# nLab Nikolai Durov

Nikolai Durov (Николай Валерьевич Дуров) is a Russian mathematician from St. Petersburg with main current interests in arithmetic geometry, currently employed at St. Petersburg Department of the Steklov Institute of Mathematics.

Durov obtained his Ph.D. in 2007 in Bonn under Gerd Faltings:

Durov’s mathematical work preceding his study in Bonn includes his work on classical Galois theory of polynomial equations; it provides essentially the third historically available method to compute algorithmically a Galois group of a given equation. His method is however statistical and some random data are included in input. The algorithm terminates with probability $1$ for all equations iff the Riemann hypothesis is true. The exposition of these results is in

• N. V. Durov, Computation of the Galois group of a polynomial with rational coefficients. I. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), Vopr.Teor. Predst. Algebr. i Grupp. 11, 117–198, 301; English translation in J. Math. Sci. (N. Y.) 134 (2006), no. 6, 2511–2548 (MR2006b:12006)
• N. V. Durov, Computation of the Galois group of a polynomial with rational coefficients. II. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 321 (2005), Vopr. Teor. Predst. Algebr. i Grupp. 12, 90–135, 298; English translation in J. Math. Sci. (N. Y.) 136 (2006), no. 3, 3880–3907 (MR2006e:12004)

Nikolai Durov is also an experienced computer programmer. He was a member of a St Petersburg State University student team winning a student world tournament in programming. His high school education was in Italy. His younger brother Pavel V. Durov is a professional programmer and main constructor behind one of the most popular internet sites in Russia.

Nikolai Durov is currently working on his habilitation thesis. His earlier publications also include

• N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309, n. 1, 318–359 (2007) (doi:jalgebra) (math.RT/0604096).

where in chapters 7–9 Durov presented a flexible theory of a class of functors which can be viewed as representing generalizations of formal schemes but over an arbitrary ring, and with weaker assumptions. This theory is then applied to a problem in Lie theory and deformation theory; an interesting chapter on symplectic Weyl algebras is included. In chapter 10 an alternative method using Hopf algebras rather than geometry is presented.

Recently he introduced the notion of a vectoid and the related notion of an algebrad which is a generalization of the notions of a symmetric and a non-symmetric operad:

• Nikolai Durov, Classifying vectoids and generalisations of operads, arxiv/1105.3114, the translation of “Классифицирующие вектоиды и классы операд”, Trudy MIAN, vol. 273
• Classifying vectoids and generalizations of operads, talk at The International Conference “Contemporary Mathematics” June 12, 2009, video: link

Other sources:

• Computation of derived absolute tensor square of the ring of integers, talk at 2nd annual conference-meeting MIAN–POMI “Algebra and Algebraic Geometry”, St. Petersburg, December 25, 2008, link
• Arithmetic intersection theory and homotopical algebra, seminar 2007
• N. V. Durov, Топологические реализации алгебраических многообразий (Topological realizations of algebraic varieties), preprint POMI 13/2012 (in Russian) abstract, pdf.gz
• N. V. Durov, МУЛЬТИПЛИКАТИВНЫЕ МОНОИДЫ ${\mathbb{F}}_p$-АЛГЕБР И АБСОЛЮТНЫЕ ТЕНЗОРНЫЕ ПРОИЗВЕДЕНИЯ КОНЕЧНЫХ ПОЛЕЙ, (Multiplicative monoids of ${\mathbb{F}}_p$-algebras and absolute tensor products of finite fields), preprint POMI 12/2012 (in Russian) abstract pdf.gz
Revised on September 20, 2016 09:03:29 by Zoran Škoda (161.53.130.104)