This year's Fields Medals should be of great interest for this crowd: most of the winners study concepts very familiar to computer scientists, while their methods touch upon the deepest, most intricate areas of pure mathematics. From sphere packings, prime numbers and lattices to percolation theory to information complexity to point configurations and much more. They are studied using representation theory, matroids, automorphic forms, real algebraic geometry and probabilistic methods among many other tools. People should check both the popular expositions in the link and the laudatio for interesting information.
[paraphrased from a comment I made in another submission]
I always found it weird how it was dominated by the french, with very few germans in between (only 2!). What are they doing differently? It's not that math is unpopular here, a lot of students are studying mathematics and every university has a math department. But they are absolutely dominating and I can't really explain it.
France has an insane system of hyper-elite universities, the Grandes écoles (https://en.wikipedia.org/wiki/Grande_%C3%A9cole). I have a friend who was in one, it's even crazier than it looks from the outside.
We have nothing comparable in Germany. If that's good or bad I don't know, but regarding super-elite competitions like the Fields-Medal I don't think there is another way.
It's cultural. France has a long tradition of selecting and funnelling the brightest math students to elite schools and institutions. In particular, there's a system of "preparatory schools", 2 years right after high school, where students work hard to get to the best possible "Grande Ecole" (engineering school). Not only the incentive is high for the students to perform well, but maths teachers in these classes are extremely qualified.
I should add that the most selective "Grande Ecole" produce more managers than engineers. Maths is mostly a mean of selecting elite, since most of these students won't work in science or engineering.
Ironically, France students on average are getting worse and worse at maths. This system works well for some elite, but not for the rest of the country. I like to think it's the same for gastronomy. France has great restaurants, but a lot of French eat mostly pizza, Mac Donald's and junk food.
Danish is worse, or at least as bad. Sixty is three twenties, so fifty is half three twenties. Eighty is four twenties, so seventy is half four twenties. Ninety is half five twenties, but a hundred is just a hundred. Sanity, at last! (The twenties have been abbreviated over the years to just an -s ending, so sixty is “tres” and so on. But still …)
I didn’t say it makes much sense. But it fits with how we talk about time in Norwegian and Danish both: When English speakers say half past five, we say half six. Which is a half hour before six. In the same way, I suppose half three twenties (halvtres in Danish) is short for half a twenty short of three twenties. So yes, there’s logic to this madness.
20 x 4 = 4 x 20 by commutativity. But the name of the number 80 is also basically 4 x 20. I guess the English equivalent would be something like ‘what is one thousand times one hundred?’ ‘One hundred thousand because multiplication is commutative’ but that is incredibly unfunny. I guess in English you think of one hundred thousand as being decomposed into a few numbers but maybe in French you just think of quatre-vingts as a single concept and will do a double-take when hearing the reason for the answer before noticing that the name of the number can be decomposed into quatre and vingt. Obviously if the joke was ever funny before it isn’t now.
It is at least partially due to the education system: MPSI -> MP -> ENS Cachan / Lyon / Ulm. Very hard entrance exams to make sure they get the best students. Small promotion (IIRC around 40 students per year for ENS Ulm FIMFA) = homogeneous level.
Can anyone give a little insight into how some of these discoveries can have engineering applications? There are some abstracts, but has anyone looked at this and seen potential uses in computer science? I'm surprised there aren't more comments on this on Hacker News considering it's the Fields Medal.
> I'm surprised there aren't more comments on this on Hacker News considering it's the Fields Medal.
Many would argue that looking for engineering applications in mathematics is not why mathematicians do mathematics. That mathematics does sometimes have engineering applications is a mysterious and beautiful side effect. It isn't the motivation behind doing mathematics.
Indeed, a possibly apocryphal tale holds that matrix algebra, while a beautiful mathematical construct, was considered by its inventor to be utterly useless. We know better today.
One of the winners says that there are no concrete applications, but that some of the tools developed ("interpolation formulas") could potentially be useful for solving differential equations and signal processing, but doesn't go into detail.
I'm only in my late 20s but have managed to rid myself of the pursuit/worry/desire/envy of being a great mathematician.
What I worry about now is how to approach the subject with my children.
Because on one hand I do not want them to not be ambitious as younglings.
But I also do not want them to experience the crushing disappointment of realizing they are not talented, as that was one of the hardest bitter pills of my life.
When I read things like this I feel happy that I went to high school with some truly extraordinary people. I already grew up knowing I wasn’t that talented compared to the real talented people.
