Tag Archives: algebraic geometry

How many points does a random curve over F_q have?

So asks a charming preprint by Achter, Erman, Kedlaya, Wood, and Zureick-Brown.  (2/5 Wisconsin, 1/5 ex-Wisconsin!)  The paper, I’m happy to say, is a result of discussions at an AIM workshop on arithmetic statistics I organized with Alina Bucur and Chantal David earlier this year.

Here’s how they think of this.  By a random curve we might mean a curve drawn uniformly from M_g(F_q).  Let X be the number of points on a random curve.  Then the average number of points on a random curve also has a geometric interpretation: it is

|M_{g,1}(\mathbf{F}_q)|/|M_{g}(\mathbf{F}_q)|

What about

|M_{g,2}(\mathbf{F}_q)|/|M_{g}(\mathbf{F}_q)|?

That’s just the average number of ordered pairs of distinct points on a random curve; the expected value of X(X-1).

If we can compute all these expected values, we have all the moments of X, which should give us a good idea as to its distribution.  Now if life were as easy as possible, the moduli spaces of curves would have no cohomology past degree 0, and by Grothendieck-Lefschetz, the number of points on M_{g,n} would be q^{3g-3+n}.  In that case, we’d have that the expected value of X(X-1)…(X-n) was q^n.  Hey, I know what distribution that is!  It’s Poisson with mean q.

Now M_g does have cohomology past degree 0.  The good news is, thanks to the Madsen-Weiss theorem (née the Mumford conjecture) we know what that cohomology is, at least stably.  Yes, there are a lot of unstable classes, too, but the authors propose that heuristically these shouldn’t contribute anything.  (The point is that the contribution from the unstable range should look like traces of gigantic random unitary matrices, which, I learn from this paper, are bounded with probability 1 — I didn’t know this, actually!)  And you can even make this heuristic into a fact, if you want, by letting q grow pretty quickly relative to g.

So something quite nice happens:  if you apply Grothendieck-Lefschetz (actually, you’d better throw in Kai Behrend’s name, too, because M_g is a Deligne-Mumford stack, not an honest scheme) you find that the moments of X still agree with those of a Poisson distribution!  But the contribution of the tautological cohomology shifts the mean from q to q+1+1/(q-1).

This is cool in many directions!

  • It satisfies one’s feeling that a “random set,” if it carries no extra structure, should have cardinality obeying a Poisson distribution — the “uniform distribution” on the groupoid of sets.  (Though actually that uniform distribution is Poisson(1); I wonder what tweak is necessary to be able to tune the mean?)
  • I once blogged about an interesting result of Bucur and Kedlaya which showed that a random smooth complete intersection curve in P^3 of fixed degree had slightly fewer than q+1 points; in fact, about q+1 – 1/q + o(q^2).  Here the deviation is negative, rather than positive, as the new paper suggests is the case for general curves; what’s going on?
  • I have blogged about the question of average number of points on a random curve before.  I’d be very interested to know whether the new heuristic agrees with the answer to the question proposed at the end of that post; if g is a large random matrix in GSp(Z_ell) with algebraic eigenvalues, and which multiplies the symplectic form by q, and you condition on Tr(g^k) > (-q^k-1) so that the “curve” has nonnegatively many points over each extension of F_q, does this give something like the distribution the five authors predict for Tr(g)?  (Note:  I don’t think this question is exactly well-formed as stated.)

 

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Squares and Motzkins

Greg Smith gave an awesome colloquium here last week about his paper with Blekherman and Velasco on sums of squares.

Here’s how it goes.  You can ask:  if a homogeneous degree-d polynomial in n variables over R takes only non-negative values, is it necessarily a sum of squares?  Hilbert showed in 1888 that the answer is yes only when d=2 (the case of quadratic forms), n=2 (the case of binary forms) or (n,d) = (3,4) (the case of ternary quartics.)  Beyond that, there are polynomials that take non-negative values but are not sums of squares, like the Motzkin polynomial

X^4 Y^2 + X^2 Y^4 - 3X^2 Y^2 Z^2 + Z^6.

So Greg points out that you can formulate this question for an arbitrary real projective variety X/R.  We say a global section f of O(2) on X is nonnegative if it takes nonnegative values on X(R); this is well-defined because 2 is even, so dilating a vector x leaves the sign of f(x) alone.

