• Homological Mirror Symmetry

  • Homological Mirror Symmetry

Schedule Talks: Monday 11:00-1:00 PM GMT+1; Wednesday, Friday 3:00-5:00 PM GMT+1. Room C3, Andrew Wiles Building (Mathematical Institute), Oxford. Starting 16 Jun, 2022.
Zoom Meeting ID: 233-759-6146. Passcode: \( \pi_3 (O(12)) = ? \) (If your answer contains \( \mathbb{Z} / 2 \), please type in Z2).
Problem sessions: TBD.
Organizers: Peize Liu, Jinghui Yang, Shuwei Wang, and Dekun Song.
All are welcome. Please notify us in advance.
Eastern Standard Time (EST)
Read

Notes

  • Week 1 (20 Jun-25 Jun): general preview on HMS (Peize); symplectic geometry (Dekun); simplicial categories, basics in complex geometry (Jinghui).
    • (Monday) Peize: A String-Theoretic Introduction to Mirror Symmetry
    (Wednesday) Jinghui: Simplicial categories. Kan extensions. Almost complex structure. Newlander-Nirenberg Theorem. (Notes in Hodge theory, see Week 4)
    (Friday) Dekun: Symplectic Manifolds. Lagrangian subspaces. Weinstein Tubular Neighborhood Theorem. Compatible almost complex structures (any two of almost complex structures, symplectic forms, Riemannian metric determine the third). Notes (27 June)
  • Week 2 (27 Jun-1 Jul): triangulated categories, derived categories, differential graded categories (Shuwei). \( A_\infty \)-algebras and \( A_\infty \)-categories (Peize).
    • (Monday) Shuwei: Brief introduction to A-side & B-side. Triangulated Categories. Localization. Derived Categories. Notes
    (Wednesday) Peize: DGAs, higher products, coalgebras, minimal models, \( A_{\infty} \)-modules. Notes
    (Friday) Shuwei: motivation for twisted complexes, differential graded categories, \( A_{\infty} \)-categories, \( A_{\infty} \)-functors, pre-triangulated categories, twisted complexes on \( A_{\infty} \)-categories, triangulated category \( H^0 \text{Tw} A \). (Notes coming soon)
  • Week 3 (4 Jul-8 Jul): Break! No meeting this week.

