Senators Kerry and Brown, Representative Capuano:

I am writing as your constituent in the 8th Congressional district of Massachusetts. I oppose H.R.3699 – Research Works Act. As a Assistant Professor of Mathematics at Northeastern University, I use free access to scientific works on a daily basis; Congress needs to strengthen our policy making publicly funded research available to the public, not undermine the policies we already have in this regard. Journal publishers are not adding significant value to scientific works, and at this point are simply a cartel to extract profit from libraries and others who want access to scientific works.

I’ve received funding from the NSF at every stage of my career, as a graduate student, postdoc and professor, and I regard it as a very minor, but very important recompense for that generous support to make my work freely available without any mandate to do so from the NSF; on the other hand, I regard it as scandalous that I am not required to do so.

Science depends on the free exchange of information by investigators all over the world; high journal prices don’t have an enormous impact (yet) on those of us who work at major universities, but science shouldn’t be restricted to such places, and Congress shouldn’t undermine federal agencies who are working to make the science the people paid for accessible to the people.

Sincerely,
Benjamin Webster

While I agree that this is a good way of thinking about what closeness is supposed to mean, and maybe the best, I would dispute that it’s the only one. It’s a very reasonable condition from the pure math side as a kind of “flatness.”

I think it’s a fairly commonly accepted principle there is a triad of the most basic “kinds of structures” on a smooth manifold (of course, there are lots of variants of these, so there there is no claim of completeness here; I don’t want to be attacked by angry contact or Finsler geometers): Riemannian, complex, and symplectic. In each case, this structure consists of a structure on the tangent spaces (a symmetric bilinear form, a complex vector space structure or an antisymmetric bilinear form) which varies smoothly; of course, since both complex and symplectic have extra conditions, I really should say almost complex and almost symplectic.

A vector space which carries a non-degenerate antisymmetric bilinear form thought of as a manifold is a almost symplectic manifold in the obvious way: identify the tangent spaces with the vector space itself and think of the anti-symmetric bilinear form as a 2-form. Call this a “constant” structure. You do the same thing in the Riemannian and complex cases.

Definition. Call a Riemannian/almost complex/almost symplectic manifold flat if it is locally isomorphic to a constant structure on a vector space.

The remarkable theorem about flat structures is that there is a tensor exactly obstructing the flatness of one of these structures. Let the “curvature tensor” of such a structure be the usual Riemann curvature tensor in the Riemannian case, the Nijenhuis tensor in the almost complex case and the exterior derivative of the associated 2-form in the almost symplectic case.

Theorem. A Riemannian/almost complex/almost symplectic manifold is “flat” if and only if its “curvature” tensor vanishes.

In the Riemannian case, I’m not even sure who ascribe this theorem to; it’s so ingrained in people’s consciousness that I’m not sure there’s a name for it. In the complex case, this is the Newlander–Nirenberg theorem, and in symplectic geometry, this is called the Darboux theorem. Of course, the terminology is chosen to make this sound “obvious” but it’s not. One way is; it’s easy to see that these quantities all vanish on a “constant” structure, but the other way is quite difficult, as one has to use some differential equations to cook up the right coordinates.

Flat Riemannian manifolds are called just that; flat almost complex and almost symplectic manifolds are called “complex” and “symplectic” with no further modifiers. I would say this is for historical reasons; the importance of non-flat Riemmannian manifolds was so important, so early in the theory (even before general relativity) that there was no reason to think of the flat case as the basic one and the non-flat as some weird variation on it, whereas I think it took a lot longer for people to recognize that almost complex manifolds had interesting uses and I honestly know of no reason to think about almost symplectic ones, but someone can correct me.

Thus, I think the Darboux theorem is a pretty good justifier of the closed definition, but the ultimate justification is that it seems to be the right level of generality; it covers the examples that interest us and any weakening of it (to, say, exact) throws out lots of examples we like. It may be that 50 years from now, people will think almost symplectic manifolds are great, and we were silly not to have studied them all this time. We’ll just have to see.

So, obviously, the first question is “what is a symplectic manifold?” Now, wikipedia will tell you it’s a manifold equipped with a non-degenerate closed 2-form. Certainly that’s right, but it doesn’t tell a novice in symplectic geometry much. Why think about such a structure?

So let me try to put a different spin on this. This isn’t all that new of a spin (in fact, Henry Cohn wrote almost exactly the same thing here), but I don’t know of anywhere symplectic manifolds are really presented like this: I want to think of a symplectic manifold as a space where one can do a particular flavor of classical mechanics.  I’m going to define a mathematical object called a phase space. This is supposed to be a set of observable facts about a physical system (a “phase”); each point might represent a specific position and specific momentum, or it might be something coarser. Informally, we want that if we specify an energy function which only depends on the phase, then we can tell how the phase evolves with time, and this evolution is “reasonable.” More formally a phase space is a manifold equipped with the following structure

• If is a smooth compactly supported function, then there is a time evolution such that . Physically, we think of this as the energy function specifying how the system evolves over time.
• Conservation of energy: .
• No conserved quantities: for any two points and , there is a chain of energy functions and times such that applying the time evolution for the ‘s in order for goes from and .
• Linearity under superposition: the flow is the exponential of a vector field , and we have that and $X_{cf}=cX_f$ for all constants $c$.
• Equilibrium: if is a critical point of , then for all .
• The assignment from energy functions to flows is equivariant under any of the flows: .