So we can ask:  is every nonnegative f a sum of squares of global sections of O(1)?  And Blekherman, Smith, and Velasco find there’s an unexpectedly clean criterion:  the answer is yes if and only if X is a variety of minimal degree, i.e. its degree is one more than its codimension.  So e.g. X could be P^n, which is the (n+1,2) case of Hilbert.  Or it could be a rational normal scroll, which is the (2,d) case.  But there’s one other nice case:  P^2 in its Veronese embedding in P^5, where it’s degree 4 and codimension 3.  The sections of O(2) are then just the plane quartics, and you get back Hilbert’s third case.  But now it doesn’t look like a weird outlier; it’s an inevitable consequence of a theorem both simpler and more general.  Not every day you get to out-Hilbert Hilbert.

Idle question follows:

One easy way to get nonnegative homogenous forms is by adding up squares, which all arise as pullback by polynomial maps of the ur-nonnegative form x^2.

But we know, by Hilbert, that this isn’t enough to capture all nonnegative forms; for instance, it misses the Motzkin polynomial.

So what if you throw that in?  That is, we say a Motzkin is a degree-6d form

expressible as

 

P^4 Q^2 + P^2 Q^4 - 3P^2 Q^2 R^2 + R^6

for degree-d forms P,Q,R.  A Motzkin is obviously nonnegative.

It is possible that every nonnegative form of degree 6d is a sum of squares and Motzkins?  What if instead of just Motzkins we allow ourselves every nonnegative sextic?  Or every nonnegative homogeneous degree-d form in n variables for n and d less than 1,000,000?  Is it possible that the condition of nonnegativity is in this respect “finitely generated?”

 

 

 

 

 

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Boyer: curves with real multiplication over subcyclotomic fields

A long time ago, inspired by a paper of Mestre constructing genus 2 curves whose Jacobians had real multiplication by Q(sqrt(5)), I wrote a paper showing the existence of continuous families of curves X whose Jacobians had real multiplication by various abelian extensions of Q.  I constructed these curves as branched covers with prescribed ramification, which is to say I had no real way of presenting them explicitly at all.  I just saw a nice preprint by Ivan Boyer, a recent Ph.D. student of Mestre, which takes all the curves I construct and computes explicit equations for them!  I wouldn’t have thought this was doable (in particular, I never thought about whether the families in my construction were rational.) For instance, for any value of the parameter s, the genus 3 curve

2v + u^3 + (u+1)^2 + s((u^2 + v)^2 - v(u+v)(2u^2 - uv + 2v))

has real multiplication by the real subfield of \mathbf{Q}(\zeta_7).  Cool!

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Puzzle: low-height points in general position

I have no direct reason to need the answer to, but have wondered about, the following question.

We say a set of points P_1, \ldots, P_N in \mathbf{A}^2 are in general position if the Hilbert function of any subset S of the points is equal to the Hilbert function of a generic set of |S| points in \mathbf{A}^n.  In other words, there are no curves which contain more of the points than a curve of their degree “ought” to.  No three lie on a line, no six on a conic, etc.

Anyway, here’s a question.  Let H(N) be the minimum, over all N-tuples P_1, \ldots, P_N \in \mathbf{A}^2(\mathbf{Q}) of points in general position, of

\max H(P_i)

where H denotes Weil height.  What are the asymptotics of H(N)?  If you take the N lowest-height points, you will have lots of collinearity, coconicity, etc.  Does the Bombieri-Pila / Heath-Brown method say anything here?

 

 

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Gross, Hacking, Keel on the geometry of cluster algebras

I have expressed amazement before about the Laurent phenomenon for cluster algebras, a theorem of Fomin and Zelevinsky which I learned about from Lauren Williams.  The paper “Birational Geometry of Cluster Algebras,”  just posted by Mark Gross, Paul Hacking, and Sean Keel, seems extremely interesting on this point.  They interpret the cluster transformations — which to an outsider look somewhat arbitrary — as elementary transforms (i.e. blow up a codim-2 thing and then blow down one of the exceptional loci thus created) on P^1-bundles on toric varieties.  And apparently the Laurent phenomenon is plainly visible from this point of view.  Very cool!

Experts are highly encouraged to weigh in below.

