I'm a postdoc in Chennai Mathematical Institute. I did my PhD at Jussieu and l'ENS. Before that, I studied at math department of Higher School of Economics.

Research domains: algebraic topology and geometry.

Here is my last year CV and the research statement

My papers with abstracts available at arXiv.

email: azakharov (^_^) cmi (^.^) ac (^-^) in

This spring we started mathematical physics seminar.

Miscellaneous writtings:

• Integral Cohomology of Eilenberg-Maclane spaces The cohomology of $K(Z,n)$ with field coefficients are well-known, it is a free algebra on classes given by the application of admissible Steenrod's operations to the fundamental class of degree $n$. The Pontryagin's ring of integral homology was described by Milgram by means of the iterated bar construction. On the other hand the case of cohomology is subtler, for example its ring structure localized at $2$ is tricker than an naive attempt to dualize Milgram's answer. Following classical Cartan's computations, we study diagonal approximation in the bar construction and relate it to the Cartan's model, following identification of the integral cohomology of Eilenberg-Maclane spaces and the cohomology of a certain de Rham complex. The proof of functoriality is given.
  
  
• Some equivalences of derived exponential functors Assume you have a co/commutative Hopf algebra $H$ over integers. The cohomology of its iterated classifying stack can be though as a non abelian derived functor. Evaluation on a complex $K\in Ch^{>0}(Z)$ following by the Dold--Kan correspondence produces a bialgebra $H(K)$. It is immediate from the construction that the multiplication on $H(K)$ depends on the multiplication on $H$, the same holds for comultiplication. Surprisingly it turns out that $E_1$-structure on $H(K)$ depends only on the coalgebra structure of $H$ in a functorial manner! In a work-in-progress with Dmitry Kubrak and Grigorii Shuklin we are going to generalize this observation.
  
  
• On Kaledin's class Kaledin introduced a complete cohomological obstruction for triviality of a deformation of a non unital associative dg-algebra $B$ over the formal completion of affine line $A^1$. In fact the role of $B$ can be played by any object which deformation theory is governed by a dg-Lie algebra in the sense of Hinich. The existence of a such obstruction found to be very useful, especially as a formality criteria for $B$. For a further reading on various applications google Coline Emprin's articles. It's curious that all existing proofs are inductive and computational, though there is a very simple geometrical picture behind the scene. In this draft we provide an elementary two-lines proof of the Lunts generalization of the classical Kaledin's assertion. So that the complexity of this problem over arbitrary base becomes transparent. Another part is the derived geometry point of view promoted in the note. It is a result of our discussions with Enrico Lampetti. Again, it is not clear if this perspective answers new questions, but certainly this setting is appealing. For example the classical Kaledin class is simply a differential of a morphism from a formal scheme (the base) to the deformation stack (corresponding to the dg-Lie algebra). In the derived setting the vanishing of the differential does not necessary imply that the map is homotopically constant. The base $A^1$ works fine only because a certain connection is automatically flat. At the time it is not clear if there is a single complete (!) cohomological obstruction of this kind for bases other than $A^1$. It would be interesting to see if prismatic decomposition approach due to Coline Emprin can be generalized and/or put in this setting.