I am a Ph.D. student under the supervision of János Pintz, working on applications of the GPY and Maynard-Tao methods. My main goal in attending the workshop is to learn from the experts of the field. The following are some questions that I was led to while following the recent breakthroughs. (i) Would it be possible to select a suitable family of tuples $\mathcal{H}_i$ and employ counting arguments or results of additive combinatorics so that the primes detected by their translates ($\gg N/(\log N)^k$ for each tuple) in $[N,2N]$ would have to intersect in a way that would yield additional information about the distribution of primes? (ii) Since we find less than $k$ primes in a $k$-tuple, what could we salvage from the method if we relaxed admissibility (by requiring, say, that the $k$-tuple fail to cover all residue classes only for primes $\leq\ell$ for a parameter $\ell$)? We would lose at least by leaving out the bad primes from the sifting procedure but on the other hand gain in tuple length if the method does go through. (iii) How small can $N$ be relative to $k$ or the diameter of $\mathcal{H}$ in the Maynard-Tao method to guarantee the existence of primes in tuples in $[N,2N]$?