## Second third of my ICM 2018 paper – Three Puzzles on Mathematics, Computation and Games. Corrections and comments welcome

Update: Here is a combined version of all three parts: Three puzzles on mathematics computations and games. Thanks for the remarks and corrections. More corrections and comments welcome.

Dear all, here is the draft of the second third of my paper for ICM 2018. Corrections and comments are very welcome! This part is around voting games and election rules, Boolean functions and their Fourier representation, noise stability and sensitivity especially of percolation, and a little circuit complexity and PCP. Corrections and comments are most welcome.

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### 6 Responses to Second third of my ICM 2018 paper – Three Puzzles on Mathematics, Computation and Games. Corrections and comments welcome

1. Jon Awbrey says:

Just a tangential association with respect to 2.2.2.  I have been exploring questions related to pivotal variables (“Differences that Make a Difference” or “Difference In ⟹ Difference Out”) by means of logical analogues to partial and total differentials.

For example, letting $\mathbb{B} = \{ 0, 1 \},$ the partial differential operator $\partial_{x_i}$ sends a function $f : \mathbb{B}^k \to \mathbb{B}$ with fiber $F = f^{-1}(1) \subseteq \mathbb{B}^k$ to a function $g = \partial_{x_i}f$ whose fiber $G = g^{-1}(1) \subseteq \mathbb{B}^k$ consists of all the places where a change in the value of $x_i$ makes a change in the value of $f.$

2. Joseph Malkevitch says:

Dear Gil,

I enjoyed looking at this essay. It might be nice to add references “locally” for each of the three essays.

Regards,

Joe

• Gil Kalai says:

Dear Joe,
This is a good point and I am really thinking about the best reference policy. Right now each part refers to 60 – 70 references and the space limitation makes this impossible. But the way the references appears in the text (names + year of publication) makes it quite easy to search so I plan to include only 5-10 references mainly for crucial recent papers for each part. Practically I will get to the bibliography when I will combine the parts next week. But later I may add the relevant references to each part in the blog posts.

3. Joseph Malkevitch says:
• Jon Awbrey says: