Introductory talks
We have a series of introductory talks, supplying the relevant background for the mini-courses.
- 1. $A_\infty$ categories. Speakers: Netanel Blaier (MIT) and Amitai Zernik (Hebrew University).
- Definition, morphisms, quasi-isomorphisms.
- Homological perturbation lemma, and its consequence: any quasi-isomorphism is invertible up to homotopy.
- Hochschild homology and cohomology, as background for the split-generation argument, and their duality in the
`weakly cyclic' situation .
Recommended reading:
- 2. The Fukaya category. Speaker: Cedric Membrez (Tel Aviv University).
- Basic definition.
- Closed-open string maps and their properties (in particular, their duality in the compact case).
- Abouzaid's split-generation criterion (e.g., in the wrapped case or the monotone case).
Recommended reading:
- 3. Coherent sheaves. Speaker: Uri Brezner (Hebrew University).
- Basic introduction and definitions.
- Introducing Beilinson's exceptional collection $\Omega^j(j)$ and showing that its derived restriction to the
hypersurface split-generates.
Recommended reading:
- 4. Matrix factorizations. Speakers: Adam Gal (Weizmann Institute) and Lena Gal (Tel-Aviv University).
- Basic definitions.
- Dyckerhoff's minimal model for the compact generator of the category of matrix factorizations of a potential with
isolated singularity. This connects with the homological perturbation lemma listed under '$A_\infty$ categories'.
- Orlov's quasi-equivalence between graded matrix factorizations of a homogeneous potential and the category of coherent
sheaves on the corresponding projective hypersurface.
Recommended reading:
- T. Dyckerhoff. Compact generators in categories of matrix factorizations.
Duke Math. J., 159(2):223–274, 2011.
- D. Orlov. Derived categories of coherent sheaves and triangulated cat-egories of singularities.
In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, volume 270 of Progr. Math., pages 503–531. Birkhäuser Boston, Inc., Boston, MA, 2009.
- D. O. Orlov. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262, 2004
- N. Sheridan. Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space.
Inventiones mathematicae, 199(1):1–186, 2015.
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