contra corona / spring’21

OnlineResources_Term 3-4 / 7-8

Research seminar “Modern Problems of Mathematical Physics” (Term 1-8) / MA06268 / 20-22

Instructor: Pavlo Gavrylenko


Quantum Integrable Systems (Term 3-4) / MA060315 / 20-22

Instructor: Anton Zabrodin


Gauge Fields and Complex Geometry (Term 3-4) / MA06178 / 20-22

Instructor: Alexei Rosly


Modern Dynamical Systems (Term 3-4) / MA060425 / 20-22

Instructors: Aleksandra Skripchenko, Sergei Lando


Quiver representations and quiver varieties (Term 3-4) / MA06259 / 20-22

Instructor: Evgeny Feigin

  • Zoom Meetings _
  •  
    Feb 3 / Lecture 1 //
    Quivers: definitions and examples. Representations of quivers: definitions and examples. Homomorphisms and isomorphisms of representations, irreducible and indecomposable representations of quivers. Krull-Schmidt theorem

    Feb 10 / Lecture 2 //
    Quotient representations for quivers, kernels and cokernels. Finite-dimensional representations of quivers as abelian category. Exact sequences, short exact sequences., split exact sequences. Examples of non split exact sequences. Sections and retractions, relation to splitting

    Feb 17 / Lecture 3 //
    Covariant and contravariant functors, functors of homomorphisms. Exact sequences and Hom functors, split exact sequences and Hom functors. Projective and injective modules. Paths, sinks, sources, oriented cycles. Simple modules S(i) and modules P(i) for quivers with no oriented cycles

    Feb 24 / Lecture 4 //
    Properties of the modules P(i) and I(i): projectivity and injectivity, description of homomorphisms spaces Hom(P(i),M). P(I) and I(i) are indecomposable. Hom spaces from P(i) to P(j), path algebra as an endomorphisms algebra of the direct sum of projectives

    Mar 3 / Lecture 5 //
    Two terms projective resolutions, explicit construction. Projective modules as direct summands of free modules. Complete classification of projective representations. Radical of P(i) as the maximal subrepresentation. Any subrepresentation of a projective representation is projective

    Mar 10 / Lecture 6 //
    Definition of Ext^1(M,N) via projective resolution, cokernels. Extensions, isomorphic extensions. Abelian group structure on the equivalence classes of extensions. An element of the space Ext^1(M,N) corresponding to an extension.

    Mar 17 / Lecture 7 //
    Variety of representations with fixed dimension vector. The product of general linear groups group action. Description of the orbits of the action. Stabilizers and dimensions of orbits. Non split sequences and inequality on the dimensions of orbits. Quadratic form of a quiver.

    Mar 24 / Lecture 8 //
    Dynkin and Eucledian quivers. A non Dynkin quiver has a Eulcedian quiver as a subquiver. Codimension of an orbit in the representation variety. Negative values of quadratic form of a quiver and infinite numebr of isoclasses of indecomposable representations

    Mar 31 / Lecture 9 //
    Kernel of the quadratic form for the Eucledian quivers. Positive definite quadratic forms and Dynkin quivers. Positive semi-definite quadratic forms and Eucledian quivers. Roots : real and imaginary, positive and negative. Dynkin quivers have finite number of roots. Positive roots for Dynkin quivers and indecomposable representations.

    Apr 7 / Lecture 10 //
    Gabriel’s theorem: indecomposable representations and roots for Dynkin quivers. Unital associative algebras: left and right ideal, maximal ideals, radical. Path algebras: basic definitions.

    Apr 14 / Lecture 11 //
    Radical of the path algebra of a quiver with no oriented cycles. Right and left modules, examples. Quiver representations and modules over the path algebras. Nakayama’s lemma. Radical of a finite-dimensional algebra is nilpotent.

    Apr 21 / Lecture 12 //
    Idempotents, indecomposable (primitive) idempotents, orthogonal idemponents. Primitive idempotents in the path algebra. Decomposition of an algebra via orthogonal idempotents. Radicals and local algebra

    Apr 28 / Lecture 13 //
    Local algebras and basic algebras. Admissible ideal, bound quiver algebras. Quiver attached to a basic algebra. Realization of a basic algebra in terms of bound quiver algebras

    May 12 / Lecture 14 //
    Quiver Grassmannians: definition and examples. Nakajima quivers varieties: framed representations, doubled quivers, moment map, stability conditions

    May 19 / Lecture 15 //
    Nakajima quiver Grassmannians: stability conditions in terms of subrepresentations of the double quiver, examples for the A_1 quiver. Universal quiver Grassmannians, projection to the product of classical Grassmannians

    May 26 / Lecture 16 //
    Expected dimension of a quiver Grassmannian, rigid representations. Quotient construction of quiver Grassmannians. Tangent space as Hom space. Stratification of quiver Grassmannians by isoclasses of representations. Cotangent bundle to a partial flag variety as Nakajima quiver variety

    Literature

    = = Ralf Schiffler, Quiver representations
    = = W. W. Crawley-Boevey, Lectures on representations of quivers, www.math.uni-bielefeld.de/~wcrawley/
    = = W. W. Crawley-Boevey, , Geometry of representations of algebras, www.math.uni-bielefeld.de/~wcrawley/
    = = V. Ginzburg, Lectures on Nakajima’s quiver varieties
    = = A. Kirillov, Jr., Quiver Representations and Quiver Varieties


Introduction to quantum theory (Term 3-4) / MA060332 / 20-22

Instructors: Vladimir Losyakov, Pavlo Gavrylenko


Quantum Field Theory (Term 3-4) / MA060316 / 20-22

Instructor: Andrei Semenov

  • Microsoft Teams

Никита Некрасов

00. Введение в локализацию для N=2 SYM / Nov 5
01. Пространства модулей инстантонов / Nov 12
02. Виртуальный характер касательного пространства (1/2) / Nov 26
03. Виртуальный характер касательного пространства (2/2) / Dec 3
04. Формулы локализации / Dec 10
05. Геометрическое квантование, локализация / Dec 17
06. Конформные блоки аффинной алгебры / Dec 24
07. Некоммутативные инстантоны (1/4) / Feb 4
08. Некоммутативные инстантоны (2/4) / Feb 11
09. Некоммутативные инстантоны (3/4) / Feb 18
10. Некоммутативные инстантоны (4/4) / Feb 25
11. Геометрия пространств Калоджеро-Мозера и ADHM / Mar 4
12. Фазовое пространство калибровочной теории в d+1 измерении / Mar 11
13. Квантовая тригонометрическая система Калоджеро-Мозера / Mar 18
14. Двумерный Янг-Миллс (1/2)/ Mar 25
15. Двумерный Янг-Миллс (2/2)/ Apr 1
16. Пространства модулей вакуумов, гиперкэлеровы многообразия, расслоения Хиггса / Apr 15
17. Энумеративная геометрия (1/2). Пространство квазиотображений / Apr 22
18. Энумеративная геометрия (2/2). Интеграл по пространству квазиотображений / Apr 29
19. Калибровочные теории в 3 и 5 измерениях / May 13
20. Локализация в топологических струнах и в инстантонах (в пятимерье) / May 20
21. Связь с теорией суперструн / June 10