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.

- homepage at Steklov
- mathnet.ru entry
- wikipedia: Nikolai Durov

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

*New approach to Arakelov geometry*, arxiv/0704.2030

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. The company is not any more in their control. Nikolai and Pavel together established communication platform and company Telegram (see FAQ and wikipedia) where they worked on Nikolai’s envisioned Telegram Open Network (TON) which he classifies as a 5th generation blockchain project enhanced with additional DNS, proxy and (torrent-like) storage infrastructure.

- TON whitepaper, 23 page pdf, mirrors pdf pdf
- Nikolai Durov,
*Telegram Open Network*, technical overview, Dec 2017, 132 pp. pdf, mirror pdf;*Telegram Open Network Blockchain*, Sep 2018, 121 pp. pdf;*Telegram Open Network Virtual Machine*, Sep. 2018, 148 pp. pdf;*Fift: a brief introduction*(about his new Forth-like stack based programming language on TON), May 23, 2019, 87 pp. pdf is at channel t.me/Tgram/170;*Catchain consensus: an outline*, pdf

The technical overview sporadically uses the notation from type theory.

Durov’s 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 - N. V. Durov,
*Homotopy theory of normed sets I. Basic constructions*, Algebra i Analiz, 29:6 (2017), 35–98 mathnet.ru;*Homotopy theory of normed sets II. Model categories*, Algebra i Analiz, 30:1 (2018) 32–95 mathnet.ru

His paper on normed sets above (part 1 out of 3) is partly extending ideas from

- Frederic Paugam,
*Overconvergent global analytic geometry*, 2015, arXiv:1410.7971v2