Nikolai Durov is a Russian mathematician from St. Petersburg with main current interests in arithmetic geometry.
He obtained his Ph.D. in 2007 in Bonn under Gerd Faltings) with a thesis New approach to Arakelov geometry, where a complete algebraization of Arakelov geometry is suggested, as a special case of a generalized scheme theory based on gluing spectra of commutative monads in the category of sets (viewed as generalized rings) rather than spectra of rings only. Though the notion of commutativity of a monad had been known to some experts in category theory, Durov introduced them to and used them in an unexpected geometrical context. In the construction of spectra, he first studied a straightfoward genetralization of a prime spectrum? of a ring and then looked at a refinement defined with help of a categorically defined class of localizations which he calls “pseudolocalizations”. Later chapters of his Bonn thesis use homotopical algebra to define derived functors in his nonabelian setup. One of the notions he introduced in this vein is that of pseudomodel stacks. The final aim of this theory is to present a natural setup in which an arithmetic Riemann–Roch theorem of Gillet–Soulé and of Faltings could be formulated and proved in the vein of the Grothendieck–Riemann–Roch theorem. The framework of generalized schemes of Durov accomodates also versions of tropical geometry and absolute geometry or geometry over (a “field of one element”). His version of geometry over , while having many advantages, has the problem that the product of the spectrum of integers with itself (over ) is the spectrum of integers, which is a counterintuitive result (since other expected solutions were suggested).
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 for all equations iff the Riemann hypothesis? is true. The exposition of these results is in
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
where in chapters 7–9 Durov presented a flexible theory 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.