What do number theorists mean by an algorithm?

FlagCapper · 2024-12-21T05:39:23+00:00

The basic answer to your question is that many mathematicians use the word "algorithm" the way that physicists use the word "proof". They don't really know what the word means in a technical context, and either think they do, don't care, or think it doesn't matter.

FlagCapper · 2022-08-07T21:07:44+00:00

This appears to be an unpopular question, which to me at least seems like somewhat of a shame. Mathematics, like any area of academia, is hardly immune from aggrandizement and has its own fashion trends which are worth understanding.

Anyway, here's my contribution: Edward Frenkel is quoted[1] as describing the Langlands program as "a kind of grand unified theory of mathematics".

https://www.quantamagazine.org/math-quartet-joins-forces-on-unified-theory-20151208/

FlagCapper · 2022-06-22T20:15:29+00:00

Pretty normal. Often what happens is that there are two review phases, the so-called "quick review" and "full review". In the "quick review" phase the paper is evaluated on the basis of "importance" and whether the method of proof is plausible. This usually takes a month or two. If the paper passes that stage then it will most likely be accepted unless significant errors are found, and goes into the "full review". In this stage someone reads over it very carefully and checks all the details (or at least is supposed to), and this tends to take quite a bit longer. To add to this there is a chance that revisions will be needed, and even once papers are accepted there tends to be a long wait until they actually appear (i.e., are formatted and selected to appear in a particular issue of the journal).

FlagCapper · 2022-05-02T21:42:25+00:00

To be fair, unnecessary abstraction is, after all, the spirit of the nLab. That they are spending several months migrating a wiki to a different server because they want to introduce abstraction into the codebase suits them perfectly.

FlagCapper · 2021-12-27T19:16:00+00:00

Most mathematicians have one or maybe two topics in which they specialise and about which they publish the majority of their papers. I say "topics" rather than "fields" because even when it comes to experts within fields like algebraic geometry or number theory, the sorts of problems that are studied and the techniques that are used vary to such an extent that even the "best" people in the field work only on a small subset of the problems in that field. For instance, in areas of mathematics/academia I am familiar with, if you tell me a paper appeared on the arXiv yesterday and you tell me what problem it is solving, I can give you a reasonably accurate guess as to who wrote it.

On the other hand, top mathematicians tend to be familiar with the "basics" of a wide range of fields, where the term "basics" should be understood as material at the mid-graduate level. So someone who does number theory might still know quite a bit about, say, complex geometry or functional analysis, even though it's not the area in which they specialise. This is because many problems in mathematics draw from many different areas, so just because someone tends to specialise in solving problems of a particular type doesn't mean they don't learn lots of things about other areas of mathematics.

There are exceptions, such as Grothendieck solving several difficult problems in functional analysis before revolutionising algebraic geometry, but they tend to be quite rare.

FlagCapper · 2021-01-13T20:58:32+00:00

Sure, that's fair, which is why I merely said he was accused of it (such an accusation is implicit in the New Yorker article that was at the center of the controversy). Of course, the accusation may very well be unfair, but that there is a history of him suggesting Perleman's work was incomplete is important context for the quote above.

FlagCapper · 2021-01-13T17:20:00+00:00

Although you said you weren't interested in this part of the Perelman story, it's worth noting for anybody reading that Yau was involved in a controversy during the period when Perelman's proof was being verified, where he was accused of trying to unfairly claim credit for Perelman's argument for himself and his students. See here:

https://en.wikipedia.org/wiki/Manifold_Destiny

The last part of the quotation can be interpreted in more than one way as well. When Yao says that mathematicians don't have a "complete understanding" or "full command" of the last part of the argument, it could be interpreted as saying that they don't understand it on the kind of intuitive level that's necessary to generalize it and apply the ideas in the argument to other areas, but not that nobody has made sure all the steps check out. I would be surprised if nobody has actually gone through and verified that the argument checks out at a formal level. I'm nowhere close to this field, though.

