Alexander Varchenko

Background

From 1964 to 1966 Varchenko studied at the Moscow Kolmogorov boarding school No. 18 for gifted high school students, where A. N. Kolmogorov and Ya. A. Smorodinsky were lecturing mathematics and physics. Varchenko graduated from Moscow State University in 1971. He was a student of V. I. 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.

Varchenko was an invited speaker at the International Congresses 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.

Research

In 1971, Varchenko proved that a family of complex quasi-projective algebraic sets with an irreducible base form a topologically locally trivial bundle over a Zariski open subset of the base. This statement, conjectured by O.Zariski, had filled up a gap in the proof of Zariski’s theorem on the fundamental group of the complement to a hypersurface published in 1937. In 1973, Varchenko proved R.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.

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. Using the formula, Varchenko constructed a counterexample to V. I. Arnold's semicontinuity conjecture that the brightness of light at a point on a caustic is not less than the brightness at the neighboring points.

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

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. Such an integral depends on the parameter – the value of the function. The integral has two properties: how fast it tends to zero, when the parameter tends to the critical value, and how the integral changes, when the parameter goes around the critical value. The first property was used to define the Hodge filtration of the asymptotic mixed Hodge structure and the second property was used to define the weight filtration.

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.

V. Schechtman and Varchenko identified in the KZ differential equations with a suitable Gauss-Manin connection and constructed multidimensional hypergeometric solutions of the KZ equations. In that construction the solutions were labeled by elements of a suitable homology group. Then the homology group was identified with a multiplicity space of the tensor product of representations of a suitable quantum group and the monodromy representation of the KZ equations was identified with the associated R-matrix representation. This construction gave a geometric proof of the Kohno-Drinfeld theorem on the monodromy of the KZ equations. A similar picture was developed for the qKZ type difference equations in joint works with G.Felder and V.Tarasov.

In the second half of 90s G.Felder, P.Etingof, and Varchenko developed the theory of dynamical quantum groups. Dynamical equations, compatible with the KZ type equations, were introduced in joint papers with G. Felder, Y. Markov, V. Tarasov. In applications, the dynamical equations appear as the quantum differential equations of the cotangent bundles of partial flag varieties.

In, E. Mukhin, V. Tarasov, and A. Varchenko proved the Shapiro conjecture 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.

It is classically known that the intersection index of the Schubert varieties in the Grassmannian of the N-dimensional planes coincides with the dimension of the space of invariants in a suitable tensor product of representations of the general linear group GL_N. In, E.Mukhin, V.Tarasov, and 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. As an application, they showed that if the Schubert varieties are defined with respect to distinct real osculating flags, then the varieties intersect transversally and all intersection points are real. This property is called the reality of Schubert calculus.

Books