## Gauge Theory and Representation Theory, Day I

So today was the first day of Gauge Theory and Representation Theory. I had a bit of irritation getting there (late train combined with the locals telling me that the Institute was in the opposite direction, causing me to meander about Princeton for about an hour before I found it) but I did eventually manage to make it. Anyway, on to the talks (though I admit to understanding virtually nothing of them…hopefully I’ll do better tomorrow.)

The first talk was Alexander Beilinson from U. Chicago. This one I understood very little of, especially in the motivation, but he did define things called factorization lines over complex curves with a line bundle, which seemed to me to have the purpose of being able to be factored into a tensor product over open covers. He concluded by stating that two groupoids are equivalent, though I didn’t quite catch how he defined them.

More interesting to me was Dennis Gaitsgory of Harvard. He ignored local geometric Langlands, which his talk title contained, and focused on localization of Kac-Moody modules. Now, I don’t know what those are either (sensing a pattern?) but as a result of his talk, I did learn what $D$-modules are. Let $V$ be an affine variety, it has coordinate ring $k[x_1,\ldots,x_n]/I$ for some ideal $I$. Then, take the module of linear differential operators with coefficients in the coordinate ring. This is an instance of a $D$-module. More generally, you can sheafify this whole construction, and for a variety $X$, you get a category of $D$-modules. This is definitely something I need to learn more about, as is what he did next: he took the derived category of the category of $D$-modules. This is where he lost me, but it has pointed me in the direction of some new things to learn (and post about as I do).

Third for the day was Imperial College’s Richard Thomas. I don’t remember the names, but he was working on things that connect Gromov-Witten Theory to GV Theory and MNOP Theory. The thread connecting them is counting curves in Calabi-Yau Threefolds and obtaining a set of numbers that both determines and is determined by the Gromov-Witten invariants.

Finally came Gregory Moore from Rutgers. His was the only really physics based talk today, and he talked about BPS (I don’t remember what the names are) wall crossing, which is closely related to what Richard Thomas was talking about, and connecting it to physics problems. Well, at least to supergravity, string theory (mostly Type IIA) and the like. I didn’t follow much of it, though.

That was all for the first day, I’ll post again tomorrow.