Peng Zhou

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2023-02-13

Vivek gave a talk, and talked about stuff during dinner.

  • What you can do with skein-on-brane, and higher genus open Gromov-Witten invariant.

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2023-02-10

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2023-02-08

  • Localization for KRLW algebra over a downstairs skeleton.
  • Realizing the skeleton of hypertoric quotient.
  • Comparison of Honda-Tian-Yuan and Mak-Smith.

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2023-02-01

complete intersection model.

2023-01-27 End of AIM workshop

  • [AK]: Vector Bundle on $\P^2$
  • [EG]: Affine Springer Fiber
  • [WL]: Ruling and Stratification

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2023-01-25

I am in a group of mathematician. I watched this beautiful moving sequences of youtube videos.

2023-01-22 $xy^2$

What happens when you try to do Fukaya-Seidel category with the function $f = xy^2$ on $\C^2$?

Well, you would say first, let's compute the regular fiber, which is $(\C^*)$, parameterized by $y$. Then, you ask, what is the monodromy. I don't think there is anything special, so let's say, the monodromy is trivial.

But, is it really trivial? hmmm, what if we set the fiberwise superpotential like $W = x+y$, we get some interesting fiberwise stop and wrapping. But I have no idea why we want these fiberwise wrapping. Anyway, on the fiber over $c$, we get $W = c/y^2 + y$.

Suppose I build a U brane on the base, and compute its endomorphism.

Let me try something else. $xy(y-a)$. How about this function? Is it better in terms of being more generic? partial $x$ gives $y(y-a)=0$, and partial y gives $x (2y-1)=0$, so we get either $(0,0)$ or $(0,a)$ as singular point, all with singular value $0$. OK, so we have two critical points over the same critical value, it seems they don't talk to each other.

Is that what happens with $f=xy^2$? Not quite, here is the trouble, the generic fiber was $\C_y \RM \{y=0, y=a\}$, And it is quite different from $a=0$. Maybe you can kill something, so that it is ok.

What's the story for $f=xy$? Say, we are on the boundary $|x|^2+|y|^2=1$, and we have something stupid, like $xy > 0$ intersecting with it. This says, we have $|x|, |y|$ free, but $\theta_x = -\theta_y$. So, the stop on the boundary is like, an open cylinder in $S^3$, which contracts to $S^1$, and which gives the stop (or the thimble as cone over the stop).

Now, we do $2 \theta_y + \theta_x = 0$. Good, suppose we now cone over this Legendrian knot, what can I say?

2023-01-12 smoothing nodal curve

Discussed with Xiaohan about how to glue and smooth a nodal curve (possibly open).

2023-01-08

Heard a talk by Tim Logvinenko, about generalized braid group action. There are many interesting ideas, very concrete diagramatic. Of course, Kapranov can say, it is all in his old work with Schechtmann on perverse schober on $Sym^n \C$, but I still like this work.

Here are the series of papers by Kapranov-Schechtmann,

I doubt a bit, about Kapranov's comment that extending the braid group action to GBr action, is just direct image functor. I need to really learn how this works, for Ed Segal's construction.

Vadim's talk note is useful.

2022-12-14 Wed

  • Cross my t and dot my i.

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blog/start.txt · Last modified: 2023/07/16 17:08 by pzhou