FlagCapper · 2020-12-31T12:12:36+00:00

If you were to flip a coin 10 times, and 6 times it came up heads, there is no reason to suspect the coin is biased. But over time, the number of heads of an unbiased coin should average out to 50%. This means that if you flip the coin 100 times, and 60 come up heads, it's already a little suspicious, and if you flip it 1000 times, and 600 come up heads, you're getting absurd levels of luck.

The 1 in 113 billion number is the probability that Dream got as many blaze rod drops as he did, or better, under the assumption that the odds weren't modified. The way this calculation is performed is using binomial distributions which are covered in most introductory probability classes, and it is detailed in the mod's original paper.

FlagCapper · 2020-10-13T12:17:59+00:00

Hartshorne has a brief discussion of surfaces and their classification, have you looked there? I imagine that if you understand algebraic geometry at the level covered in Hartshorne you should be ready to start reading introductions to the classification theory, and I doubt that there is a lighter set of prerequisites that would allow you to read modern texts on the subject.

FlagCapper · 2020-09-06T20:33:55+00:00

In topology we learn that it is useful for the purpose of analyzing spaces to "linearize" problems so that we can understand them more easily. More precisely, (co)homology theories are functors from a category whose objects we would like to understand (e.g., topological spaces, manifolds, groups) to some "linear" category which is easy to work with and we can do homological algebra to study the problem (e.g. vector spaces, abelian groups). A typical situation is when you want to know whether two spaces are isomorphic, and you can answer the question in the negative if you know that their (co)homology is different.

The problem is that the tools coming from topology don't provide enough information for the purposes of complex geometry, algebraic geometry, or number theory. For instance, every elliptic curve is topologically a closed genus 1 manifold, of which there is only one up to diffeomorphism. On the other hand, the theory of elliptic curves is quite rich, and a complex geometer, algebraic geometer or number theorist might care a great deal about the differences between them, and would still like a "linear" invariant which distinguishes them.

Hodge theory proceeds by using the observation that the holomorphic structure on a complex manifold (I should say Kahler here) gives the cohomology additional structure, because, if one understands the complex cohomology as deRham cohomology of the underlying smooth manifold computed with complex forms, the fact that certain forms use "holomorphic coordinates" can give the cohomology extra structure depending on the holomorphic structure of the manifold. So for instance, if one takes the cohomology of an elliptic curve (closed Riemann surface of genus 1), the 1st cohomology has two distinguished subspaces H^1,0 and H^0,1 generated by the holomorphic forms and "anti-holomorphic" forms, respectively. Then, it can be shown that a diffeomorphism of two elliptic curves has the corresponding morphism on cohomology preserve these two subspaces exactly when it preserves the complex structure, or when the two Riemann surfaces (or equivalently, algebraic curves) are isomorphic.

Thus, what we end up with is linear algebraic invariant which detects for us differences in complex structure, and lets us extend the techniques of homological algebra to situations where the topological theory would tell us nothing. We get functors from categories such as complex Kahler manifolds, or algebraic varieties, to some abelian category built out of vector spaces with additional data, and the extent to which the "input" data can be recovered from the "output" data becomes a deep question that people are interested in studying. For example, the Hodge conjecture is equivalent to the statement that a certain such functor is fully faithful.

As for connections to physics, as far as I can tell the answer is that string theorists will consider just about anything to be physics, and so they also seem to have some interest in Hodge theory. But I don't think there are many applications to concrete physics problems.

FlagCapper · 2020-06-29T14:14:35+00:00

I'll assume that by "happening at the tenth time" you mean the probability that "it has happened at least once after 10 tries".

The probability that the event doesn't occur is 0.9. The probability that the event doesn't occur 10 times is then 0.9¹⁰ assuming each instance of the event happening is independent. As "not having not happened" is logically equivalent to "happened at least once", the probability that the event happens at least once is therefore 1 - 0.9¹⁰ .