Now I’m at peace with being just a person. I don’t expect more from myself than just trying my best. I only get disappointed if my effort is lacking.
I had similar feelings of lack of talent when I was doing my higher maths/stats. Then I realized that I don't really love either; that realization prompted me to drop out and through a series of serendipitous events, I ended up learning computers. That is when I realized that I really love programming. And that ensured that I don't have any crushing disappointments with respect to the probable lack of my talent in programming; even though I am never going to be even close to being the best in the field, its fine - I just love it.
Honestly if you are a mathematician you are already at the top of humanity in terms of cognition. Saying that is not talent is quite something. But I guess I can understand if you see yourself in relation to other mathematicians.
It's interesting that the Field's medal award is quite a bit lower than the Nobel Prize's (~$15K vs ~$1mm) and is only awarded every four years, yet it still garners a lot of interest, speculation, and possibly envy from non-winners. Is there some kind of side-benefit to winning beyond the renown (which is probably a double-edged sword)?
Have to be under 40 too, so it isn't really an equivalent to the Nobel. Maths culture is weird though, long held belief that it is a young person's game.
There is the Abel prize, which so far has been awarded only to pretty old people. But it has only been around since 2003, so it does not quite have the weight of history behind it.
This is the shortcoming Abel prize is trying to fulfill, more like a lifetime achievement and as others mentioned Andrew Wiles is one of the recipients of the prize
I do like that the Fields Medal goes to people nearer to the start of their careers than the end like the Nobel awards.
I would be curious about, say, the next five people that didn’t make the cut. Is it common that they still go on to get a Fields Medal?
Also I wonder if there was ever a joint result that led to both authors being awarded a Fields Medal. I can’t find one but strong collaborations are not uncommon, so…
She was a star way before the current events. She solved a long-standing open problem in fundamental maths. This is the holy grail in maths research. The main resulting paper is single-author, so there is no doubt who did the work.
She got her PhD from Bonn in 2013 and became a full prof at EPFL in 2018 (in EU this type of progression takes 10-20 years, and this is extremely impressive even by US standards). It was very clear to everyone at that time that she would be big.
Thank you for the answer, that's what I expected (as an answer, I mean).
And to respond to a HN-er above who was mentioning that he/she found my question "distasteful" and "insinuating", yes, the reality today is that lots of people involved on one side of the current conflict in Eastern Europe have been receiving prizes and media attention just because of their nationality at this present moment in history, not necessarily because of their actual work in their respective fields. Glad to see that the "judges" (I have no idea who gives up this award, or how) behind the Field Medals have steered clear of all that and have sticked to merit.
I find this question distasteful, since it's asked with nothing more than a vague insinuation of Viazovska potentially not being "that good" as motivation. Here's the long citation:
"A very long-standing problem in mathematics is to find the densest way to pack identical
spheres in a given dimension. It has been known for some time that the hexagonal packing of
circles is the densest packing in 2 dimensions, while in 1998 Hales gave a computer assisted
proof of the Kepler conjecture that the faced centered cubic lattice packing gives the densest packing in 3 dimensions. The densest packing wasn't known in any other dimension until in
2016 Viazovska proved that the E8 lattice gave the densest packing in 8 dimensions and, very shortly afterwards, together with Cohn, Kumar, Miller and Radchenko, proved that the Leech lattice gave the densest packing in 24 dimensions. Viazovska's approach built off work of Cohn and Elkies, who had used the Poisson summation formula to give upper bounds on the possible density of sphere packings in any dimension. Their work had suggested that in 8 and
24 dimensions there might exist a radial Schwartz function with very special properties (for
instance it and its Fourier transform should vanish at the lengths of vectors in the respective lattice packings) which would give an upper bound equal to the lower bound coming from the known lattice packings. Viazovska invented a completely new method to produce such
functions based on the theory of modular forms.
Viazovska has developed these ideas in other directions. With Radchenko she proved the
unexpected result that any even Schwartz function such that it and its Fourier transform
vanish at the square root of every non-negative integer must be identically zero. In fact they showed that any even Schwartz function can be written [...] for certain special functions a_n and b_n.
With Cohn, Kumar, Miller and Radchenko she showed that the E8 and Leech lattice not only
gave optimal sphere packings in dimensions 8 and 24, but that they minimize energy for every
potential function that is a completely monotonic function of squared distance."
I agree that it is a distasteful question. But what is much more distasteful that it wouldn't be out of the realm of possibility for the answer to that question to be yes. Not that I am in any position to make that assessment.
