615:. 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
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558:, 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.
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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
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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
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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
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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
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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.
491:'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.
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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
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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.
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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.
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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.
691:
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.
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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.
474:
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
547:(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
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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.
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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.
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588:-dimensional planes coincides with the dimension of the space of invariants in a suitable tensor product of representations of the general linear group
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Felder, G.; Tarasov, V.; Varchenko, A. (1999). "Monodromy of solutions of the elliptic quantum
Knizhnik-Zamolodchikov-Bernard difference equations".
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Varchenko, A. (1983). "On the
Semicontinuity of the Spectra and Estimates from Above of the Number of Singular Points of a Projective Hypersurface".
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Tarasov, V.; Varchenko, A. (1997). "Geometry of q-hypergeometric functions as a bridge between
Yangians and quantum affine algebras".
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60:
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Mukhin, E.; Tarasov, V.; Varchenko, A. (2009). "The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz".
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Etingof, P.; Varchenko, A. Why the Boundary of a Round Drop Becomes a Curve of Order Four (University Lecture Series), AMS 1992,
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Varchenko, A. (1984). "Estimate of the Number of Zeros of a Real Abelian Integral Depending on a Parameter and Limit Cycles".
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Etingof, P.; Varchenko, A. (1998). "Solutions of the quantum dynamical Yang–Baxter equation and dynamical quantum groups".
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Felder, G.; Rimányi, R.; Varchenko, A. (2018). "Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology".
368:
in 1982. From 1974 to 1984 he was a research scientist at the Moscow State University, in 1985–1990 a professor at the
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650:, "for contributions to singularity theory, real algebraic geometry, and the theory of quantum integrable systems".
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Markov, Y.; Felder, G.; Tarasov, V.; Varchenko, A. (2000). "Differential Equations Compatible with KZ Equations".
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Rimányi, R.; Tarasov, V.; Varchenko, A. (2012). "Partial flag varieties, stable envelopes and weight functions".
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Rimányi, R.; Tarasov, V.; Varchenko, A. (2012). "Partial flag varieties, stable envelopes and weight functions".
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in singularity theory, in particular, he gave a formula, relating Newton polygons and asymptotics of the
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781:"Theorems of Topological Equisingularity of Families of Algebraic Manifold and Polynomial Mappings"
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Tarasov, V.; Varchenko, A. (2002). "Duality for Knizhnik-Zamolodchikov and Dynamical Equations".
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Varchenko, A. (1980). "The Asymptotics of Holomorphic Forms Determine a Mixed Hodge Structure".
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Schechtman, V.; Varchenko, A. (1991). "Arrangements of Hyperplanes and Lie Algebra Homology".
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Asymptotics of Integrals and Algebro-Geometric Invariants of Critical Points of Functions
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A. Varchenko (1969). "The branching of multiple integrals which depend on parameters".
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It may require cleanup to comply with Knowledge (XXG)'s content policies, particularly
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Felder, G.; Varchenko, A. (1996). "On representations of the elliptic quantum group
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Varchenko, A. (1976). "Newton Polyhedra and Asymptotics of Oscillatory Integrals".
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Theorems on Topological Equisingularity of Families of Algebraic Sets and Maps
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In 1969 Varchenko identified the monodromy group of a critical point of type
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singular points of a projective hypersurface of given degree and dimension.
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316:, born February 6, 1949) is a Soviet and Russian mathematician working in
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Varchenko's homepage on the web-site of the University of North Carolina
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Zariski, O. (1937). "On the Poincaré group of projective hypersurface".
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of a function of an odd number of variables with the symmetric group
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1009:"Monodromy representations of braid groups and Yang-Baxter equations"
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Sottile, Frank (2010). "Frontiers of reality in Schubert calculus".
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1576:"Schubert calculus and representations of the general linear group"
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In, Evgeny Mukhin, Tarasov, and Varchenko proved the conjecture of
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were lecturing mathematics and physics. Varchenko graduated from
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Symmetry, Integrability and Geometry: Methods and Applications
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is to decide if there exists an upper bound for the number of
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It is classically known that the intersection index of the
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which is the Weyl group of the simple Lie algebra of type
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Varchenko, A. (1975). "Versal Topological Deformations".
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A major contributor to this article appears to have a
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may be too technical for most readers to understand
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646:He was named to the 2023 class of Fellows of the
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534:Vadim Schechtman and Varchenko identified in the
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340:From 1964 to 1966 Varchenko studied at the Moscow
1574:Mukhin, E.; Tarasov, V.; Varchenko, A. (2009).
635:(section of algebraic geometry) and in 1990 in
494:Varchenko was among creators of the theory of
1628:"ICM Plenary and Invited Speakers since 1897"
1530:Bulletin of the American Mathematical Society
639:(a plenary address). In 1973 he received the
487:published in 1937. In 1973, Varchenko proved
8:
1739:Fellows of the American Mathematical Society
1581:Journal of the American Mathematical Society
1040:Drinfeld, V. (1990). "Quasi-Hopf algebras".
374:University of North Carolina at Chapel Hill
282:University of North Carolina at Chapel Hill
61:Learn how and when to remove these messages
731:Love and Math: The Heart of Hidden Reality
510:Varchenko introduced the asymptotic mixed
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186:Learn how and when to remove this message
168:Learn how and when to remove this message
111:Learn how and when to remove this message
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1632:International Congress of Mathematicians
629:International Congress of Mathematicians
627:Varchenko was an invited speaker at the
1276:{\displaystyle E_{\tau ,\eta }(sl_{2})}
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608:{\displaystyle \operatorname {GL} _{N}}
344:for gifted high school students, where
360:. Varchenko defended his Ph.D. thesis
1391:J. Math. Phys., Analysis and Geometry
551:'s equivariant enumerative geometry.
364:in 1974 and Doctor of Science thesis
93:make it understandable to non-experts
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1734:University of North Carolina faculty
1714:21st-century American mathematicians
1709:20th-century American mathematicians
538:(or, KZ equations) with a suitable
554:In the second half of 90s Felder,
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727:Edward Frenkel (1 October 2013).
342:Kolmogorov boarding school No. 18
42:This article has multiple issues.
536:Knizhnik–Zamolodchikov equations
148:. Please discuss further on the
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1659:. American Mathematical Society
370:Gubkin Institute of Gas and Oil
306:Alexander Nikolaevich Varchenko
50:or discuss these issues on the
1729:Moscow State University alumni
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245:Moscow State University (1971)
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1684:Mathematics Genealogy Project
1613:10.1090/s0894-0347-09-00640-7
1553:10.1090/s0273-0979-09-01276-2
1013:Annales de l'Institut Fourier
762:Funkcional. Anal. I Prilozhen
648:American Mathematical Society
356:in 1971. He was a student of
314:Александр Николаевич Варченко
912:Soviet Mathematics - Doklady
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641:Moscow Mathematical Society
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735:. Basic Books. pp.
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567:real algebraic geometry
565:and Michael Shapiro in
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255:Varchenko's theorem
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