Peng Zhou

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• Discussion with Xin Jin about her proof

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I was discussing with Yixuan yesterday. He mentioned a few works by Siu-Cheong Lau are noteworthy.

• Localized mirror functor. No, not just probing the space using a compact immersed Lagrangian, but with deformation, allowing immersed Lagrangians.
• Nakajima quiver variety coming from Floer theory

What is the Crane-Frenkel 4d TQFT? What is the small quantum group? Why we need to be small? Just so the Verma can be defined?

I have been thinking about Liouville sector. The condition on stop is very harsh, in the sense that the Liouville flow need to preserve the boundary. I am not sure if GPS themselves constructed these required structures.

• Construct Liouville structure?
• Examples

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• colimit of algebras

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• Mina relates Cherns-Simon's partition function $CS(\Sigma \times S^1)$ with our rank of $K$-theory formula.
• Lecture note on Ringel-Hall algebra.

We had a long detour into Cautis-Kamnitzer's constructions, at least we had some familiarity with flag variety.

But, I still do not know about disk with 3 or more stops.

• Disk with 2 stops and $k$-strands, is assigned to $D-mod(pt/GL(k) )$ which is derived equivalent to $NH_k-Mod$.
• Disk with 2 stops, goes to the monoidal category $U_-$.
• Disk with 3 stops with $k$ strands, we have $k+1$ many objects serving as generators, and they form an full exceptional collection.

Today, I did

1. more study on the slices of affine Grassmannian
2. Return to CKL

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I still want to understand what Cautis-Kamnitzer-Licata did.

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I watched a bunch of Tim Logvinenko's video, talking about generalized braid category and its representations. The origin of the story is trying to understand Caustic-Kamnitzer's construction, namely, what acts on the ambient space's coh category rather than the slice's.

I didn't understand CK's construction, so I probably should go back and read.

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