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Shende and Tsimerman on equidistribution in Bun_2(P^1)

Very nice paper just posted by Vivek Shende and Jacob Tsimerman.  Take a sequence {C_i} of hyperelliptic curves of larger and larger genus.  Then for each i, you can look at the pushforward of a random line bundle drawn uniformly from Pic(C) / [pullbacks from P^1] to P^1, which is a rank-2 vector bundle.  This gives you a measure \mu_i on Bun_2(P^1), the space of rank-2 vector bundles, and Shende and Tsimerman prove, just as you might hope, that this sequence of measures converges to the natural measure.

I think (but I didn’t think this through carefully) that this corresponds to saying that if you look at a sequence of quadratic imaginary fields with increasing discriminant, and for each field you write down all the ideal classes, thought of as unimodular lattices in R^2 up to homothety, then the corresponding sequence of (finitely supported) measures on the space of lattices converges to the natural one.

Equidistribution comes down to counting, and the method here is to express the relevant counting problem as a problem of counting points on a variety (in this case a Brill-Noether locus inside Pic(C_i)), which by Grothendieck-Lefschetz you can do if you can control the cohomology (with its Frobenius action.)  The high-degree part of the cohomology they can describe explicitly, and fortunately they are able to exert enough control over the low-degree Betti numbers to show that the contribution of this stuff is negligible.

In my experience, it’s often the case that showing that the contribution of the low-degree stuff, which “should be small” but which you don’t actually have a handle on, is often the bottleneck!  And indeed, for the second problem they discuss (where you have a sequence of hyperelliptic curves and a single line bundle on each one) it is exactly this point that stops them, for the moment, from having the theorem they want.

Error terms are annoying.  (At least when you can’t prove they’re smaller than the main term.)

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Y. Zhao and the Roberts conjecture over function fields

Before the developments of the last few years the only thing that was known about the Cohen-Lenstra conjecture was what had already been known before the Cohen-Lenstra conjecture; namely, that the number of cubic fields of discriminant between -X and X could be expressed as

\frac{1}{3\zeta(3)} X + o(X).

It isn’t hard to go back and forth between the count of cubic fields and the average size of the 3-torsion part of the class group of quadratic fields, which gives the connection with Cohen-Lenstra in its usual form.

Anyway, Datskovsky and Wright showed that the asymptotic above holds (for suitable values of 12) over any global field of characteristic at least 5.  That is:  for such a field K, you let N_K(X) be the number of cubic extensions of K whose discriminant has norm at most X; then

N_K(X) = c_K \zeta_K(3)^{-1} X + o(X)

for some explicit rational constant $c_K$.

One interesting feature of this theorem is that, if it weren’t a theorem, you might doubt it was true!  Because the agreement with data is pretty poor.  That’s because the convergence to the Davenport-Heilbronn limit is extremely slow; even if you let your discriminant range up to ten million or so, you still see substantially fewer cubic fields than you’re supposed to.

In 2000, David Roberts massively clarified the situation, formulating a conjectural refinement of the Davenport-Heilbronn theorem motivated by the Shintani zeta functions:

N_{\mathbf{Q}}(X) = (1/3)\zeta(3)^{-1} X + c X^{5/6} + o(X^{5/6})

with c an explicit (negative) constant.  The secondary term with an exponent very close to 1 explains the slow convergence to the Davenport-Heilbronn estimate.

The Datskovsky-Wright argument works over an arbitrary global field but, like most arguments that work over both number fields and function fields, it is not very geometric.  I asked my Ph.D. student Yongqiang Zhao, who’s finishing this year, to revisit the question of counting cubic extensions of a function field F_q(t) from a more geometric point of view to see if he could get results towards the Roberts conjecture.  And he did!  Which is what I want to tell you about.

But while Zhao was writing his thesis, there was a big development — the Roberts conjecture was proved.  Not only that — it was proved twice!  Once by Bhargava, Shankar, and Tsimerman, and once by Thorne and Taniguchi, independently, simultaneously, and using very different methods.  It is certainly plausible that these methods can give the Roberts conjecture over function fields, but at the moment, they don’t.

Neither does Zhao, yet — but he’s almost there, getting

N_K(T) = \zeta_K(3)^{-1} X + O(X^{5/6 + \epsilon})

for all rational function fields K = F_q(t) of characteristic at least 5.  And his approach illuminates the geometry of the situation in a very beautiful way, which I think sheds light on how things work in the number field case.

