ALG's main page at the Bogomolov Lab

Alexey Gorodentsev

at the Laboratory of Algebraic Geometry,
Faculty of Mathematics,
Higher School of Economics

Content:

My Math: some papers
Math for students: lecture notes and other teaching materials (mostly in russian)
Math for schools: sheets with problems, programs, and tests (in russian only)
My pictures are stored now at RasFokus, ArtHero, and Picasa.

My Math

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 talk «ALAG» at St.Petersbubrg's conference «Science of the Future» (September 18, 2014)
Transparencies of my KIAS Lectures on ALAG (April - May 2001)
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.

Math for students

Below are some printed matter supplying my courses given at the Independent University of Moscow, at the Faculty of Mathematics in the Higher School of Economics, and at some other places.

NEW! HSE Algebra, 1st year (2024/2025, in Russian). Previous versions of this course: 2023/2024, 2022/2023.
MIPT Representation Theory (2023 Spring term, in Russian).
HSE Algebra, 2nd year (2010/2011, in Russian).
HSE Linear Algebra and Geometry, 2021/22 (in Russian). Previous versions of this course (all in Russian): 2020/2021, 2019/2020, Spring 2019 (2nd term only), 2011/12 (slightly different version)
IUM Algebra-1, 2020/21 (in Russian). Previous versions of this course (all in Russian): 2013/14., Autumn 2005 (1st term), Spring 2006 (2nd term), Spring 2001 (linear algebra and geometry), and very old materials written before 1996. My textbook for 1st year students in Russian: version of May 2011, 4.5 Mb PDF.
IUM Algebra, 2nd year, 2021/22 (in Russian). Previous versions of this course (all in Russian): 2014/15, Fall 2006, Spring 2007. There is also a stand alone part of this course «Introduction to commutative and homological algebra» (Spring 2008). My textbook for 2nd year students in Russian: version of June 2015, 2.1 Mb PDF.
IUM Geometry, 1st year (2016/17, in Russian).
HSE Geometric introduction to algebraic geometry. (Fall 2016, in Russian). Pre-requisits: first 3 terms of the standard basic courses of algebra, calculus, and geometry/topology given at Faculty of Math (HSE) or at IUM. A shorted version of this course, the six lectures at the Ural State University (Ekaterinbourg, October 8–13, 2012) are here.
HSE Introduction to the Theory of Categories and Homological Algebra, Fall 2019 (in Russian). A previous version of this course is here.
HSE Sheaves and Homological Algebra (Spring 2018). Introduction to the theory of sheaves and supplying homological algebra. Pre-requisits: first 3 terms of the standard basic courses of algebra, math. logic, calculus, geometry, and topology given at Faculty of Math (HSE) or at IUM. The previous versions of this course were given in Fall 2015 (mixed in English and in Russian) and in Spring 2017 (in Russian).
Math in Moscow: Algebraic Geometry. A Start Up Course» (Fall 2017, in English). The old good stable version (unchanged after Fall 2009) is also available from here.
Math in Moscow: Introduction to commutative and homological algebra (in English). Some lecture notes, exercises, programs, and examples of written exams.
São Carlos winter classes on Geometry (July 2013, in English). Intensive mini-course of Geometry and Linear algebra given at the São Carlos branch of the Universidade de São Paulo (Brasil).
Quantum systems and geometry: there and back again (in Russian). IUM course joint with A.S.Losev. Developed for the second year students (both physicists and mathematicians) who are going to start speak the both languages: physical and geometrical.
HSE, Exercises on Discrete Mathematics 1 (Fall 2018, in Russian). Printed matter for seminars.

Math for schools

From time to time I teach advanced math in Moscow schools. Some printed matter for these lessons are presented here. Mostly, they are so called "sheets of problems" (individual tasks for independent solution), all in Russian.

Numbers and Sums: this is a textbook for junior schoolboys; in the most part it was written by I.Artamkin, S.Khoroshkin, and A.Kulakov with several minor remarks added by me, A.Khokhlov, and M.Prokhorov.
Algebra & Geometry 8-9: several fragments from the regular math classes I taught in last 1980th and early 1990th at 109th and 1018th schools in a collaboration with V.Dolotin, A.Khokhlov, and my wife Tanya.
Calculus 9-11: complete course of calculus designed for studying during the last 3 years in (powerful enough) math. schools. It was given at 57th school during 1999-2002 and then partially repeated in 179th school. These materials would never appear without closed collaboration with R.Anno, S.Dorichenko, O.Kryazheva, D.Mikhalin, E.Smirnov, N.Stepanov, and K.Trushkin.
Calculus for physical class of 179th school: printed matter on the course given at ITEP during 2005-2007 in a collaboration with N.Amburg and A.Rubin.