Here's a connection that I found interesting.

A positive solution to the Kadison-Singer problem would follow from a positive solution to a stronger conjecture I called ${\rm KS}_2$ (and the eventual solution found by Marcus, Spielman, and Srivastava did do it this way).

The ${\rm KS}_2$ conjecture states that there exists a universal constant $\delta < 1/2$ such that, for any $n$ and $k$, if vectors $v_1, \ldots , v_n \in \mathbb{C}^k$ satisfy $\sum_{1 \leq i \leq N} |\langle u, v_i\rangle|^2 \leq 1$ for every unit vector $u$, then there is a subset $X \subseteq \{1, \ldots, n\}$ such that $\sum_{i \in X} |\langle u, v_i\rangle|^2 \in [\frac{1}{2} - \delta, \frac{1}{2} + \delta]$ for every unit vector $u$.

The first comment is that we don't need to consider all unit vectors $u$, only finitely many $u_j$ forming an $\epsilon$-net in the unit sphere of $\mathbb{C}^k$. Depending on how small $\delta$ is, $\epsilon$ might not even need to be very small. So fix unit vectors $u_1, \ldots, u_M$ and choose $X \subseteq \{1, \ldots, n\}$ randomly. For each $1 \leq j \leq M$ let $E_j$ be the event that we get $\sum_{i \in X} |\langle u_j, v_i\rangle|^2 \in [\frac{1}{2} - \delta, \frac{1}{2} + \delta]$ for that $j$.

If we could say that the compound event $E_1 \cap \cdots \cap E_M$ has positive probability, the problem would be solved. The connection to the Lovász local lemma is that if the $u_j$'s aren't too close together, then the $E_j$ should be approximately independent in some sense. I guess the intuition here is that the $v_i$'s that matter the most for one $u_j$ are those nearly parallel to that $u_j$, so if the $u_j$ are sufficiently far apart then success on one $u_j$ shouldn't have too much effect on success on some other $u_j$.

I got this idea from Nets Katz and John Shareshian, and we weren't able to make it work, but I still wonder if there is some form of the local lemma which would succeed. Conversely, I guess you could say that since ${\rm KS}_2$ was actually proved, we do have something resembling the local lemma in this setting.

Here's a connection that I found interesting.

A positive solution to the Kadison-Singer problem would follow from a positive solution to a stronger conjecture I called ${\rm KS}_2$ (and the eventual solution found by Marcus, Spielman, and Srivastava did do it this way).

The ${\rm KS}_2$ conjecture states that there exist universal constants $\delta < 1/2$ and $\epsilon > 0$ such that, for any $n$ and $k$, if vectors $v_1, \ldots , v_n \in \mathbb{C}^k$ have euclidean norm at most $\epsilon$ and satisfy $\sum_{1 \leq i \leq N} |\langle u, v_i\rangle|^2 \leq 1$ for every unit vector $u$, then there is a subset $X \subseteq \{1, \ldots, n\}$ such that $\sum_{i \in X} |\langle u, v_i\rangle|^2 \in [\frac{1}{2} - \delta, \frac{1}{2} + \delta]$ for every unit vector $u$.

The first comment is that we don't need to consider all unit vectors $u$, only finitely many $u_j$ forming a $\sigma$-net in the unit sphere of $\mathbb{C}^k$. Depending on how small $\delta$ is, $\sigma$ might not even need to be very small. So fix unit vectors $u_1, \ldots, u_M$ and choose $X \subseteq \{1, \ldots, n\}$ randomly. For each $1 \leq j \leq M$ let $E_j$ be the event that we get $\sum_{i \in X} |\langle u_j, v_i\rangle|^2 \in [\frac{1}{2} - \delta, \frac{1}{2} + \delta]$ for that $j$.

If we could say that the compound event $E_1 \cap \cdots \cap E_M$ has positive probability, the problem would be solved. The connection to the Lovász local lemma is that if the $u_j$'s aren't too close together, then the $E_j$ should be approximately independent in some sense. I guess the intuition here is that the $v_i$'s that matter the most for one $u_j$ are those nearly parallel to that $u_j$, so if the $u_j$ are sufficiently far apart then success on one $u_j$ shouldn't have too much effect on success on some other $u_j$.

I got this idea from Nets Katz and John Shareshian, and we weren't able to make it work, but I still wonder if there is some form of the local lemma which would succeed. Conversely, I guess you could say that since ${\rm KS}_2$ was actually proved, we do have something resembling the local lemma in this setting.