Geometrically speaking, to count cubic extensions of F_q(t) is to count trigonal curves over F_q.  And the moduli space of trigonal curves has a classical unirational parametrization, which I learned from Mike Roth many years ago:  given a trigonal curve Y, you push forward the structure sheaf along the degree-3 map to P^1, yielding a rank-3 vector bundle on P^1; you mod out by the natural copy of the structure sheaf; and you end up with a rank-2 vector bundle W on P^1, whose projectivization is a rational surface in which Y embeds.  This rational surface is a Hirzebruch surface F_k, where k is an integer determined by the isomorphism class of the vector bundle W.  (This story is the geometric version of the Delone-Fadeev parametrization of cubic rings by binary cubic forms.)

This point of view replaces a problem of counting isomorphism classes of curves (hard!) with a problem of counting divisors in surfaces (not easy, but easier.)  It’s not hard to figure out what linear system on F_k contains Y.  Counting divisors in a linear system is nothing but a dimension count, but you have to be careful — in this problem, you only want to count smooth members.  That’s a substantially more delicate problem.  Counting all the divisors is more or less the problem of counting all cubic rings; that problem, as the number theorists have long known, is much easier than the problem of counting just the maximal orders in cubic fields.

Already, the geometric meaning of the negative secondary term becomes quite clear; it turns out that when k is big enough (i.e. if the Hirzebruch surface is twisty enough) then the corresponding linear system has no smooth, or even irreducible, members!  So what “ought” to be a sum over all k is rudely truncated; and it turns out that the sum over larger k that “should have been there” is on order X^{5/6}.

So how do you count the smooth members of a linear system?  When the linear system is highly ample, this is precisely the subject of Poonen’s well-known “Bertini theorem over finite fields.”  But the trigonal linear systems aren’t like that; they’re only “semi-ample,” because their intersection with the fiber of projection F_k -> P^1 is fixed at 3.  Zhao shows that, just as in Poonen’s case, the probability that a member of such a system is smooth converges to a limit as the linear system gets more complicated; only this limit is computed, not as a product over points P of the probability D is smooth at P, but rather a product over fibers F of the probability that D is smooth along F.  (This same insight, arrived at independently, is central to the paper of Erman and Wood I mentioned last week.)

This alone is enough for Zhao to get a version of Davenport-Heilbronn over F_q(t) with error term O(X^{7/8}), better than anything that was known for number fields prior to last year.  How he gets even closer to Roberts is too involved to go into on the blog, but it’s the best part, and it’s where the algebraic geometry really starts; the main idea is a very careful analysis of what happens when you take a singular curve on a Hirzebruch surface and start carrying out elementary transforms at the singular points, making your curve more smooth but also changing which Hirzebruch surface it’s on!

To what extent is Zhao’s method analogous to the existing proofs of the Roberts conjecture over Q?  I’m not sure; though Zhao, together with the five authors of the two papers I mentioned, spent a week huddling at AIM thinking about this, and they can comment if they want.

I’ll just keep saying what I always say:  if a problem in arithmetic statistics over Q is interesting, there is almost certainly interesting algebraic geometry in the analogous problem over F_q(t), and the algebraic geometry is liable in turn to offer some insights into the original question.

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This Week’s Finds In Number Theory

Twenty years ago yesterday, John Baez posted the first installment of This Week’s Finds in Mathematical Physics.  In so doing, he invented the math blog, and, quite possibly, the blog itself.  A lot of mathematicians of my generation found in John’s blog an accessible, informal, but never dumbed-down window beyond what we were learning in classes, into the messy and contentious ground of current research.  And everybody who blogs now owes him a gigantic debt.

In his honor I thought it would be a good idea to post a “This Week’s Finds” style post of my own, with capsule summaries of a few papers I’ve recently noted with pleasure and interest.  I won’t be able to weave these into a story the way John often did, though!  Nor will there be awesome ASCII graphics.  Nor will any of the papers actually be from this week, because I’m a little behind on my math.NT abstract scanning.

If you run a math blog, please consider doing the same in your own field!  I’ll link to it.

Update:  It begins!  Valeria de Palva offers This Week’s Finds In Categorical Logic.  And Matt Ward, a grad student at UW-Seattle, has This Week’s Finds in Arithmetic Geometry.

1)  “On sets defining few ordinary lines,” by Ben Green and Terry Tao.

The idea that has launched a thousand papers in additive combinatorics:  if you are a set approximately closed under some kind of relation, then you are approximately a set which is actually closed under that kind of relation.  Subset of a group mostly closed under multiplication?  You must be close to an honest subgroup.  Subset of Z with too many pair-sums agreeing?  You have an unusually large intersection with an authentic arithmetic progression.  And so on.