  • Week 4 (13 Jul-15 Jul): Hodge theory on closed Riemannian and Kähler manifolds (Jinghui).
    • (Monday) Break! No meeting this Monday!
    (Wednesday) Jinghui: de Rham complexes, Dolbeault complexes, Kähler manifold, principle symbols, elliptic operators, formal adjunctions. Notes on Hodge Theory
    (Friday) Jinghui: Hodge decompositions, Hodge star operator, Poincaré duality, Lefschetz opertor, Kähler identities, Lefschetz decomposition, Hodge-Riemann bilinear relations, Hodge index theorem, Serre duality. Notes see above.
  • Week 5 (18 Jul-22 Jul): Floer homology (Dekun).
    • (Monday) Dekun: Morse homology, Lagrangian Floer cohomology, moduli spaces, bubbling and Gromov compactness, orientations. Notes
    (Wednesday) Break! No meeting today.
    (Friday) Dekun: more on Floer homology, chain complexes \( CF(L_0, L_1) \), boundaries, higher products, compactification of marked discs, nodal pointed discs, gluing. Notes see Week 6 Wednesday.
  • Week 6 (25 Jul-29 Jul): moduli spaces, Floer homology (continued), operads (Dekun). derived categories, Kan extension, derived functors as Kan extensions, Ext and Tor (Peize).
    • (Monday) Emergency Break! No meeting today.
    (Wednesday) Dekun: recap on last week's lecture, more technicalities, spectral sequences, Kuranishi structures, TFTs, cobordisms, operads. Notes
    (Friday) Peize: derived categories, Kan extensions, derived functors, connection between classical derived functor (via proj/inj res) and one via Kan extension, examples of derived functor, Ext, Tor, tensor-hom adjunction. Notes
  • Week 7 (1 Aug-5 Aug): review of scheme theory, (quasi-)coherent sheaves, smoothness, perfect complexes, Serre duality, Grothendieck-Verdier duality, Bondal-Orlov reconstruction theorem, Fano varieties, \( A_{\infty} \)-structure on \( \text{Coh}(X) \) (Peize). Algebraic operads (Shuwei).
    • (Monday) Peize: schemes, sheaves of modules, quasi-coherent sheaves, bounded derived category of coherent sheaves \( D^b \text{Coh}(X) \), Kronecker quivers, \( D^b \text{Coh}(\mathbb{P}^1) \simeq D \text{Rep} Q \). Notes see Friday.
    (Thursday) Shuwei: (ns, symmetric) operads, \( \mathsf{As}, \mathsf{Com}, \mathsf{Lie} \), free operads, quadratic data, rewriting test, representing operad by rooted tree. Notes
    (Friday) Peize: smoothness, perfect complexes, Serre duality, Serre functor, Grothendieck-Verdier duality. Notes (Note: this note serves as a completion of the lecture note in Monday)
  • Week 8 (8 Aug-12 Aug): Morse theory and the symplectic quotient (Dekun); algebraic operads (continued, Shuwei); Bondal-Orlov reconstruction from derived categories (Peize).
    • (Monday) Dekun: equivariant cohomology, Morse theory, Kirwan map, symplectic quotient, examples on projective spaces and Delzant spaces, GIT, moduli of holomorphic bundles, Yang-Mills theory. Notes
    (Wednesday) Peize: (very) ampleness, (anti-)Fano variety, Calabi-Yau variety, Bondal-Orlov reconstruction theorem, Fourier-Mukai transforms, Orlov's theorem, Gabriel reconstruction theorem. Notes (Note: this note serves as a completion of the lecture notes of last week)
    (Friday) Shuwei: recap on algebraic operads, \( A_{\infty} \)-operad and \( \mathsf{As} \), Koszul duality, cooperads, cofree cooperads, quadratic data, operadic cobar construction, Maurer-Cartan equation, Koszul dual operads, bar construction, homotopy transfer theorem. Notes. Typed Notes
  • Week 9 (15 Aug-19 Aug): homotopical algebra, algebraic K-theory, intro to Dennis trace method (Jinghui). quiver and quiver representations (Shuwei).
    • (Monday) Jinghui: review of simplicial category, geometric realization, nerve of categories, homotopical algebra, homology, Quillen's theorem A & B. Notes
    (Thursday) Jinghui: algebraic K-theory, motivation for computing algebraic K-theory, Dennis trace map, Hochschild homology, negetive cyclic homology, (equivariant) spectra, intro to Bökstedt trace mathod. Notes
    (Friday) Shuwei: recap on \( A_{\infty} \)-category and twisted complexes, quivers, path algebra/category and modules over them, quiver representations, classification theorem for linear quiver, Kronecker quiver, intersection theory. Notes
  • Week 10 (22 Aug-26 Aug): Derived categories, Beilinson resolutions, Baer-Bondal, quiver representations (Peize). Intro to homotopy type theory (Jinghui). Brief computation recipes to HMS for 3-torus (Dekun).
    • (Monday) Peize: review of Orlov's theorem & Gabriel reconstruction theorem, \( D^b \textup{Coh}(\mathbb{P}^n) \) , Beilinson resolutions, Baer-Bondal theorem, exceptional sequences, tilting objects, quiver representations, semi-orthonogal decompositions. Notes
    (Wednesday) Jinghui: intro to Martin-Löf type theory, various examples of types, homotopical interpretation, hoTT, \( \pi_n(S^k) \), some easy computation on homotopy groups of spheres, Serre SS and Atiyah-Hirzebruch SS.
    (Thursday) Dekun: review on Fukaya categories, Floer homology of 2-torus, Dehn surgery method, Hochschild cohomology and its computation, periodicity phenomenon in the computation, relation with elliptic curves. Main reference: Yanki Lekili, Timothy Perutz. (2011). Fukaya categories of the torus and Dehn surgery.
  • Seminar paused due to personal reasons.


Problem Sets

Read

References



Useful Links


Mathematical Institute