All of these are hopefully intuitive properties for a physically system to have.

For example, if we let for some manifold , we can think of this as the phase space for a single particle running around in (or more generally particles in and ), where the covector measures momentum.  This case, we can split our position into space and momentum coordinates ; the time derivative of is a vector on and the time derivative of is a convector.  On the other hand, for any function , the differential along the space coordinates is a covector, and along the momentum coordinates is a vector.   Hamilton’s equations rewrite Newton’s laws of motion as

This gives a rule for obtaining , and the flow is obtained by integrating this vector field.

Now, I hope you’ve all guessed what the coming theorem is:

Theorem. A phase space is the same thing as a symplectic manifold.

So, given a phase space, how does one find the symplectic structure? Well, by the equilibrium condition, the vector $X_f$ at a point depends only on $df$: if and are equal at a point, then and agree there too, since it is an equilibrium of $f-g$. Thus, by linearity, we have a linear map from the cotangent to the tangent bundle of , which captures the assignment from energy functions to vector fields. By the lack of conserved quantities, this must be an isomorphism.

Of course, an isomorphism between a vector bundle and its dual can be thought of as an element of its tensor square ; if are coordinates in a neighborhood in , then we have a coordinate independent 2-tensor given by

That is, if we let be the matrix coefficients of and the matrix coefficients of its inverse,

I’d like to show that this is a 2-form, that is, that (and thus ) has an anti-symmetric matrix for any basis and its dual.

So, by conservation of energy applied to the function , we have that . Furthermore, applied to , we have that , so indeed is a 2-form.

We’re almost to a symplectic manifold. We have a non-degenerate 2-form, we just need to know why its closed. Conveniently, we have one axiom we haven’t used: the equivariance of the assignment from energies to flows under the flows themselves. We can how restate this in terms of : it says that is invariant under all of the flows corresponding to functions. In terms of the vector fields , we say that the Lie derivative of along is trivial. This can be restated more compactly: there’s a formula for the Lie derivative of a 2-form which is

By definition, we have that though of as a 1-form is just . In particular, this 1-form is closed, and we just have

which is the same as saying that .

Hooray! That finishes the proof one direction: the proof of the other direction can be found (in scattered pieces) in any text on symplectic geometry. The vector field is called the Hamiltonian vector field of , and you’ll most often find these properties phrased in terms of this or its associated Poisson bracket .  Thus, conservation of energy becomes antisymmetry and equivariance becomes the Jacobi identity (note to Lie algebraists: this is the identity that Jacobi actually knew.  He had no idea what a Lie algebra or group was).

This ends our first installment; I’ll continue as I come across bits of exposition that I think actually add to the exposition in Cannas da Silva.

Let be a smooth hypersurface in , over the field with elements. The Weil conjectures are conjectures about the number of points of over . Specifically, they say that there should be some matrix such that

and that the eigenvalues of should be algebraic integers of norm .
Here I am using the Lefschetz hyperplane theorem to know what is for .

This is, of course, a famously hard theorem. The claim about the eigenvalues is the hardest part, but simply the existence of a matrix for which this formula holds is already quite hard; the first proof was due to Dwork.

What I am going to show you is that there is a much easier proof of the above formula modulo ; a proof of the sort that could be appear in Ireland and Rosen. Many of the terms above disappear mod , so our goal is just to show that there is some matrix such that

Some polyhedral notation

Let have degree . Let be the simplex in . We will use the standard shorthand to mean the monomial , for . For any polytope , we’ll write for the lattice points in .

Let be the defining equation of , so is of the form . Let

.

The rows and columns of will be indexed by the lattice points in the interior of . We’ll write for the interior of .

Remark: I’m going to stick to the case of hypersurfaces in projective space, but this argument generalizes to hypersurfaces in any toric variety, and those of you who are used to toric varieties will recognize that I am choosing my notation accordingly.

The Chevalley-Warning trick

We start with a trick which may be familiar from the proof of the Chevalley-Warning theorem.

Notice that, for a nonnegative integer, we have

.

Let be any polynomial , where ranges through some finite subset of . We deduce that

Let be the hypersurface in affine space defined by the polynomial . For , we have

.

We now compute in two ways and get:

Write

So the above formula is

The second equality is just thinking about which exponents of the form could occur in .

