Alexey Pokrovskiy: On the Manickam-Miklós-Singhi Conjecture

Suppose that we have a set of numbers $x_1,\dots, x_n$ which have nonnegative sum. How many subsets of $k$ numbers from $\{x_1,\dots, x_n\}$ must
have nonnegative sum? By choosing $x_1 = n - 1$ and $x_2 = ... = x_n = -1$, it is easy to see that there are such sets with only  ${n-1 \choose k-1}$ nonnegative $k$-sums. Manickam, Miklós, and Singhi conjectured that for $n$ at least $4k$ this is optimal i.e.  that for $n$ at least $4k$ there
are always at least ${n-1 \choose k-1}$ nonnegative $k$-sums in any set of n numbers.
We will survey different approaches that have been taken in trying to prove this conjecture, and relationships it has with other problems.