This new paper considers the case of sets in R^2 with few ordinary lines; that is, sets S such that most lines that intersect S at all intersect S in three or more points.  How can you cook up a set of points with this property?  There are various boring ways, like making all the points collinear.  But there’s only one interesting way I can think of:  have the points form an “arithmetic progression” …,-3P,-2P, -P, P,2P,3P, …. in an elliptic curve!  (A finite subgroup also works.)  Then the usual description of the group law on the curve tells us that the line joining two points of S quite often passes through a third.  Green and Tao prove a remarkable quasi-converse to this fact:  if a set has few ordinary lines, it must be concentrated on a cubic algebraic curve!  This might be my favorite “approximately structured implies approximates a structure” theorem yet.

2) “Asymptotic behavior of rational curves,” by David Bourqui.  Oh, I was about to start writing this but when I searched I realized I already blogged about this paper when it came out!  I leave this here because the paper is just as interesting now as it was then…

3) “The fluctuations in the number of points of smooth plane curves over finite fields,” by Alina Bucur, Chantal David, Brooke Feigon, and Matilde Lalin;

“The probability that a complete intersection is smooth,” by Alina Bucur and Kiran Kedlaya;

“The distribution of the number of points on trigonal curves over F_q,” by Melanie Matchett Wood;

“Semiample Bertini theorems over finite fields,” by Daniel Erman and Melanie Matchett Wood.

How many rational points does a curve over F_q have?  We discussed this question here a few years ago, coming to no clear conclusion.  I still maintain that if the curve is understood to vary over M_g(F_q), with q fixed and g growing, the problem is ridiculously hard.

But in more manageable families of curves, we now know a lot more than we did in 2008.

You might guess, of course, that the average number of points should be q+1; if you have to reason to think of Frobenius as biased towards having positive or negative trace, why not guess that the trace, on average, is 0?  Bucur-David-Feigon-Lalin prove that this is exactly the case for a random smooth plane curve.  It’s not hard to check that this holds for a random hyperelliptic curve as well.  But for a random trigonal curve, Wood proves that the answer is different — the average is slightly less than q+2!

Where did the extra point come from?

Here’s one way I like to think of it.  This is very vague, and proves nothing, of course.  The trigonal curve X has a degree-3 map to P^1, which is ramified at some divisor D in P^1.  If D is a random divisor, it has one F_q-point on average.  How many F_q-points on X lie over each rational point P of D?  Well, generically, the ramification is going to be simple, and this means that there are two rational points over D; the branch point, and the unique unramified point.  Over every other F_q-point of D, the Frobenius action on the preimage in X should be a random element of S_3, with an average of one fixed point.  To sum up, in expectation we should see q rational points of X over q non-branch rational points of P^1, and 2 rational points of X over a single rational branch point in P^1, for a total of q+2.

(Erman and Wood, in a paper released just a few months ago, prove much more general results of a similar flavor about smooth members of linear systems on P^1 x P^1 (or other Hirzebruch surfaces, or other varieties entirely) which are semiample; for instance, they may have a map to P^1 which stays constant in degree, while their intersection with another divisor gets larger and larger.)

Most mysterious of all is the theorem of Bucur and Kedlaya, which shows (among other things) that if X is a random smooth intersection of two hypersurfaces of large degree in P^3, then the size of |X(F_q)| is slightly less than q+1 on average.  For this phenomenon I have no heuristic explanation at all.  What’s keeping the points away?

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Mochizuki on ABC

[Update:  Lots of traffic coming in from Hacker News, much of it presumably from outside the usual pro number theory crowd that reads this blog.  If you’re not already familiar with the ABC conjecture, I recommend Barry Mazur’s beautiful expository paper “Questions about Number.”]

[Re-update:  Minhyong Kim’s discussion on Math Overflow is the most well-informed public discussion of Mochizuki’s strategy.  (Of course, it is still very sketchy indeed, as Minhyong hastens to emphasize.)   Both Kim’s writeup and discussions I’ve had with others suggest that the best place to start may be Mochizuki’s 2000 paper “A Survey of the Hodge-Arakelov Theory of Elliptic Curves I.”]

Shin Mochizuki has released his long-rumored proof of the ABC conjecture, in a paper called “Inter-universal Teichmuller theory IV:  log-volume computations and set-theoretic foundations.”

