For those who don't know, Constantinos Daskalakis, one of the winners profiled, proved that finding a Nash equilibrium (for example, in an economy) is a PPAD-complete problem: if anyone discovers an efficient algorithm for finding Nash equilibria, such an algorithm could be used for efficiently solving all other problems in the PPAD complexity class. PPAD problems are widely considered to be intractable. No algorithm is known for solving them with running time bounded by a polynomial function of input/problem size. The implications in many fields, starting with economics, are clearly significant. Daskalakis's prize is well-deserved.
There is a fantastic introductory MIT video lecture on this work by Daskalakis himself available on YouTube.[a] His enthusiasm and passion in that video lecture are contagious -- matched only by his ability to explain his work in an intuitive manner. Highly recommended for those HNers who have computer-science or economics backgrounds and are interested in computational complexity but are not yet familiar with Daskalakis's work.
How do PPAD-complete problems relate to P=NP/NP-complete problems/etc., if they do at all? Your description of PPAD-complete problems reminds me a lot of descriptions I've read of NP-related problems, but this is the first time I've heard of the term "PPAD".
Very -- very! -- informally: in PPAD-complete problems, a solution is known to exist (e.g., there is always a Nash equilibrium), but no one knows of an efficient algorithm for finding the solution. In other complexity classes widely believed to be intractable, such as NP-complete, the problems are decision problems that ask a yes/no question, and no one knows of an efficient algorithm for answering the question. By "efficient algorithm," I again mean an algorithm whose running time is bounded by a polynomial function of input or problem size.
so I guess, one way to put it, is to look at the Traveling Salesman Problem?
The decision problem asks “given a set of cities is there a tour shorter than length n?”. This is NP-Complete.
However the pop culture version of the problem is “given a set of cities, what is a tour of the shortest possible length?”. This would be the PPAD-complete version of the problem, since that solution does exist, it’s just unknown and not known to be tractable within polynomial time?
Can the Nash Equilibrium problem have a decision version whose answer is either true or false?
TSP is NP-hard for optimisation version and NP-complete for decision version. Consider how you would test that a minimal solution is minimal vs testing whether a given solution has less than a given cost. PPAD is less complete than these two classes.
The decision problem for Nash Equilibria might be, does a second equilibrium exist?
Aah gotcha. So you’re saying it’s easier to verify the solution to a Nash equilibrium problem than it is to verify the solution to the TSP optimization problem?
It bums me out that the fields medal is largely considered the top prize in mathematics, yet it has age restrictions. To people outside the field, this means that significant developments may go underreported.
Edit: Let me address the down votes / polarization on this comment: Ageism in the most esteemed prize of a particular field seems obviously wrong to me. Is there a better alternative available? (That said: kudos to the winners! Truly great achievements all around.)
It is in fact age limited precisely to combat ageism of the early 20th century. Otherwise only elders would get it (would they outlive the queue of their masters), when it's irrelevant to their career development (but beneficial to the prize's prestige). It was intentionally set up with this reasoning - not to be about lifetime contribution (which really compounds linearly in mathematics, there is no age dropoff "in the data" if you will) but making career easier for a very few young upstarts.
You're right. It does seem ageist, when seen in isolation from the outside. It seems to suggest that the field of mathematics only and solely cares about the work of young mathematicians.
Yet, is it perhaps possible that this position may signify a person possessed of a wonderful opportunity to become more educated? The Fields Medal is not intended to be the capstone recognition of a mathematcian's career any more than the Clark Medal is in economics. It's intended to award outstanding achievement for a younger contributor in a field that infamously tends to worship the more senior. The IMU awards other medals as well, including one for lifetime achievement.
In short, you're completely right to be bummed out! But perhaps you could take this glorious opportunity to learn more.
Hence my ask in the edit: which specific prize / award / recognition can I follow as an amateur math afficionado that doesn't suffer from age bias?
You didn't mention one, beyond saying that it exists or that it's given in special circumstances... Which is the challenge re:Fields being the most prestigious.
This is a strange attitude to have. The prize was specifically designed to highlight the work of promising young Mathematicians so why is this an issue? It would be like complaining that the Little League World Series is ageist because it doesn't allow older competitors. Besides there is also the Abel Prize which is equally if not more prestigious, is open to any living Mathematician and has a far more substantial monetary prize attached.
I agree. If you look at what a mathematician does they sit around and think. You don't have that sort of luxury if you are from a poor family and have to worry about basic survival. In many cases (not all) the people that complete major accomplishment X at young age Y is from an upper middle class to upper class family with lots of life advantages. Not everyone have these advantages early in life and may eventually get to those positions due to their talents, but it may take them longer given their life situations. There are poor but extremely talented underrepresented people out there the world may never know. We tend to celebrate talent from the elite class, not the poor. The elite are given resources they need to succeed, the poor are often overlooked. This can cause a major talent drain from truly exceptional but unknown folks in the world.
Is this a problem only in countries where you have to pay a lot to study? I did not had to pay for my higher education and the people that had a high talent and potentials achieved good results, there was no need to work to make money for studying and if you are good you got positions at University or get jobs at high paying companies.
In countries like the US it starts at birth. Families in rich areas will send their children to elite, selective high schools that cost upward $30,000+. Some of these schools have strong math and science curriculum and are considered direct feeders to elite universities, such as the ivies. These poorer kids could be high IQ'd just like the richer ones, but with less access to resources, may be very behind their richer peers. These richer kids will get into the elite schools, while the poor kids may go to lower end ones or not go to colleges at all. Rich parents can hire their children SAT tutors, ensure their children go to good STEM schools, have various college prep resources for their children, while the poor parents don't know anything about college, sent their children to lesser high schools, and have children that may be just as bright, but essentially ones that have no chance to be admitted to the places they otherwise might have gotten into. College admissions at elite colleges favor children from the elite that had the resources to prepare their children for such pedigree. There are exceptions to the rule, but many poor families face deep struggle. This is regardless of however high IQ'd their children may be.
