a function of an odd number of variables with the symmetric group S n + 1 {\displaystyle S_{n+1}} which is the Weyl group of the simple Lie algebra of type A n {\displaystyle A_{n}} .

From 1964 to 1966 Varchenko studied at the Moscow Kolmogorov boarding school No. 18 for gifted high school students, where Andrey Kolmogorov and Ya. A. Smorodinsky were lecturing mathematics and physics. Varchenko graduated from Moscow State University in 1971. He was a student of Vladimir Arnold. 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. From 1974 to 1984 he was a research scientist at the Moscow State University, in 1985–1990 a professor at the Gubkin Institute of Gas and Oil, and since 1991 he has been the Ernest Eliel Professor at the University of North Carolina at Chapel Hill.

Alexander Nikolaevich Varchenko (Russian: Александр Николаевич Варченко, born February 6, 1949) is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.