Introductory Enumerative Geometry

The main goal of this reading group would be to explore the broad topic of enumerative geometry.

 

We start off by going through the book “An Invitation to Modern Enumerative Geometry” by Andrea Ricolfi, most likely covering the basics of Donaldson-Thomas theory. In practice, this would mean going through Ricolfis Chapters on Localization Techniques, the geometry of the Hilbert / Quot schemes and lastly DT/PT theory for Calabi Yau Threefolds. I can see this taking up about 6 weeks if people are familiar with the subject, on a weekly schedule. This then concludes the first half of the reading group, with students now being familiar with the basics of modern Enumerative Geometry and (virtual) sheaf counting. Depending on the audience, we could also take it easier, and work thoroughly through the book instead, which might end up taking up the entire semester (which would also be fine and a lot of fun).

 

After that (if we get here), I would be happy to explore different counting theories: Gromov-Witten theory / relative GW theory / logarithmic- and tropical GW theory. This part of the reading group would then depend on the interest of the participants. Optimally, we would touch on the subject of logarithmic geometry, starting in the second half, and discuss curve / sheaf counting in the log setting. If there is interest, we would then build on log geometric principles a la Gross-Siebert to discuss logarithmic counts and touch on the subject of refined invariants and their relation to relative Gromov-Witten counts. Another direction would be to stick to Donaldson-Thomas theory and explore generalized DT theory / Attractor DT invariants / Calabi Yau 4fold geometries.

 

In principle, the second half of this group should be dedicated to working through smaller papers together and exploring the different flavours and ideas that come up in modern enumerative geometry.

 

Staff that could give informal input includes Prof. Szabo, Prof. Hollands and Prof. Alim, as they are all familiar with enumerative geometry.