Gromov-Witten/Donaldson Thomas theory and Birational/Symplectic invariants for algebraic surfaces

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. 

1. Gromov-Witten invariants
– Definition, examples via algebraic geometry I
– Virtual Fundamental class I (definition).  
– Virtual fundamental class II (computation in some cases)
2. Computing Gromov-Witten invariants
– Three level GW classes
– Genus zero invariants of the projective plane
3. Quantum cohomology (small and big)
4. Donaldson-Thomas invariants for surfaces and threefolds
– Definition, examples I
– Virtual fundamental class I
– Connections to GW invariants: MNOP and stable pair invariants
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
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
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
8. What does DT theory tell us about rationality of surface?
– Overall statement of Orlov conjecture
– Donaldson invariants and Seiber-witten classes
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

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