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.