I'm not the person who made the statement. They asked the question. That's exactly the opposite of making an assessment. That person tries to find someone who CAN make that assessment to answer the question.
> I agree that it is a distasteful question. But what is much more distasteful that it wouldn't be out of the realm of possibility for the answer to that question to be yes.
So you said it is not out of the realm of possibility for it to be the case, that is an assessment. By your own statement, you are in no position to judge, so how do you know it is possible?
How do I know it's possible? Because it's decided by people. People are sensitive for these kinds of things. What I meant with assessment in this case is that I cannot judge whether this is happened in this specific case. As in I don't know enough about mathematics to make an assessment. Not I don't know enough about committees to say it's possible or not. I thought that would be clear from the post but I can see how you misunderstood it.
The short answer is that we don't really know; they are related to some of the most mysterious aspects of representation theory and number theory.
What we know is that these dimensions support lattice structures with exceptional symmetries. The intricate geometry of these symmetries interacts with number theory to provide spectral data (automorphic forms) that have very rare properties; this is the starting point for Maryna's investigations.
Okounkov's popular scientific exposition "The magic of 8 and 24", linked in the article, goes into considerable detail on the constructions.
This is not an innocent question, this user is a Kremlin tool, reading his comments on reddit is pretty disgusting, here's for example blaming Ukraine for the fact Russia is bombing hospitals: https://old.reddit.com/r/europe/comments/vp08m0/the_health_c...
He justifies bombing of children hospital in Odessa (far from the frontline) by the fact that Ukraine sometimes uses evacuated buildings on the frontline as bases.
The commenter stated that the Ukrainian Army often puts troops and military hardware near civilian infrastructure (true). They did not assert that this was the case for the specific Odessa hospital in question (merely that it was a possibility), or that this justifies Russian bombings.
What else do you find "implicit"? Does this Ukrainian Military set-up [1] in yet another school is implicit or explicit? Or using school buses [2] for military evacuation? And I wish those had been the only examples, they're not. And last but not least, do you find the salute accompanying these "Slava Ukraini" chants [3] explicit or implicit? They look pretty explicit to me.
Maryna Viazovska received the Fields medal for her groundbreaking work on the optimality of the E8 and Leech lattices for sphere packings; she has been expected to win the Fields medal for 3 years now, long before the war.
Politics plays an important role in those awards, for example this year's awards were due to be presented at the ICM in Saint Petersburg, Russia but the ceremony was moved to Helsinki. But for this particular case, Maryna was on everyone's list and she has done exceptional work.
The choice of venue and the choice of Fields medal recipients are unrelated decisions, taken by different groups of people. Of course the former is much more strongly influenced by world events. And indeed, it took only a few days after the invasion to realize that the planned ICM in Saint Petersburg was impossible.
I wasn't even thinking about that, but then I watched all videos, and in hers she spends more time talking about the war than she does talking about her work. I understand she's been affected but that feels out of place here.
Marie Skłodowska-Curie, who got her second Nobel for Radium and Polonium discovery, named Polonium to bring attention to the Russian occupation of her Motherland - Poland. I don't see anything out of place with Maryna Viazovska talking about war as it may be hard to see anything more important than your Motherland, your people, being totally destroyed.
You misunderstood my comment. I said it felt out of place *here*, in a video dedicated to her mathematical work, hosted on mathunion.org (the clue is in the URL). Doesn't mean she cannot talk about the war in general.
As for Marie Curie, I don't think they had Internet and 24-hour news channels back then. I don't need mathematicians to bring attention to the war in Ukraine when everyone is constantly talking about it already.
By then the occupation of Poland had been going for more than a century, so it wasn't about news. And in case of Maryna it also doesn't seem to be about news. It is her well-deserved chance and her choice to say what she thinks is needed to be said on such a momentous occasion.
[paraphrased from a comment I made in another submission]
We have nothing comparable in Germany. If that's good or bad I don't know, but regarding super-elite competitions like the Fields-Medal I don't think there is another way.
I should add that the most selective "Grande Ecole" produce more managers than engineers. Maths is mostly a mean of selecting elite, since most of these students won't work in science or engineering.
Ironically, France students on average are getting worse and worse at maths. This system works well for some elite, but not for the rest of the country. I like to think it's the same for gastronomy. France has great restaurants, but a lot of French eat mostly pizza, Mac Donald's and junk food.
In German 25 is five and twenty. 67 is seven and sixty. Annoying but bearable.