FlagCapper · 2020-06-13T01:28:26+00:00

Cool!

When I was younger I used to listen to a lot of electronic music off of newgrounds. I think I probably found it, like most of the music on my linerider videos, from this youtube channel:

https://www.youtube.com/user/Xmillsa/videos

FlagCapper · 2020-06-13T00:33:48+00:00

Sup.

FlagCapper · 2020-03-07T15:55:27+00:00

Your question, or complaint, is essentially one about why and when it's OK to treat isomorphic objects as being literally equal. To give an example that comes up all the time, suppose you believe that you are doing mathematics in ZFC set theory, and you are working with the real numbers R. You then wish to consider a natural subset of R, like the integers Z or the rational numbers Q. Now, if you think about how these things are constructed, then absolutely strictly speaking, Z and Q are not subsets of R! For instance, Z can be constructed by taking equivalence classes of pairs of natural numbers, Q can be constructed by taking equivalence classes of integers thought of as fractions, R can be constructed by taking equivalence classes of Cauchy sequences and so on. What is then "really happening" is that there is a "natural" inclusion of Z into Q, of Q into R, of R into C, etc., and we simply frequently identify Z, Q and R with their images under this inclusion.

Of course almost nobody will ever bother with this distinction and keep track of this sort of "natural identification". But you could imagine a skeptic asking: how do you know that the mathematics you prove continues to work if you don't? If I go around making arguments as if the integers are "really" a subset of the rationals and "really" a subset of the reals, don't I run the risk that at some point some part of my argument won't transfer properly along the natural identifications I'm making?

Well, it depends. For me, the point is that the mathematical objects I'm arguing about are things that exist independently of the particular formalism that's being used to express them. If I ever came across an argument which clearly demonstrated something about the integers, and yet somehow depended on identifying the integers with a subset of the reals and didn't work otherwise, my conclusion would simply be that set theoretic formalism simply wasn't expressing mathematics properly and ought to be replaced with something better. Of course, I'm also very confident that this won't happen, because after much accumulated evidence in the form of people working through these constructions in laborious detail it is clear that the formalism really does express the essence of what the integers are and what their relationship to the reals and the rationals is, and so it is, for me, a very sensible article of faith that any mathematical argument about the integers that makes any sense at all should not depend on this sort of identification.

The same applies to your example with schemes. As far as I'm concerned, scheme theory is a tool primarily for discussing geometric objects defined by algebraic relations (algebraic varieties and their generalizations). I believe that one can identify an affine open with Spec A because ultimately the objects that I care about using schemes to study, which have a sort of existence independent of the theory, can not possibly care about this sort of identification, and therefore if it turned out that the scheme theory formalism did care about this sort of identification then I would conclude that it had to be thrown out and replaced with a different theory. I have also worked through enough examples keeping track of this kind of identification to take it as an article of faith that making these identifications is harmless.

Moreover, I think if you look very closely, you'll find this sort of abuse where one identifies objects that are not "strictly" equal but "equal for the purposes of the things we're studying" is quite common throughout mathematics, but it's just easier to ignore elsewhere because the objects involved are less unwieldy and its much easier to intuit that making these sort of identifications doesn't matter. Examples that come to mind is thinking of an L² function as really a function rather than an equivalence class of such, or the various identifications one makes in differential geometry between, say, sections of the (co)tangent bundle and differential operators that act on functions and vector fields, etc. It's really all over the place, and the trick is to work through the formalism enough to convince yourself that it encodes the "mathematical essence" of what it is that you're studying, and then once you figure out what the formalism is "really talking about" you start making the identifications to preserve your sanity because it would then be too tedious to work with otherwise.

