# Alexey L. Gorodentsev

- Professor:Faculty of Mathematics
- Programme Academic Supervisor:Mathematics

- Tenured Professor

- Alexey L. Gorodentsev has been at HSE University since 2008.

### Education, Degrees and Academic Titles

- 1999Associate Professor
- 1989
Candidate of Sciences* (PhD) in Physical and Mathematical Sciences

Lomonosov Moscow State University - 1988
Doctoral programme

Lomonosov Moscow State University - 1985
Degree

Lomonosov Moscow State University

According to the International Standard Classification of Education (ISCED) 2011, Candidate of Sciences belongs to ISCED level 8 - "doctoral or equivalent", together with PhD, DPhil, D.Lit, D.Sc, LL.D, Doctorate or similar. Candidate of Sciences allows its holders to reach the level of the Associate Professor.

### Courses (2019/2020)

If you are looking for the actual content of my courses (programs, lecture notes, training aids etc) click here.

- An Introduction to the Theory of Categories and Homological Algebra (Optional course (faculty); Faculty of Mathematics; 1, 2 module)Rus
- Geometry (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Research Seminar "Mathematics Practical Training" (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Past Courses

### Courses (2018/2019)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1, 2 module)Rus
- Categories and Functors (Optional course (faculty); Faculty of Mathematics; 1, 2 module)Rus
- Discrete Mathematics (Bachelor’s programme; Faculty of Mathematics; 1 year, 1, 4 module)Rus
- Geometry (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus

### Courses (2017/2018)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1, 2 module)Rus
- Algebraic Geometry: a Start-up Course (Optional course (faculty); Faculty of Mathematics; 1, 2 module)Eng
- Mathematics Practical Training (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-4 module)Rus
- Sheaves and Homological Algebra (Optional course (faculty); Faculty of Mathematics; 3, 4 module)Eng

### Courses (2016/2017)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1, 2 module)Rus
- Algebra (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Mathematics Practical Training (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-4 module)Rus

### Courses (2015/2016)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-3 module)Rus
- Algebra (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Mathematics. Licenciatus (Bachelor’s programme; Faculty of Mathematics; 3 year, 4 module)Rus
- Mathematics Practical Training (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Research Seminar of Master’s Programme (Master’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Sheaves and Homological Algebra (Optional course (faculty); Faculty of Mathematics; 1, 2 module)Eng

### Courses (2014/2015)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-3 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 4 year, 3, 4 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 3 year, 3, 4 module)Rus
- Algebraic Geometry: a start-up course (Master’s programme; Faculty of Mathematics; 2 year, 1, 2 module)Eng
- Homological Algebra and Derived Categories (Master’s programme; Faculty of Mathematics; 1 year, 3, 4 module)Rus
- Research Seminar "Computations in Homological Algebra" (Master’s programme; Faculty of Mathematics; 1 year, 3, 4 module)Rus
- Research Seminar "Homological Methods in Math. Physics and Representation Theory 1" (Master’s programme; Faculty of Mathematics; 1 year, 1, 2 module)Rus
- Research Seminar "Homological Methods in Math. Physics and Representation Theory 2" (Master’s programme; Faculty of Mathematics; 1 year, 3, 4 module)Rus

### Courses (2013/2014)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-3 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 4 year, 3, 4 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 3 year, 3, 4 module)Rus
- Geometric Introduction to Algebraic Geometry 1 (Master’s programme; Faculty of Mathematics; 1 year, 1, 2 module)Eng
- Geometric Introduction to Algebraic Geometry 2 (Master’s programme; Faculty of Mathematics; 1 year, 3, 4 module)Eng
- Geometry (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-3 module)Rus
- Mathematics Practical Training (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Mathematics Practical Training (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-4 module)Rus

### Courses (2012/2013)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-3 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 4 year, 3, 4 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 3 year, 3, 4 module)Rus
Invariants and Representations of Classical Groups 1 (Master’s programme; Faculty of Mathematics; field of study "010100.68 Математика", field of study "010100.68 Математика"; 1 year, 1, 2 module)Rus

- Invariants and Representations of Classical Groups 1 (Bachelor’s programme; Faculty of Mathematics; 3 year, 1, 2 module)Rus
- Invariants and Representations of Classical Groups 1 (Master’s programme; Faculty of Mathematics; 2 year, 1, 2 module)Rus
Invariants and Representations of Classical Groups 2 (Master’s programme; Faculty of Mathematics; field of study "010100.68 Математика", field of study "010100.68 Математика"; 1 year, 3, 4 module)Rus

- Invariants and Representations of Classical Groups 2 (Bachelor’s programme; Faculty of Mathematics; 4 year, 3, 4 module)Rus
- Invariants and Representations of Classical Groups 2 (Bachelor’s programme; Faculty of Mathematics; 3 year, 3, 4 module)Rus
- Invariants and Representations of Classical Groups 2 (Bachelor’s programme; Faculty of Mathematics; 2 year, 3, 4 module)Rus
- Invariants and Representations of Classical Groups 2 (Master’s programme; Faculty of Mathematics; 2 year, 3, 4 module)Rus
Research Seminar "Algebra and Geometry 1" (Master’s programme; Faculty of Mathematics; field of study "010100.68 Математика", field of study "010100.68 Математика"; 1 year, 1, 2 module)Rus

