

Top mathematical achievements of the last 5ish years, maybe - rndn
http://richardelwes.co.uk/2015/06/18/the-top-10-mathematical-achievements-of-the-last-5ish-years-maybe/

======
fdej
The result of Ono and Bruinier is significant, but I don't know if it really
belongs on this list.

Euler's pentagonal number theorem already gives a simple algebraic description
of the partition numbers. The so-called "finite algebraic formula" of Ono and
Bruinier is only more "finite", "algebraic" or "formula" in a very artificial,
press-release-exaggerated sense.

Sort of like claiming that any of the contorted "prime formulas" that various
people have come up with
([http://mathworld.wolfram.com/PrimeFormulas.html](http://mathworld.wolfram.com/PrimeFormulas.html))
provide a better finite description of the primes than the good old sieve of
Eratosthenes.

Now, this comparison is not quite fair. Unlike those prime formulas, the Ono-
Bruinier result is actually mathematically significant in that it does give
you more information about the partition numbers. It's a nice result, but is
it more interesting than hundreds of other interesting results in mathematics
in the last 5ish years? Even if we just look at the extremely narrow field of
mathematics that is the study of partition numbers, there are other nice
recent achivements such as Radu's proof of Subbarao's conjecture.

Ono is certainly an excellent mathematician, but he might be even more
excellent at getting extreme amounts of publicity for his results. I don't
want to give the impression that I'm sour about this. On the contrary, I think
it's awesome that mathematics can get this kind of publicity, and Ono earned
it (other mathematicians should learn from his example!), but it's something
to keep in mind.

Likewise, about HOTT. It's very promising, but is it really an "achivement"
yet? Who knows, the hype could turn out to be justified. I guess if you want
to assess scientific progress as recent as in the last five years, you have to
try to predict the future as well.

~~~
Garlef
Yes: HOTT is an acchievement. It differs from the other items on the list,
though, as it belongs to the "theory building" branch of mathematics and not
to the "problem solving" branch.

In the end, the current implementation of HoTT might not be the one standing
the test of time; But the central idea behind the programme is here to stay: A
refactoring of mathematics. (Higher) category theory is currently the best
tool to uncover deep connections between different areas of mathematics. But
doing higher category theory using set theory can be compared to writing a
webapp directly in machine code: Possible, but cumbersome. A top level
language providing suitable abstractions is needed and HoTT is one proposal.

It can be compared to the ongoing shift from imperative to functional
languages.

------
aruss
I'm somewhat surprised that Gentry's fully homomorphic encryption wasn't
mentioned. It's definitely more applied math than anything else, but if you're
going to mention RSA and factoring you might as well mention the biggest
crypto breakthrough in the last 5-6 years.

------
claudius
The list at the bottom regarding computational achievements is possibly
slightly more relatable and also contains this nice gem:

    
    
        The most impressive feat of integer-factorisation using a quantum
        computer is that of 56,153=233 × 241. The previous record was 15.

------
BasDirks
Mochizuki's work is interesting. I wonder how long that will take to verify.

------
agounaris
worst website background ever...

------
hessenwolf
Four extra lines in a nice IMRaD format would have been a nice addition to
start the discussion. It looks like the reader still has a lot of work to do.

~~~
gjm11
I can't imagine what four extra lines would have made things much easier for
the reader. These results are from a variety of fields of mathematics;
understanding the context for any one of them is a substantial piece of work.
They aren't part of some single scientific experiment that could be presented
"in a nice IMRaD format".

What sort of four extra lines did you have in mind?

(Note that each of Elwes's top 10 comes with a link to further information,
generally in reasonably accessible form.)

~~~
hessenwolf
1\. What the question is and why it is important.

2\. Broadly, what area of mathematics was used - how did they go about solving
it.

3\. Do we have a concrete proof? For a general case?

4\. Where are we now and what is next?

~~~
gjm11
So, four extra lines _for each item_? That would (much more than) double the
length of the piece. Except that for several of them, much of that information
is already there.

Still, let's have a go.

10\. _Mochizuki and the abc conjecture_. The abc conjecture says, roughly,
that you rarely have a+b=c where a,b,c are all products of large powers of
prime numbers. You can think of this as saying that the additive and
multiplicative structures of the integers are kinda-independent. The abc
conjecture, if true, would imply lots and lots of other conjectures that
number theorists have made. (In particular, it "almost implies" Fermat's Last
Theorem.) Mochizuki claims to have proved it using a very complicated thing
he's created called "inter-universal Teichmueller theory". Teichmueller theory
is all about spaces whose structure is based on the complex numbers, which you
can think of as being obtained as follows: start with the integers; allow
yourself to form fractions (giving the rational numbers); "fill in the gaps"
(forming the real numbers); "add roots of algebraic equations" (forming the
complex numbers). But there are some not-so-obvious ways to "fill in the gaps"
where instead of the usual limiting process where you do things with smaller
and smaller errors (sqrt(41) ~= 6, 6.4, 6.40, 6.403, 6.4031, etc.) you try to
do things that are right "modulo large powers of p" for some prime p; e.g.,
sqrt(41) = 1, 21, 821, 3821, 03821, 203821, etc.; but you should really think
of this as exhibiting one thing mod powers of 2 and another mod powers of 5.
Doing something comparable to Teichmueller theory using these gives you
"p-adic Teichmueller theory", and then "inter-universal Teichmueller theory"
is a further generalization that I don't understand. No one is quite sure
whether Mochizuki's proof is correct; it's based on a huge amount of abstruse
stuff he's invented that no one was very interested in before he claimed to
have used it to prove the abc conjecture, and getting one's head around all
that takes time. The next step is for the mathematical community (or at least
some small bit of it) to understand IUTT well enough to check Mochizuki's
proof. I believe that's happening, but it won't be quick.

Hmm, that's pretty long (despite not managing to do more than gesticulate
vaguely in the direction of the relevant mathematics). I'll do the others, but
I think it'll be one per comment rather than a single monstrously long
comment.

~~~
schoen
Thank you for doing all of these, that's a terrific contribution.

~~~
hessenwolf
Isn't it? I haven't enjoyed somebody being disliking a comment of mine so much
:). I've been reading these all day.

~~~
gjm11
You're welcome. By the way, I like your username.

~~~
hessenwolf
A fellow reader :)

(I just ego-surfed my hn name. The twitter guy is somebody else.)

