Alexander Varchenko
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Alexander Nikolaevich Varchenko (russian: Александр Николаевич Варченко, born February 6, 1949) is a Soviet and Russian mathematician working in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
,
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
.


Education and career

From 1964 to 1966 Varchenko studied at the Mosco
Kolmogorov boarding school No. 18
for gifted high school students, where
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
an
Ya. A. Smorodinsky
were lecturing mathematics and physics. Varchenko graduated from
Moscow State University M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...
in 1971. He was a student of
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–A ...
. 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 The University of North Carolina is the multi-campus public university system for the state of North Carolina. Overseeing the state's 16 public universities and the NC School of Science and Mathematics, it is commonly referred to as the UNC Sy ...
at
Chapel Hill Chapel Hill or Chapelhill may refer to: Places Antarctica * Chapel Hill (Antarctica) Australia *Chapel Hill, Queensland, a suburb of Brisbane *Chapel Hill, South Australia, in the Mount Barker council area Canada * Chapel Hill, Ottawa, a neighbo ...
.


Research

In 1969 Varchenko identified the monodromy group of a critical point of type A_n of a function of an odd number of variables with the symmetric group S_ which is the Weyl group of the simple Lie algebra of type A_n. 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 , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...
, had filled up a gap in the proof of Zariski's theorem on the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the complement to a complex algebraic
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
published in 1937. In 1973, Varchenko proved
René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w ...
'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 polygon In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of interest was ''essentially'' the field of formal Lau ...
s in singularity theory, in particular, he gave a formula, relating Newton polygons and asymptotics of the
oscillatory integral In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators fo ...
s 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 In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. 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 cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
s 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. Vadim Schechtman and Varchenko identified in the
Knizhnik–Zamolodchikov equations In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affin ...
(or, KZ equations) with a suitable
Gauss–Manin connection In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s ...
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
quantum KZ equations In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a co ...
(or, qKZ-type difference equations) in joint works with Giovanni Felder and Vitaly Tarasov. The weight functions appearing in multidimensional hypergeometric solutions were later identified with stable envelopes in
Andrei Okounkov Andrei Yuryevich Okounkov (russian: Андре́й Ю́рьевич Окунько́в, ''Andrej Okun'kov'') (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematic ...
's equivariant enumerative geometry. In the second half of 90s Felder, Pavel 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, Evgeny Mukhin, Tarasov, and Varchenko proved the conjecture of Boris Shapiro and Michael Shapiro in
real algebraic geometry In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial ...
: if the
Wronski determinant In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian o ...
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 In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
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 \operatorname_N. In, Mukhin, 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 In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of ...
.


Recognition

Varchenko was an invited speaker at the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
in 1974 in
Vancouver Vancouver ( ) is a major city in western Canada, located in the Lower Mainland region of British Columbia. As the List of cities in British Columbia, most populous city in the province, the 2021 Canadian census recorded 662,248 people in the ...
(section of algebraic geometry) and in 1990 in
Kyoto Kyoto (; Japanese: , ''Kyōto'' ), officially , is the capital city of Kyoto Prefecture in Japan. Located in the Kansai region on the island of Honshu, Kyoto forms a part of the Keihanshin metropolitan area along with Osaka and Kobe. , the ci ...
(a plenary address). In 1973 he received the
Moscow Mathematical Society The Moscow Mathematical Society (MMS) is a society of Moscow mathematicians aimed at the development of mathematics in Russia. It was created in 1864, and Victor Vassiliev is the current president. History The first meeting of the society wa ...
Award. He was named to the 2023 class of Fellows of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, "for contributions to singularity theory, real algebraic geometry, and the theory of quantum integrable systems".


Books

* Arnolʹd, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts. Monographs in Mathematics, 82. Birkhäuser Boston, Inc., Boston, MA, 1985. xi+382 pp. * Arnolʹd, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Monographs in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1988. viii+492 pp. * Etingof, P.; Varchenko, A. Why the Boundary of a Round Drop Becomes a Curve of Order Four (University Lecture Series), AMS 1992, * Varchenko, A. Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups. Advanced Series in Mathematical Physics, 21. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. x+371 pp. * Varchenko, A. Special functions, KZ type equations, and representation theory. CBMS Regional Conference Series in Mathematics, 98. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003. viii+118 pp.


References


External links

*
Varchenko's homepage on the web-site of the University of North Carolina
{{DEFAULTSORT:Varchenko, Alexander Russian mathematicians 20th-century American mathematicians 21st-century American mathematicians 1949 births Living people Moscow State University alumni University of North Carolina faculty Fellows of the American Mathematical Society