# things I learned last week

• How the graph construction of Barak, Gopalan, Hastad, Meka, Raghavendra and Steurer (which shows the near-optimality of the “Cheeger-type” bound in Arora-Barak-Steurer) works.
• That Kuperberg, Lovett and Peled finally showed that, for every constant $t$, there is a sample space of size poly$n$ of permutations $\{ 1,\ldots,n \} \rightarrow \{ 1,\ldots, n \}$ such that a uniformly sampled permutation from the sample space is $t$-wise independent. This was open even for $t=4$.
• That proving the following “quadratic uncertainty principle” is an open question, and probably a very difficult one: suppose that $q_1,\ldots,q_m$ are n-variate polynomials of total degree at most 2 and $c_1,\ldots,c_m$ are real coefficients such that for every $(x_1,\ldots,x_n) \in \{ 0,1\}^n$ we have

$x_1 \cdot x_2 \cdots x_n = \sum_i c_i \cdot (-1)^{q_i(x_1,\ldots,x_n)}$

prove that $m$ must be exponentially large in $n$. (If the $q_i$ are all linear, then the standard uncertainty principle gives us $m \geq 2^n$.)

• That women can be real men, and that they should so aspire.
• That the rich really are different from you and me

More here.

1. No, the $x_i$ are in $\{0,1\}$, otherwise you can get their product with one degree-one polynomial