I just saw this an hour ago and so I have very little to say, beyond what I wrote on Google+ when rumors of this started circulating earlier this summer:

I hope it’s true:  my sense is that there’s a lot of very beautiful, very hard math going on in Shin’s work which almost no one in the community has really engaged with, and the resolution of a major conjecture would obviously create such engagement very quickly.

Well, now the time has come.  I have not even begun to understand Shin’s approach to the conjecture.  But it’s clear that it involves ideas which are completely outside the mainstream of the subject.  Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.

Let me highlight one point which is clearly important, which I draw from pp.3–6 of the linked paper.

WARNING LABEL:  Of course my attempt to paraphrase is based on the barest of acquaintance with a very small section of the work and is placed here just to get people to look at Mochizuki’s paper — I may have it all wrong!

Mochizuki argues that it is too limiting to think about “the category of schemes over Spec Z,” as we are accustomed to do.  He makes the inarguable point that when X is a kind of thing, it can happen that the category of Xes, qua category, may not tell us very much about what Xes are like — for instance, if there is only one X and it has only one automorphism. Mochizuki argues that the category of schemes over a base is — if not quite this uninformative — insufficiently rich to handle certain problems in Diophantine geometry.  He wants us instead to think about what he calls the “species” of schemes over Spec Z, where a scheme in this sense is not an abstract object in a category, but something cut out by a formula.  In some sense this view is more classical than the conventional one, in which we tend to feel good about ourselves if we can “remove coordinates” and think about objects and arrows without implicitly applying a forgetful functor and referring to the object as a space with a Zariski topology or — ptui! – a set of points.

But Mochizuki’s point of view is not actually classical at all — because the point he wants to make is that formulas can be intepreted in any model of set theory, and each interpretation gives you a different category.  What is “inter-universal” about inter-universal Teichmuller theory is that it is important to keep track of all these categories, or at least many different ones.  What he is doing, he says, is simply outside the theory of schemes over Spec Z, even though it has consequences within that theory — just as (this part is my gloss) the theory of schemes itself is outside the classical theory of varieties, but provides us information about varieties that the classical theory could not have generated internally.

It’s tremendously exciting.  I very much look forward to commentary from people with a deeper knowledge than mine of Mochizuki’s past and present work.

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August math linkdump

  • Algebraists eat corn row by row, analysts eat corn circle by circle.  Yep, I eat down the rows like a typewriter.  Why?  Because it is the right way.
  • This short paper by Johan de Jong and Wei Ho addresses an interesting question I’d never thought about; does a Brauer-Severi variety over a field K contain a genus-1 curve defined over K?  They show the answer is yes in dimensions up to 4 (3 and 4 being the new cases.)  In dimension 1, this just asks about covers of Brauer-Severi curves by genus 1 curves; I remember this kind of situation coming up in Ekin Ozman’s thesis, where certain twists of modular curves end up being covers of Brauer-Severi curves.  Which Brauer-Severi varieties are split by twisted modular curves?
  • Always nice to see a coherent description of the p-adic numbers in the popular press; and George Musser delivers the goods in Scientific American, in the context of recent work in cosmology by Harlow, Shenker, Stanford, and Susskind.  Two quibbles:  first, if I understood Susskind’s talk on this stuff correctly, the point is to model things by an infinite regular tree.  The fact that when the out-degree is a prime power this happens to look like the Bruhat-Tits tree is in some sense tangential, though very useful for getting an intuitive picture of what’s going on — as long as your intuition is already p-adic, of course!  Second quibble is that Musser seems to suggest at the end that p-adic distances can’t get arbitrarily small:

On top of that, distance is always finite. There are no p-adic infinitesimals, or infinitely small distances, such as the dx and dy you see in high-school calculus. In the argot, p-adics are “non-Archimedean.” Mathematicians had to cook up a whole new type of calculus for them.

Prior to the multiverse study, non-Archimedeanness was the main reason physicists had taken the trouble to decipher those mathematics textbooks. Theorists think that the natural world, too, has no infinitely small distances; there is some minimal possible distance, the Planck scale, below which gravity is so intense that it renders the entire notion of space meaningless. Grappling with this granularity has always vexed theorists. Real numbers can be subdivided all the way down to geometric points of zero size, so they are ill-suited to describing a granular space; attempting to use them for this purpose tends to spoil the symmetries on which modern physics is based.

 

 

 

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