Hanna Neumann conjecture

In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957.
In 2011, a strengthened version of the conjecture (see below) was proved independently by Joel Friedman
and by Igor Mineyev.
In 2017, a third proof of the Strengthened Hanna Neumann conjecture, based on homological arguments inspired by pro-p-group considerations, was published by Andrei Jaikin-Zapirain. 
History.
The subject of the conjecture was originally motivated by a 1954 theorem of Howson who proved that the intersection of any two finitely generated subgroups of a free group is always finitely generated, that is, has finite rank. In this paper Howson proved that if "H" and "K" are subgroups of a free group "F"("X") of finite ranks "n" ≥ 1 and "m" ≥ 1 then the rank "s" of "H" ∩ "K" satisfies:
In a 1956 paper Hanna Neumann improved this bound by showing that :
In a 1957 addendum, Hanna Neumann further improved this bound to show that under the above assumptions
She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has
This statement became known as the "Hanna Neumann conjecture".
Formal statement.
Let "H", "K" ≤ "F"("X") be two nontrivial finitely generated subgroups of a free group "F"("X") and let "L" = "H" ∩ "K" be the intersection of "H" and "K". The conjecture says that in this case
Here for a group "G" the quantity rank("G") is the rank of "G", that is, the smallest size of a generating set for "G".
Every subgroup of a free group is known to be free itself and the rank of a free group is equal to the size of any free basis of that free group.
Strengthened Hanna Neumann conjecture.
If "H", "K" ≤ "G" are two subgroups of a group "G" and if "a", "b" ∈ "G" define the same double coset "HaK = HbK" then the subgroups "H" ∩ "aKa"−1 and "H" ∩ "bKb"−1 are conjugate in "G" and thus have the same rank. It is known that if "H", "K" ≤ "F"("X") are finitely generated subgroups of a finitely generated free group "F"("X") then there exist at most finitely many double coset classes "HaK" in "F"("X") such that "H" ∩ "aKa"−1 ≠ {1}. Suppose that at least one such double coset exists and let "a"1...,"a""n" be all the distinct representatives of such double cosets. The "strengthened Hanna Neumann conjecture", formulated by her son Walter Neumann (1990), states that in this situation
The strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman.
Shortly after, another proof was given by Igor Mineyev.