Peng Zhou

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

Then, I am also going to meet with Denis Auroux. His not-so-recent work with Abouzaid is about fibered Lagrangian. There is one thing that is quite interesting to me: consider the very singular fibration $f: \C^3 \to \C$, $f=xyz$. Consider a Lagrangian, say living over the line $Re f = -1$, and in the fiber, we pick the 'positive real' slice in $(\C^*)^2$. I am not sure what does 'positive real mean, but it must be cocore to the compact Lagrangian, which I think is well-defiend $|x|=|y|=|z|$ in each fiber. OK, great, then turn on the gradient flow for $Re(f)$, do we get a nice thim

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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It's a good place to work and study, the Joshua Tree Field Station (very cool hotel).

I am still trapped by the disk with 3 stops, no punctures.

But do you understand the disk with 2 stops, how the gluing works? one disk has $k_1$ strands; another disk has $k_2$ strands, both with $2$ stops, put them together, get $k=k_1+k_2$ strands.

It is about going from $BGL(k_1)$ and $BGL(k_2)$ to $BGL(k)$. I just know $T^{k_1}$ and $T^{k_2}$ merges to $T^{k_1+k_2}$. Let's try parabolic induction: we have $$ G_1\times G_2 = GL(k_1) \times GL(k_2) \gets P=P_{k_1, k_2} \to GL(k_1+k_2)=G $$

Assume everything is fine, then we have $$ BP \to B(G_1 \times G_2), \quad BP \to BG $$ The first one is saying, if you have a principle $P$ bundle $E$ over something $M$, then you can build the associated bundle of $E \otimes_P L$.

I guess the corresponding construction on sheaf is just doing pull-push along the parabolic induction.

Is there classifying space for two step flags? well, one needs to say the ranks, so it would be the $BP_{k_1, k_2}$.

Let's keep on thinking about disk with three stops.

There are a few things in common with disk with two stops and one feature in the middle, like a hole, a puncture. One need to decide if the T-branes are on the left or on the right. In the case of punctures, or several punctures, we are looking at how the standard flag in $\C^k$ map to the target flag $0 \In \C$.

I really don't know where is that target space come from?