13 Facts About Alexander Varchenko

Alexander Nikolaevich Varchenko is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.

From 1964 to 1966 Varchenko studied at the Moscow Kolmogorov boarding school No 18 for gifted high school students, where Andrey Kolmogorov and Ya.

In 1969 Alexander Varchenko identified the monodromy group of a critical point of type of a function of an odd number of variables with the symmetric group which is the Weyl group of the simple Lie algebra of type.

In 1971, Alexander 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.

In 1973, Alexander Varchenko proved Rene 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.

Alexander Varchenko was among creators of the theory of Newton polygons in singularity theory, in particular, he gave a formula, relating Newton polygons and asymptotics of the oscillatory integrals associated with a critical point of a function.

Alexander Varchenko formulated a conjecture on the semicontinuity of the spectrum of a critical point under deformations of the critical point and proved it for deformations of low weight of quasi-homogeneous singularities.

Alexander 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.

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

In, Evgeny Mukhin, Tarasov, and Alexander Varchenko proved the conjecture of Boris Shapiro and Michael Shapiro in real algebraic geometry: if the Wronski determinant of a complex finite-dimensional vector space of polynomials in one variable has real roots only, then the vector space has a basis of polynomials with real coefficients.

In, Mukhin, Tarasov, and Alexander Varchenko categorified this fact and showed that the Bethe algebra of the Gaudin model on such a space of invariants is isomorphic to the algebra of functions on the intersection of the corresponding Schubert varieties.

Alexander Varchenko was an invited speaker at the International Congress of Mathematicians in 1974 in Vancouver and in 1990 in Kyoto.

Alexander Varchenko 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".