- Research Seminar "Algebra and Geometry 1" (Bachelor’s programme; Faculty of Mathematics; 4 year, 1, 2 module)Rus
- Research Seminar "Algebra and Geometry 1" (Bachelor’s programme; Faculty of Mathematics; 1 year, 1, 2 module)Rus
- Research Seminar "Algebra and Geometry 1" (Bachelor’s programme; Faculty of Mathematics; 3 year, 1, 2 module)Rus
- Research Seminar "Algebra and Geometry 1" (Bachelor’s programme; Faculty of Mathematics; 2 year, 1, 2 module)Rus
- Research Seminar "Algebra and Geometry 1" (Master’s programme; Faculty of Mathematics; 2 year, 1, 2 module)Rus
Research Seminar "Algebra and Geometry 2" (Master’s programme; Faculty of Mathematics; field of study "010100.68 Математика", field of study "010100.68 Математика"; 1 year, 3, 4 module)Rus

- Research Seminar "Algebra and Geometry 2" (Bachelor’s programme; Faculty of Mathematics; 1 year, 3, 4 module)Rus
- Research Seminar "Algebra and Geometry 2" (Bachelor’s programme; Faculty of Mathematics; 3 year, 3, 4 module)Rus
- Research Seminar "Algebra and Geometry 2" (Bachelor’s programme; Faculty of Mathematics; 2 year, 3, 4 module)Rus
- Research Seminar "Algebra and Geometry 2" (Master’s programme; Faculty of Mathematics; 2 year, 3, 4 module)Rus

### Courses (2011/2012)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 4 year, 1, 2 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 3 year, 1, 2 module)Rus
- Geometry (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-3 module)Rus
- Research Seminar "Toric Geometry" (Master’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Research Seminar "Toric Geometry" (Master’s programme; Faculty of Mathematics; 2 year, 1-4 module)Rus
- Research Seminar "Toric Geometry" (Bachelor’s programme; Faculty of Mathematics; 4 year, 1-4 module)Rus
- Research Seminar "Toric Geometry" (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Research Seminar "Toric Geometry" (Bachelor’s programme; Faculty of Mathematics; 3 year, 1-4 module)Rus
- Research Seminar "Toric Geometry" (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-4 module)Rus

### Courses (2010/2011)

- Algebra (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Algebra (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-4 module)Rus
- Algebraic Geometry (Bachelor’s programme; Faculty of Mathematics; 3 year, 1, 2 module)Rus
- Algebraic Geometry. Start up course (Master’s programme; Faculty of Mathematics; 1 year, 3, 4 module)Eng
- Research Seminar "Symmetric Functions, Grassmannians, Flags" (Bachelor’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Research Seminar "Symmetric Functions, Grassmannians, Flags" (Bachelor’s programme; Faculty of Mathematics; 3 year, 1-4 module)Rus
- Research Seminar "Symmetric Functions, Grassmannians, Flags" (Master’s programme; Faculty of Mathematics; 1 year, 1-4 module)Rus
- Research Seminar "Symmetric Functions, Grassmannians, Flags" (Bachelor’s programme; Faculty of Mathematics; 2 year, 1-4 module)Rus

### Student Term / Thesis Papers

- Bachelor
A. Novikov «Tangent Algebra of Saturated Triangulated Category». Faculty of Mathematics, 2018

V. Shubin «On Exceptional Bases of Derived Categories of Coherent Sheaves on Toric Fano Varieties». Faculty of Mathematics, 2018

S. Mizyuk «Toric Varieties and Gale Duality». Faculty of Mathematics, 2017

E. Statnik «Topological Invariants of Universal Vector Bundles on Moduli Spaces of Coherent Sheaves on $\mathbb P_2$». Faculty of Mathematics, 2016

- Master
E. Statnik «Hochschild Homology and Cohomology for Categories». Faculty of Mathematics, 2018

I. Nikitin «Exceptional Bases of Rank 4 Mukai Latices». Faculty of Mathematics, 2017

### Area of research

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.

### Conferences

- 2014
наука будущего (science of the future) (Санкт Петербург). Presentation: Abelian Lagrangian Algebraic Geometry (ALAG)

- 2013ESI School and Conference Geometry and Quantization GEOQUANT 2013 (Вена). Presentation: Mukai lattices
- XII международная школа по теоретической и математической физике (Севастополь). Presentation: Курс "Массивы"
- XII международная школа по теоретической и математической физике (Севастополь). Presentation: Гомологическая алгебра. Введение. Проективные резольвенты. Спектральные последовательности. Комплексы Кошуля

