During the Spring 2021 Semester Artan Sheshmani (CMSA/ I.M. A.U.) will be teaching a CMSA special lecture series on Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces.
In order to attend this series, please fill out this form.
The lectures will be held Mondays from 8:00 – 9:30 AM ET and Wednesdays from 8:00 – 9:00 AM ET beginning January 25 on Zoom.
You can watch Prof. Sheshmani describe the series here.
| Date | Topic | Video |
|---|---|---|
| 1. Gromov-Witten invariants – Definition, examples via algebraic geometry I – Virtual Fundamental class I (definition). – Virtual fundamental class II (computation in some cases) | Gromov-Witten-invariants 1 Gromov-Witten invariants 2 Gromov-Witten invariants 3 Gromov-Witten invariants 4 Gromov-Witten invariants 5 Gromov-Witten invariants 6 Gromov-Witten invariants 7 | |
| 2. Computing Gromov-Witten invariants – Three level GW classes – Genus zero invariants of the projective plane | Gromov-Witten Invariants 8 | |
| 3. Quantum cohomology (small and big) | Quantum Cohomology 1 Quantum Cohomology 2 | |
| 4. Donaldson-Thomas invariants for surfaces and threefolds – Definition, examples I – Virtual fundamental class I – Connections to GW invariants: MNOP and stable pair invariants | Donaldson-Thomas invariants 1 Donaldson-Thomas invariants 2 Donaldson-Thomas invariants 3 | |
| 5. Torsion sheaf DT theory – Sheaves on surfaces and their modularity property I – K3-fibered threefolds and S-duality conjecture – Conifold transitions and of DT invariants on nodal K3-fibrations | Torsion Sheaf DT theory 1 Torsion Sheaf DT theory 2 Torsion Sheaf DT theory 3 | |
| 6. Vafa-Witten theory as a torsion sheaf theory – Virtual fundamental class construction – Computations and proof of modularity – Instanton branch Twisted Seiberg-Witten invariants – Monopole branch and Nested Hilbert scheme invariants as quantum corrections – Higher rank flag sheaves and SU(r, C) Vafa-Witten theory for r>1 | Vafa-Witten invariants 1 Vafa-Witten invariant 2 Vafa-Witten invariants 3 Vafa-Witten invariants 4 | |
| 7. 4 folds and DT theory – Atiyah class and sheaf counting on Calabi-Yau 4 folds – Kapustin-Witten theory as a torsion sheaf theory – Modularity of DT invariants on noncompact 4 folds. – Algebraic construction of 4-fold Virtual fundamental class via localization – Degenerations and Kapustin-Witten and Vafa-Witten interaction | 4folds and DT theory 1 4folds and DT theory 2 | |
| 8. What does DT theory tell us about rationality of surface? – Overall statement of Orlov conjecture – Donaldson invariants and Seiber-witten classes | Categorical DT theory | |
| 9. A quick tour of Derived Categories – Derived category of an abelian variety – Derived Functor – Spectral sequences | ||
| 10. Derived categories of coherent sheaves – Basic structure – Spanning classes in the derived category – Derived functors in algebraic geometry – Grothendieck-Verdier duality | ||
| 11. Fourier-Mukai transforms – Orlov’s results – Passage to cohomology – Geometrical aspects of Fourier-Mukai kernel – Derived equivalence versus birationality | ||
| 12. Spherical and exceptional objects – Auto-equivalences induced by spherical objects – Braid group actions and Beilinson spectral sequence | ||
| 13. Flips and flop – Derived categories under blowup – The standard flip – The Mukai flop | ||
| 14. Rationality of surface via DT theory of 3folds and 4 folds. – Statement of conjecture – Periodic, normal, cyclic cohomology – Hochschild cohomology and torsion sheaf theory on surfaces – Passage to canonical bundle of surface and cotangent bundle of surface – Proof of correspondence I – Phantom categories and Proof of correspondence II |