However, in French, 60 is sixty. 70 is sixty ten. 80 is four twenty. 90 is four twenty ten. 99 is four twenty nineteen.
Any French person is steeled by their crazy numbering system, from an early age, allowing them to reach the highest mathematical heights as they grow.
I'm confused. Wouldn't that be... thirty? :-)
A: 80 because multiplication is commutative.
¯\_(ツ)_/¯
That's a really cool name as well, like "Standard Deluxe" or so.
Many would argue that looking for engineering applications in mathematics is not why mathematicians do mathematics. That mathematics does sometimes have engineering applications is a mysterious and beautiful side effect. It isn't the motivation behind doing mathematics.
https://youtu.be/V8icZM14v6Q?t=119
Now I’m at peace with being just a person. I don’t expect more from myself than just trying my best. I only get disappointed if my effort is lacking.
(I got out of the research game two years after my PhD, so it's not like there was any doubt.)
I would be curious about, say, the next five people that didn’t make the cut. Is it common that they still go on to get a Fields Medal?
Also I wonder if there was ever a joint result that led to both authors being awarded a Fields Medal. I can’t find one but strong collaborations are not uncommon, so…
She was a star way before the current events. She solved a long-standing open problem in fundamental maths. This is the holy grail in maths research. The main resulting paper is single-author, so there is no doubt who did the work.
She got her PhD from Bonn in 2013 and became a full prof at EPFL in 2018 (in EU this type of progression takes 10-20 years, and this is extremely impressive even by US standards). It was very clear to everyone at that time that she would be big.
Or just look at the paper: https://arxiv.org/abs/1603.04246 you can't fake that with politics.
And to respond to a HN-er above who was mentioning that he/she found my question "distasteful" and "insinuating", yes, the reality today is that lots of people involved on one side of the current conflict in Eastern Europe have been receiving prizes and media attention just because of their nationality at this present moment in history, not necessarily because of their actual work in their respective fields. Glad to see that the "judges" (I have no idea who gives up this award, or how) behind the Field Medals have steered clear of all that and have sticked to merit.
"A very long-standing problem in mathematics is to find the densest way to pack identical spheres in a given dimension. It has been known for some time that the hexagonal packing of circles is the densest packing in 2 dimensions, while in 1998 Hales gave a computer assisted proof of the Kepler conjecture that the faced centered cubic lattice packing gives the densest packing in 3 dimensions. The densest packing wasn't known in any other dimension until in 2016 Viazovska proved that the E8 lattice gave the densest packing in 8 dimensions and, very shortly afterwards, together with Cohn, Kumar, Miller and Radchenko, proved that the Leech lattice gave the densest packing in 24 dimensions. Viazovska's approach built off work of Cohn and Elkies, who had used the Poisson summation formula to give upper bounds on the possible density of sphere packings in any dimension. Their work had suggested that in 8 and 24 dimensions there might exist a radial Schwartz function with very special properties (for instance it and its Fourier transform should vanish at the lengths of vectors in the respective lattice packings) which would give an upper bound equal to the lower bound coming from the known lattice packings. Viazovska invented a completely new method to produce such functions based on the theory of modular forms.
Viazovska has developed these ideas in other directions. With Radchenko she proved the unexpected result that any even Schwartz function such that it and its Fourier transform vanish at the square root of every non-negative integer must be identically zero. In fact they showed that any even Schwartz function can be written [...] for certain special functions a_n and b_n.
With Cohn, Kumar, Miller and Radchenko she showed that the E8 and Leech lattice not only gave optimal sphere packings in dimensions 8 and 24, but that they minimize energy for every potential function that is a completely monotonic function of squared distance."
Maybe if in no position to make an assessment it is better not to make one?!
> I agree that it is a distasteful question. But what is much more distasteful that it wouldn't be out of the realm of possibility for the answer to that question to be yes.
So you said it is not out of the realm of possibility for it to be the case, that is an assessment. By your own statement, you are in no position to judge, so how do you know it is possible?
What we know is that these dimensions support lattice structures with exceptional symmetries. The intricate geometry of these symmetries interacts with number theory to provide spectral data (automorphic forms) that have very rare properties; this is the starting point for Maryna's investigations.
Okounkov's popular scientific exposition "The magic of 8 and 24", linked in the article, goes into considerable detail on the constructions.
[1] https://twitter.com/mdfzeh/status/1543222792363163658
[2] https://twitter.com/mdfzeh/status/1543306307385556992
[3] https://twitter.com/Numinis/status/1542056886299762690