At the morning session,

**Bezrukavnikov** was on noncommutative resolution of singularities. After the lecture, I asked him whether we can construct noncommutative projective schemes after Artin, Stafford, and ven den Bergh, in higher dimensional cases.

We went to a Japanese noodle restaurant for lunch, where we ate Japanese "healthy" noodles with vegetable or fish. In Japan, we have snows everywhere (in Kyoto it was first snowflakes, but soon became stronger). In Boston and New York, it was also snowing.

At the afternoon session (14:30-),

**Nakajima**'s quiver gauge theory started from the historic Donaldson theory of instantons where we have two compactifications of translations and bubbles. We dealt with Uhlenbeck's compactification. He explained that the order to take affinization and Langlands dual group (co-weight, co-root) is important, which was different from that of Gaitsgory. We will work on infinite dimensional Grassmannian, but we have a good approximation by finite dimensional geometry; so, don't be afraid. We worked on the intersection cohomology, usual tensor product (not fusion product), and level-rank duality by Igor Frenkel (due to instanton number and the order of the cyclic group acting on the space (orbifold singularities)), which will be talked in more detail.

**Gaitsgory**'s (about one third of) talk was overlapped with Nakajima's introducion to geometric Satake correspondence. He introduced

1) something like a Tannakian category (rigid tensor category, which can be identified with a representation of algebraic group)

and

2) the notion of "compactly generated" category

and

3) many limits... (direct, inverse 2-limit, ...)

and

4) non-degenerated Killing bilinear form

and

5) deformation 1-parameter $\kappa$ for the deformation of quantum groups

To make a good algebraic theory, he made $\kappa$ an irrational parameter at the Kashiwara conference, but there are some progresses after that. He claimed the last theorem / conjecture that the Whittaker sheaf (or rather, Whittaker category) $Whit_{\kappa} (G)$ is equivalent to $KL_{{\kappa}^{-1}}$ (${}^L G$). (not in Riemann-Hilbert correspondence)

In the last note, he used some representation of Heisenberg algebra and semi-infinite cohomology depending on "$

**n (\kappa)**$", which should be a nilradical part (to construct the W-algebras) and I asked that point, because the audience was very silent after the lecture. He said "Yes, it is a nilpotent, Laurent." I am awaiting for the

**global** quantum geometric Langlnads correspondence of tomorrow's talk. (I know

**local** quantum geometric Langlands and affine Kac-Moody algebras, but I am not satisfied with the localization functor.)

*See you tomorrow!*