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

    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

  • Youtube Videos

    Vladimir Losyakov, Quantum Mechanics / Jan 21 Квантование оператора момента Introduction
    Vladimir Losyakov, Quantum Mechanics / Jan 21 / 1. Физические системы и наблюдаемые
    Pavlo Gavrylenko, Quantum Mechanics / Jan 28 / 2. Лагранжева механика
    Vladimir Losyakov, Quantum Mechanics / Jan 28 / 3. Электромагнитная волна

    Vladimir Losyakov, Quantum Mechanics / Feb. 4 / 4. Электромагнитные волны, поляризаторы, фотоэффект

    Pavlo Gavrylenko, Quantum Mechanics / Feb. 4 / 5. Гамильтонова механика

    Павел Гавриленко — Квантовая механика. Feb. 11 / 6. Скобка Пуассона, теорема Нётер

    Vladimir Losyakov, Quantum Mechanics / Feb. 11 / 7. Начальные и конечные состояния, Гильбертово пространство

    Vladimir Losyakov, Quantum Mechanics / Feb. 18 / 8. Гильбертово пространство, разложение единицы, операторы

    Pavlo Gavrylenko, Quantum Mechanics / Feb. 18 / 9. Континуальный интеграл

    Pavlo Gavrylenko, Quantum Mechanics / Feb. 25 / 10. Континуальный интеграл

    Vladimir Losyakov, Quantum Mechanics / Квантовая механика. Feb. 25 / 11. Каноническое квантование

    Vladimir Losyakov, Quantum Mechanics / Mar. 4 / 12. Временная эволюция

    Pavlo Gavrylenko, Quantum Mechanics / Mar. 4 / 13. Уравнение Шредингера из функционального интеграла

    Pavlo Gavrylenko, Quantum Mechanics / Mar. 11 / 14. Представление Гейзенберга, представление взаимодействия

    Vladimir Losyakov, Quantum Mechanics / Mar. 11 / 15. Представления канонических коммутационных соотношений

    Pavlo Gavrylenko, Quantum Mechanics / Mar. 18 / 16. Уравнения Гейзенберга и их решение. Отступление про Алгебры Ли

    Pavlo Gavrylenko, Quantum Mechanics / Mar. 18 / 17. Когда квантовые формулы совпадают с классическими. Moyal product

    Vladimir Losyakov, Quantum Mechanics / Mar. 25 / 18. Спектр оператора координаты в квантовой механике

    Pavlo Gavrylenko, Quantum Mechanics / Mar. 25 / 19. Про экспоненты и коммутаторы

    Vladimir Losyakov, Quantum Mechanics / Apr 1 / 20. Измерение координаты и импульса
    Vladimir Losyakov, Quantum Mechanics / Apr 1 / 20 (продолжение). Измерение координаты и импульса

    Pavlo Gavrylenko, Quantum Mechanics / Apr 1 / 21. Сохраняющиеся величины (квантовая терема Нётер)

    Pavlo Gavrylenko, Quantum Mechanics / Apr 8 / 21. Подготовка к контрольной

    Vladimir Losyakov, Quantum Mechanics / Apr 8 / 22. Собственные состояния в разных потенциалах

    Vladimir Losyakov, Quantum Mechanics / Apr 15 / 23. Атом водорода в классической механике
    Pavlo Gavrylenko, Quantum Mechanics / Apr 15 / 24. Квантование оператора момента


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

Instructor: Andrei Semenov

  • Microsoft Teams

Никита Некрасов

Введение в локализацию для N=2 SYM / Nov 5
Пространства модулей инстантонов / Nov 12
Виртуальный характер касательного пространства (1/2) / Nov 26
Виртуальный характер касательного пространства (2/2) / Dec 3
Формулы локализации / Dec 10
Геометрическое квантование, локализация / Dec 17
Конформные блоки аффинной алгебры / Dec 24

Некоммутативные инстантоны (1/4) / Feb 4
Некоммутативные инстантоны (2/4) / Feb 11
Некоммутативные инстантоны (3/4) / Feb 18
Некоммутативные инстантоны (4/4) / Feb 25
Геометрия пространств Калоджеро-Мозера и ADHM / Mar 4
Системы Калоджеро-Мозера и их связь с калибровочными теориями / Mar 11
Квантовая тригонометрическая система Калоджеро-Мозера / Mar 18
Двумерный Янг-Миллс (1/3)/ Mar 25
Двумерный Янг-Миллс (2/3)/ Apr 1
Пространства вакуумов в калибровочных теориях (1/2) / Apr 15
Пространства вакуумов в калибровочных теориях (2/2) / Apr 22