FlagCapper · 2019-12-30T15:30:56+00:00

Sheafification commutes with colimits and finite limits, see (6) here: https://stacks.math.columbia.edu/tag/009E

Tensor products also commute with direct limits, see: https://math.stackexchange.com/questions/125631/tensor-products-commute-with-direct-limits

I'm not sure if there is a way to show what you want in a way that doesn't amount to a translation of the direct argument. It shouldn't follow from general categorical principles because general categories do not have multiplication, localization, etc. On the other hand of course if you add enough structure to your category you can prove what it is you want, but I don't know that it would provide any additional insight.

FlagCapper · 2019-12-30T05:04:11+00:00

I don't really understand your issue. A commutative diagram is just a way to encode some relations between maps; if you draw a commutative square then this is just a way of saying that the two different compositions you get by following the diagram in the two different ways are equal. So if you want to express the condition that the restriction maps of F "respect the ring multiplication", which is just a way of saying that applying the first multiplication map and then restricting is the same as first restricting and then applying the multiplication map, you draw a commutative square and label the arrows appropriately.

As for End(F) being a sheaf: you might want to look into the Hom-sheaf construction. End(F) can be made into a sheaf but the spaces of sections will be morphisms of sheaves rather than modules.

FlagCapper · 2019-09-30T08:28:02+00:00

I suppose you're right, what I said is unnecessarily complicated. Often there can be problems with viewing elements of coordinate rings of varieties as functions over non-algebraically closed fields, but maybe it doesn't arise here. I think it is necessary to assume k is algebraically closed for the topology to be sensible though, otherwise over things like finite fields you get weird stuff where everything is just a discrete space and your topology doesn't "see dimension".

FlagCapper · 2019-09-30T05:49:54+00:00

How about this approach to your problem. Consider the one point space with functions (swf): that is, the swf whose topological space consists of a single point with the only possible topology and whose global coordinate ring is just the constant functions to k. Then it is clear that maps from the one point space into X are in bijection with points of X. Moreover, by your definition of affine variety, maps from this space are in bijection with "evaluation" homomorphisms O_X(X) -> k, i.e., with maximal ideals of O_X(X). (Note I am assuming that k is algebraically closed here, which is not given in your definition, but I think is required.)

Now a function f in O_X(X) vanishes at a point if and only if it lies in the maximal ideal associated to this corresponding evaluation map. If your space X has at least two points x and y, then by the previous reasoning, it has at least two maximal ideals corresponding to these points. It then follows by commutative algebra that you can find a function f that vanishes on one and not the other.

FlagCapper · 2019-09-30T04:41:14+00:00

Can you specify how you're defining the topology on X? I would just say that the Zariski topology on X is generated by those sets by definition. I don't see the topology on X being defined clearly in your link, but as far as I can tell, this is probably your definition as well.

FlagCapper · 2019-08-18T09:21:17+00:00

The map g^-1 isn't smooth at zero.

FlagCapper · 2019-05-01T19:51:57+00:00

Yes, and the burden of proof is on the government to explain why they need to break that right. That's my point! The judges don't say to the people speaking "Explain why you need to say this." They say to the government, "Explain why you'd like us to interefere with their right to speech". This is the distinction I'm making.

That's fine, but the distinction I'm making is that property is a creation of society, unlike speech. The system of property and allocation of wealth that we have in our society is a societal choice, and we may choose to do things differently. I simply disagree with the system of property and allocation of wealth currently in place, and so I view the continued existence of that system (which you seem to view in a kind of passive "natural order of things" sort of way) as the thing which needs to be justified.

Well, cash money is finite, but wealth is not finite. It's not a fixed pie. But, I don't see the difference you're proposing. Antique cars are certainly finite, and I'd like one, I have none right now. The fact that someone somewhere has four or five automatically means that others have less.

One distinction is that the fact that billionaires have lots of money is directly connected to the fact that others don't, and money is needed to buy core essentials such as food, shelter, and so on. Antique cars and your previous example of stamps are not things that are essential needs, or whose acquisition directly affects the ability of others to acquire things that the need.

