Alexey L. Gorodentsev

My research interests are mostly in geometry and homological algebra of vector bundles and sheaves on algebraic, analytic, and symplectic manifolds as well as applications of all this staff to mathematical physics.

• Helix theory. This is a cohomological technique for orthogonal decompositions of derived and triangulated categories in terms of exceptional bases. It was developed firstly in my PhD theses (1986/87, see also Gorodentsev A.L., Rudakov A.N. Exceptional vector bundles on projective spaces. Duke Math. J. 54 (1987) p.115-130) in order to describe exceptional vector bundles on Fano varieties. Now the helices are quite popular as one of "mirror symmetry bypasses" between algebraic geometry of coherent sheaves and differential geometry of Picard - Lefschetz pencils.

  Transparencies of my KIAS Lectures on helices   (May - June 2001)
  The Helix theory    (joint with S.A.Kuleshov; Moscow Math. J. 4:2 (2004), p.377-440)
  Mukai lattices    (Transparencies of my talk at the workshop GEOQUANT-2013, Wien, August 27, 2013)

   

ALAG. This is an abbreviation for Abelian Lagrangian Algebraic Geometry developed by A.N.Tyurin (see for example math.AG/9902027 and math.AG/9909094) as a geometrical framework for geometric quantization.

Transparencies of my KIAS Lectures on ALAG    (April - May 2001)

St.Petersburg lecture on ALAG (September 2014)
English   and    Russian   versions of "ALAG" paper explaining basic ALAG principles (joint with A.N.Tyurin).

• t-stability. A concept of stability comes from Geometric Invariant Theory where it is used to construct compact orbit spaces for algebraic group actions. t-stability generalizes this approach in a context of generic linear triangulated categories. This allows to consider moduli space problems in very general setup.

  "t-stabilities and t-structures on triangulated categories" (joint with S.A.Kuleshov and A.N.Rudakov): in english and in russian.
  "On finest and modular t-stabilities" (joint with S.A.Kuleshov, in english).

   

• Syzygies of projective varieties. There are classical problems of geometry and homological algebra: how to describe a minimal free resolution for the graded coordinate algebra of projective variety in the category of modules over the polynomial ring, and what kind of extra algebraic structures does this resolution admit.

  "On syzygies of highest weight orbits" (joint with A.S.Khoroshkin and A.N.Rudakov, in english). We study a minimal resolution for the coordinate algebra of projective orbit of the highest weight vector in an irreducible representation of semisimple complex Lie group. We show that the syzygies are isomorphic to the cohomologies of appropriate graded Lie superalgebra and illustrate this by several examples including the complete computation of syzygies for grassmannians Gr(2,N) in the standard Plucker embedding.

since 2008 professor at the Faculty of Mathematics, HSE, Moscow
since 2000 senior researcher at the Mathematical Physics Group, ITEP, Moscow
since 1992 professor at the Independent University of Moscow

http://gorod.bogomolov-lab.ru/index_rus.html