I agree with all of this, but don't see what it has to do with age. The best 50-year-old mathematicians also heavily tend to come from privileged backgrounds. Privileged upbringing is simply one required component of what it takes to be among the best mathematicians of your generation.
I think this rule is just wrong. Some mathematicians come from privilege, some do not. Even for the ones who do, it's rare that they come from _great_ privilege for the time. They are the children of mayors, not kings -- professors and engineers, not the financial aristocracy.
Now it seems you're moving the goalposts on what counts as privilege. Do you have any quantitatively backed reason to believe Fields medallists are more likely to come from the financial aristocracy than say Abel prize winners?
I don’t have access to a biography of all the Field’s medalists. I am simply pointing out that there is a wide distribution of financial privilege in a list of famous mathematicians that I did not choose. I would argue this is unexceptional, but I point it out because I have trouble with the hypothesis that one must come from privilege to get anywhere in math. If anything I should be asking for the quantitative proof of the hypothesis.
There’s a commonly held belief (myth?) that mathematics is a young person's pursuit in the sense of great discoveries. If you look at Erdös you see most of his great contributions occurring when he was younger and he was prolific throughout his life.
I think the downvotes come from your sentence about significant developments going under reported. I don’t think this happens at all. If someone discovers something great or profound in math no one cares about the age of the discoverer. Wiles was given a special prize because he was past the age of 40 when he proved Fermat's Last Theorem.
When the mathematical knowledge increases, does the general character of the mathematical problems stay the same in the terms of required work and time? It's possible the problem set that young people can solve before they turn 40 will decrease after some time.
This seems better to me than the Nobel Prize where it may take Decades for your work to be recognized and is kind of a nice retirement gift for old academics. The Fields though can forever change your career trajectory when you win it.
> Accustomed to meeting the highest of standards, he saw his dissertation as mediocre. Quietly, Venkatesh started eyeing the exit ramps, even taking a job at his uncle’s machine learning startup one summer to make sure he had a fallback option.
But keep in mind this, as another comment here mentions:
> Accustomed to meeting the highest of standards, he saw his dissertation as mediocre. Quietly, Venkatesh started eyeing the exit ramps, even taking a job at his uncle’s machine learning startup one summer to make sure he had a fallback option.
Scholze's win has been predicted for quite a while now, as it turns out. He's a number theorist of stunning originality primarily known for developing a new kind of geometry, that of perfectoid spaces, for arithmetic purposes.
Here's an interview with him that will be accessible to nonspecialists:
At a higher level, here's an appraisal of his work by a professional in a closely related area:
It's not often that contemporary mathematics provides such a clear-cut example
of concept formation as the one I am about to present: Peter Scholze's
introduction of the new notion of perfectoid space. The 23-year old Scholze
first unveiled the concept in the spring of 2011 in a conference talk at the
Institute for Advanced Study in Princeton. I know because I was there. This
was soon followed by an extended visit to the Institut des Hautes Études
Scientifiques (IHES) at Bûres- sur-Yvette, outside Paris — I was there too.
Scholze's six-lecture series culminated with a spectacular application of the
new method, already announced in Princeton, to an outstanding problem left over
from the days when the IHES was the destination of pilgrims come to hear
Alexander Grothendieck, and later Pierre Deligne, report on the creation of the
new geometries of their day. Scholze's exceptionally clear lecture notes were
read in mathematics departments around the world within days of his lecture —
not passed hand-to-hand as in Grothendieck's day — and the videos of his talks
were immediately made available on the IHES website. Meanwhile, more killer
apps followed in rapid succession in a series of papers written by Scholze,
sometimes in collaboration with other mathematicians under 30 (or just slightly
older), often alone. By the time he reached the age of 24, high-level
conference invitations to talk about the uses of perfectoid spaces (I was at a
number of those too) had enshrined Scholze as one of the youngest elder
statesmen ever of arithmetic geometry, the branch of mathematics where number
theory meets algebraic geometry.) Two years later, a week-long meeting in 2014
on Perfectoid Spaces and Their Applications at the Mathematical Sciences
Research Institute in Berkeley broke all attendance records for "Hot Topics"
conferences.
- Michael Harris, "The Perfectoid Concept: Test Case for an Absent Theory"
Is the Fields Medal going to still be relevant in 10 years, when most of the the major mathematical discoveries are made by deep learning and deep reinforcement learning systems? Already systems are learning to reason about concepts [1] and
of course there is classical work on proof checkers [2]. It's very likely that the 2028 Fields medal will be awarded to a programmer, not some mathematical super-genius (assuming that the committee is fair, and not biased against machines).
If you're wondering why you're being down voted, it's because very few people believe mathematical discoveries and proofs are going to be automated any time soon. FCT was exceptional in that the theorists were able to reduce the theoretical proof to a brute force check that no human wanted to do. To be honest, making these claims, especially with such certainty, comes off as rather crank-y.
Mathematics require the exact type of abstract thinking machines suck at. Machines can execute things fast, learn things fast (provided we have well stablished rules), but than can't (at our current moment in time) come up with useful abstractions to help solve new problems.
There is a fantastic introductory MIT video lecture on this work by Daskalakis himself available on YouTube.[a] His enthusiasm and passion in that video lecture are contagious -- matched only by his ability to explain his work in an intuitive manner. Highly recommended for those HNers who have computer-science or economics backgrounds and are interested in computational complexity but are not yet familiar with Daskalakis's work.
[a] https://www.youtube.com/watch?v=TUbfCY_8Dzs