### Publications^{22}

- Book Gorodentsev A. L.
*Algebra II. Textbook for Students of Mathematics*Vol. II: Algebra II, Textbook for Students of Mathematics. Cham : Springer, 2017. doi - Book Gorodentsev A. L.
*Algebra I Textbook for Students of Mathematics*/ Пер. с рус. Cham : Springer, 2016. doi - Chapter Городенцев А. Л. Учителя // В кн.: Поколения ВШЭ. Учителя об учителях. М. : Издательский дом НИУ ВШЭ, 2013. С. 118-123.
- Preprint Gorodentsev A. L., Losev A. S., Lysov V.
*Functorial A$_\infty$-Coproduct of Combinatorial Simplicial Chains Transferred to Itself under Barycentric Subdivision*/ The Erwin Schrodinger International Institute for Mathematical Physics. Series "ESI preprints". 2010. No. 2257. - Article Gorodentsev A. L., Kuleshov S. On Finest and Modular t-stabilities //
*American Mathematical Society Translations: Series 2*. 2007. Vol. 221. P. 121-146. - Article Gorodentsev A. L., Rudakov A. N., Khoroshkin A. On Syzygies of Highest Weight Orbits //
*American Mathematical Society Translations: Series 2*. 2007. Vol. 221. P. 79-120. - Article Кулешов С., Городенцев А. Л. Helix Theory. // Moscow Mathematical Journal. 2004. Т. 4. № 2. С. 377-440.
- Article Кулешов С., Рудаков А. Н., Городенцев А. Л. t-stabilities and t-structures on triangulated categories. // Известия РАН. Серия математическая. 2004. Т. 68. № 4. С. 117-150.
- Article Тюрин А., Городенцев А. Л. ALAG (Abelian Lagrangian Algebraic Geometry) // Известия РАН. Серия математическая. 2001. Т. 65. № 3. С. 15-50.
- Article Gorodentsev A. L. Top Chern classes of universal bundles on the complex projective plane and correlation functions of asymptotically free SYM N=2 QFT. Algebraic geometry, 11. //
*Journal of Mathematical Sciences*. 2001. Vol. 106 . No. 5 - Article Шапиро Б., Gorodentsev A. L. On associated discriminants. //
*Beitrage zur Algebra und Geometrie*. 1998. Vol. 39. No. 1. P. 53-74. - Article Леенсон М., Городенцев А. Л. How to calculate the correlation function in twisted SYM N = 2, Nf = 4 QFT on projective plane. // Journal of Mathematical Sciences. 1997. Т. 84. № 5
- Article Городенцев А. Л. Исключительные расслоения на поверхностях с подвижным антиканоническим классом // Известия РАН. Серия математическая. 1988. Т. 52. № 4. С. 740-757.
- Article Городенцев А. Л. Перестройки исключительных расслоений на Р„. // Известия РАН. Серия математическая. 1988. Т. 52. № 1. С. 1-13.
- Article Gorodentsev A. L., Rudakov A. N. Exceptional vector bundles on projective spaces //
*Duke Mathematical Journal*. 1987. Vol. 54. No. 1. P. 115-130.

### Positions

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

## HSE Staff Members Receive University Awards

On January 18, several university staff members were awarded HSE medals and notes of acknowledgement from the Rector.

## HSE Staff Members Awarded Status of Tenured Professor

On June 22, several HSE lecturers and staff members were awarded the status of Tenured Professor at a meeting of HSE Academic Council. Sixteen HSE staff members became Distinguished Professors at the Higher School of Economics for the first time.

## Springer Nature, HSE and Skoltech to Publish Book Series in Mathematics

Moscow Lectures, a new series of books in English, is set to be published by Springer Nature. The series is issued jointly by HSE and Skoltech, and its Editor-in-Chief is Alexey Gorodentsev, Professor at the HSE Faculty of Mathematics. Twelve volumes are currently in preparation and the first volume will be published at the beginning of June 2018. The series builds on the outstanding research and education in the field of mathematics in Moscow. It is aimed at graduate and undergraduate students, as well as lecturers and researchers, across the globe.

## After Turning Down Offers in New York and Berlin, Chinese Student Pursues Mathematics Training at HSE

For DaoDao Yang, a native of Dali, China, choosing a programme in mathematics came down to the decision of where he felt most challenged and most comfortable. This ultimately meant turning down offers at New York University and the Berlin Mathematical School to accept an offer from the Master’s programme at the Higher School of Economics. In an extensive interview with the HSE news service, Dao Dao recently spoke about what led him to choose HSE for his postgraduate study; his impressions of the programme, its professors and his fellow students; and his future plans.

## 'I’m Glad I Listened to My Heart'

A career in science and technology might sound appealing to the average Russian, but a recent survey showed that most Russians believe men are better suited for this line of work than women. The young women interviewed for this story, a mathematician and a programmer, are of a different opinion.

## ‘Studying with Enthusiastic People Around is Fun’

In 2015 eight graduates from the HSE Faculty of Mathematics started PhD programmes at MIT, McGill, Caltech, Pennsylvania State University and other top universities. HSE 4th year student Dmitry Pirozhkov who has got a place at Columbia University talked to the HSE News Service about how to get onto a prestigious PhD programme and about his expectations and academic plans.