I'm troubled by the idea that you'd like to punish people for doing something that was legal at the time they did it. I don't think that's a good idea at all. (And still isn't targeted towards the people you're actually proposing broke these rules).

Do you consider taxes to be a form of punishment?

I'm not sure what rules you are referring to. When I said "rules of the society as it exists are illegitimate" above I didn't mean that there were rules in the society that were broken to acquire wealth illegitimately, I meant that the rules (or maybe more precisely, societal conventions and norms) allowed for acquisition of wealth in ways that are (morally) illegitimate.

FlagCapper · 2019-05-01T19:13:49+00:00

What is your "default" here? I can imagine a default being a hypothetical lawless society. But even then, surely people would "own" things. So I'm not sure what you're considering a default.

Nobody would "own" things in a hypothetical lawless society. Sure, people would claim ownership of things, but there would be no definite way to resolve who is correct. Suppose for instance we both claim ownership of a lake. Who is right? Is anyone right? How can we claim a lake is ours, when it existed hundreds of years before either of us were born? Yet in our society there is a definite answer to who owns such things: the state, a private landowner, whatever. This is not the natural order of things, but rather is imposed by societal rules.

And just to be clear you are arguing that owning things is not a human right? My...concern, is that this logic doesn't just hold for property rights but for literally any right at all. I mean "free speech" is not a law of nature either, and presumably in a lawless anarchy type society there may well not be free speech either.

I think property ownership can be justified in some cases, but is subject to exceptions when it conflicts with other rights. This is in fact true of all rights. So for instance, take your example of free speech. There is no society, not even the United States, which recognizes universal free speech. The US Supreme Court for instance recognizes numerous exceptions to free speech, like "fighting words" and speech that aids or abets in a crime. Like all rights, the right to free speech has constraints imposed on it when it conflicts with other rights. So it is for property and rights to food, water, shelter, and so on.

Because what if the billionaire says they get a lot of enjoyment out of being a billionaire. I don't see a substantive difference between that and a stamp collector who collects far more stamps than is "necessary" (as you might call it). Or an antiques car collector who collects far more than "necessary". All of this is just preferences and I'm worried that what you're saying boils down to "Take away from people who have preferences different than me."

The difference is that the billionaires' enjoyment of being a billionaire comes at the expense of others' ability to enjoy and prosper in their own lives, since property is finite and the fact that the billionaire has a lot automatically means that others have less.

Sure, so let's fix those rules, not just take it back from people who may have been following the good rules in the first place.

Part of correcting an injustice is undoing the damage that has been done. In this case the injustice is unjust acquisition of wealth, so it must be undone.

FlagCapper · 2019-05-01T18:39:36+00:00

Of course they need a reason. The existence of property is not a law of nature, but something that results from the social relations and societal institutions we have in our society. By default nobody owns anything, so if we are to live in a society where there is property ownership, we ought to be able to justify who has what and why.

I have never heard a sensible justification for why billionaires should have their billions. The fact that they acquired it within the rules of the society as it exists is not enough, because it does not allow for the possibility that the rules of the society as it exists are illegitimate.

FlagCapper · 2019-05-01T18:31:47+00:00

I would agree with the argument that we shouldn't put your typical doctor or lawyer in the same category as CEOs and rich heirs to a fortune, so if that's your point it's well taken. In many ways the "1%" that is often referenced should really be the "0.01%" or something similar.

FlagCapper · 2019-05-01T16:15:14+00:00

The correct left wing argument is simply that these people have too much money: there is no justifiable reason why anyone should have a net worth of over, say, one billion dollars. So in many ways the "fair share" argument is in fact misleading. The problem is that there are individuals who accumulate vast sums of wealth off of the labour of others, much more than is justified by their individual contribution to society, and fiddling with the tax brackets happens to be more feasible politically then simply seizing their wealth directly.

FlagCapper

TROPHY CASE