Alexander Goncharov

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Alexander Goncharov

Alexander B. Goncharov (born April 7, 1960 ) is a Soviet - American mathematician who works in algebraic geometry , arithmetic algebraic geometry and geometry.

Life

Goncharov received the gold medal at the Mathematics Olympiad in 1976 . He studied at Lomonosov University in Moscow , where he attended the seminars of Israel Gelfand , Alexander Beilinson and Yuri Manin . In 1982 he graduated. From 1985 to 1992 he was in the cybernetics group of the Academy of Sciences of the USSR in Moscow. In 1987 he received his doctorate there with Israel Gelfand (“Generalized conformal structures on manifolds”). In 1990 he went to the USA. In 1991 he was a Harvard Prize Fellow at Harvard University , in 1992/3 at MSRI (and 2001), in 1990 and 1992 visiting scholar at MIT , where he was a lecturer from 1993 to 1995. From 1996 to 1998 he was a professor at the Max Planck Institute for Mathematics in Bonn and, from 1996, an associate professor at Brown University . In 1999 he became a professor at Brown University, since 2010 he has been a professor at Yale University , but regularly works as a visiting researcher in Europe (Max Planck Institute for Mathematics in Bonn, Institut des Hautes Études Scientifiques near Paris).

He worked on the theory of motifs and polylogarithms in connection with the study of motivic fundamental groups of algebraic curves . Here he found a connection between the motivic fundamental groups of the projective straight line without zero, the point in infinity and the Nth roots of unity and the geometry of modular varieties for the linear group GL (N) for all N. He also found connections between the motivic fundamental groups and Feynman integrals from quantum field theory . He proved special cases of Don Zagier's conjectures about the connection between polylogarithms and values ​​of Dedekind zeta functions at special integer places and expanded them to conjectures named after him ( conjecture of Goncharov ), which establish connections between algebraic K-theory and the cohomology of motivic complexes .

He also dealt with higher Teichmüller theory and its quantization , as well as with integral geometry .

In 1992 he received the EMS Prize of the European Society of Mathematicians at the First European Congress of Mathematicians in Paris. In 1994 he was invited speaker at the International Congress of Mathematicians ("Polylogarithms in arithmetic and geometry").

Works (selection)

  • Geometry of configurations, polylogarithms, and motivic cohomology. Adv. Math. 114 (1995) no. 2, 197-318.
  • (with AM Levin) Zagier's conjecture on L (E, 2). Invent. Math. 132 (1998) no. 2, 393-432.
  • Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12 (1999), no. 2, 569-618.
  • (with P. Deligne) Groupes fondamentaux motiviques de Tate mixed. Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 1, 1-56.
  • (with VV Fock) Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. No. 103: 1-211 (2006).
  • (with VV Fock) The quantum dilogarithm and representations of quantum cluster varieties. Invent. Math. 175 (2009), no. 2, 223-286.
  • (with H. Gangl, A. Levin) Multiple logarithms, algebraic cycles and trees, in Pierre Cartier u. a. Frontiers in Number Theory, Physics and Geometry , Volume 2, Springer Verlag 2007
  • (with VV Fock) Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Standard. Great. (4) 42 (2009), no. 6, 865-930.
  • (with R. Kenyon) Dimers and cluster integrable systems. Ann. Sci. Éc. Standard. Great. (4) 46 (2013), no. 5, 747-813
  • (with T. Dimofte, M. Gabella) K-Decompositions and 3d Gauge Theories. ArXiv
  • (with J. Golden, M. Spradlin, C. Vergu, A. Volovich) Motivic Amplitudes and Cluster Coordinates. ArXiv

Web links

Individual evidence

  1. ↑ Don Zagier Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields , Arithmetic algebraic geometry (Texel, 1989), Progress in Mathematics, Volume 89, 1991, Birkhäuser Verlag, pp. 391-430
  2. ↑ Goncharov Geometry of configurations, polylogarithms, and motivic cohomology , Advances in Mathematics, Volume 114, 1995, pp. 197-318
personal data
SURNAME Goncharov, Alexander
ALTERNATIVE NAMES Goncharov, Alexander B.
BRIEF DESCRIPTION Ukrainian-American mathematician
DATE OF BIRTH April 7, 1960