Firstly, the bounded gaps between primes project has made a considerable amount of progress since the value of H = 285278. The latest result has the following parameters:
v08ltu’s bounds, leading to a value of H = 12012.
Tables of admissible k-tuples had already been precomputed, so Terence Tao was able to consult them to find that this gives an upper bound of H = 12012. This is also dangerously close to Ben Green’s estimate of 10000 as being the smallest value of H provable using these sieve methods. Again, updated information can be found on the polymath page.
In other news, a book has been published refounding mathematics upon a recent hybridisation of homotopy theory and type theory. It is intended to be more machine-checkable than ZFC set theory, which has been used as the foundation of mathematics since Nicolas Bourbaki*. Also, category theory cannot be founded in ZFC set theory, since many of the categories are proper classes (too large to be sets).
* Nicolas Bourbaki was a set of mathematicians, rather than an individual mathematician.