latest dicoveries in the field

What are the latest discoveries in the field of mathematics?

     People make new mathematical “discoveries” everyday. Well, I suppose it really depends on what you mean by the term “discoveries.” If proving results and building theories counts, then we are on the same page.

Below are some notable, recent mathematical discoveries, which constitute a far-from-exhaustive list, rife with personal bias. That is the nature of such lists. Mathematics is vast, and most discoveries come from niche fields with very little exposure to even the general mathematical public. Nonetheless, some of the most famous discoveries are included.

The abc Conjecture (Mochizuki)

This conjecture, despite its simple statement, is incredibly important in number theory. The proof itself, as usual, is the real story. Not only is Mochizuki's proof is truly massive, fleshing out a mathematical theory its author has called "Inter-Universal Teichmüller Theory" (IUTT or IUT or IU-Tech or, using Fesenko’s terminology, “arithmetic deformation theory”). Unfortunately, Mochizuki’s proof has yet to be confirmed, so it could be flawed. Few, if any, fully understand it.

Here is a one particular statement of the conjecture: For each ϵ>0

, there are only finitely many triples (a,b,c) of coprime positive integers for which a+b=c

that satisfy the equation

c>d1+ϵ,

where d

is the product of the distinct prime factors of the product abc

.

The Twin Prime Conjecture (Zhang, Tao, et al.)

This conjecture is somewhat simpler to state but no less remarkable or important for number theory. Terry Tao, the mathematician heading some of the recent development on this, is also better known than Mochizuki, who was unheard of by most until a few years ago due to the obscurity of his preferred field of anabelian geometry, among other reasons. Tao is a Fields medalist who is commonly thought to be one of the greatest mathematicians of this century.

Anyway, about the conjecture and the progress. Here is the statement of the conjecture itself: There are infinitely many twin primes (i.e., numbers with no factors other than one and themselves in pairs where they are separated by only two such as 3 and 5).

The progress on this is thanks to a previously unknown mathematician named Yitang Zhang, who proved that there were infinitely many primes with prime-gap approximately 70 million (the first finite gap ever to be established). This was a big deal. More recently, the hugely collaborative Polymath Project, which is partly maintained on Tao's blog, has substantially tightened the bound. Thus far it has been found that there are infinitely many primes of separation of 6, assuming some other conditions.

The Navier-Stokes Problem (Tao – Again)

There's a reason I said Tao might be one of the greatest mathematicians around. Here we see he is involved in another substantial development.

This problem belongs to the field of mathematical physics and has to do with fluid mechanics. The Navier-Stokes equation is an extraordinary complex differential equation for which the smoothness and existence problem for the three-dimensional case is unsolved. This is also a Millennium Prize Problem, meaning that the Clay Mathematics Institute offers one million dollars to whoever solves it completely in the form they present it.

The partial result that Tao published handles an averaged version of the problem. This not only gives some insight into the full problem by solving a simpler version, but also provides a methodology of proof that could assist in solving the full problem.

The Cobordism Hypothesis (Introduced: Baez & Dolan; Proved: Lurie)

This is an older result than the other three, and it is much more difficult to describe even vaguely, but it is no less important.

The result has to do with an insightful classification of topological quantum field theories (yes, physics again) using, among other things, the abstract nonsense known as category theory. The proof comes from a very well-known Harvard mathematician and MacArthur fellow.