Finally, we shift from affine space to projective space. We have . So

An example

Let’s look at the polynomial over . The polytope is a line segment of length , so there are two interior lattice points, namely and . We have, in part,

(All coefficients are reported modulo .)
So the sum in is . and we deduce that the number of roots of in is . Sure enough, has no roots in .

Similarly,

So, as before, we deduce that the number of roots of in is also and, indeed, the polynomial has no roots in that field either.

Finally,

So the number of roots of in is and, indeed, all three roots of the polynomial are in this field.

If you compute for higher and higher values, you’ll see that the coefficients of and cycle through , and with period .

So we are to show that there is some matrix over whose powers have traces , and , repeating cyclically. Certainly, it’s true in this case (a diagonal matrix with entries and works). But why is it true in general?

The matrices

Let’s not just look at the coefficients of , for . Let’s look at the coefficient of , for and . For, example, continuing the previous example, we’ll be looking at the coefficients of , , and . We’ll organize them into a matrix, with rows indexed by and columns by . Call this matrix .

In the above example, , so . We also get that , , and the values repeat from there. (I highly recommend taking a computer algebra system and having it work out these powers for you. It’s really fun to watch them go!)

It is now obvious what we should prove. We should show that . Then, taking , we will have and , as desired.

Finishing the proof

I feel guilty spelling out the proof. It is so much more fun for you to find it yourselves. Really, once you know what you should be proving, there are only a few reasonable things to try. We adopt the convenient notation for the coefficient of in the polynomial .

Okay, here it is. Set . Let . So we have . Since the coefficients of our polynomials are in , we have , or .

So, for any and , we have

We just have to think through what ranges over in this sum.

Well, had better be a lattice point, or there will be no term in . Also, has to be in , as it is to be an exponent of . Set . Then . So lies on the interior of the line segment between , which is in , and , which is in . So is in . We conclude that the only nonzero terms in the above sum come from , and

This is exactly the equation for multiplying matrices. QED.

Some concluding thoughts

I first learned about the Weil conjectures from the introduction to Freitag and Kiehl. This made it seem like an amazing, and thoroughly unmotivated insight, that there should be some cohomology groups around such that the traces of the Frobenius action give the point counts. Looking at examples like this makes the idea seem much more natural. After all, what is the equation but a statement that we have a representation of the group here? And, in our example, the matrices repeat with period three — and the Frobenius for the cubic in our above example has order three! Once you see the matrices, it is hard for the modern mind not to look for the group representation.

Of course, this is very anachronistic of me; the modern mathematical mind looks for the group action BECAUSE in part of the success of that method in proving the Weil conjectures. But, to my mind, every bit of demystification helps.

Those who are familiar enough with the theory may be bothered that the size of my matrices is the number of lattice points in the interior of , which is , not . This is because the argument I am giving here is the low-tech version of Fulton’s fixed point formula, not of the Lefschetz fixed point formula. Unfortunately, Fulton’s theorem only works modulo — if you want to count points modulo higher powers of , you’ll need to work with larger matrices.

Which brings me to a suggestion for someone who really knows this -adic material, and wants to turn out an awesome blog post. It is my vague understanding that Dwork’s great accomplishment was to figure out how to generalize this argument to higher powers of . If someone wanted to write up how this works to count points modulo , in the same sort of elementary way, I’d love to read it. UPDATE I have since realized that rationality of the zeta function modulo is not a good approximation to Dwork’s proof. See comments below.

For each positive integer , the Farey sequence is the increasing sequence of rationals in with denominator at most . For example:

1. .
2. 
3. 
4. 
5. 

It is a standard exercise in basic problem-solving classes to prove that they have the following two remarkable properties:

1. Two rationals in the unit interval are neighbors in some Farey sequence if and only if they satisfy .
2. If and are neighbors in the Farey sequence , then they will remain neighbors in successive Farey sequences until they are separated by the fraction in the sequence .

These properties are typically proved using direct algebraic methods, but I’d like to describe a way to look at them geometrically. The geometric context is provided by Ford circles. Given a pair of coprime integers , the Ford circle is the circle of radius centered at in the complex plane (except when , where I will decree that it is the line together with an additional point called ). There is a minor ambiguity in identifying circles, since and are the same Ford circle. If we ignore the infinite case, the circle is tangent to the real line at the rational point , and each rational number is contained in a unique circle.

There is an immediate connection between Ford circles and Farey fractions: the Farey sequence is in bijection with the set of Ford circles that are tangent to the real line on the interval and have radius at least . A less immediate connection is that Ford circles only intersect at tangent points (whose locations can be explicitly computed). We end up with the following geometric interpretation of the two properties of Farey sequences:

1. If we have rationals , then is nonempty (and indeed a singleton) if and only if . That is, Farey neighbors correspond precisely to tangent pairs of Ford circles. Here, we adopt the convention that fractions are in lowest terms, and negative signs never appear in denominators.
2. If the Ford circles and are tangent to each other, then the Ford circle is the unique circle that is tangent to the real line and the other two Ford circles.

The purpose of this post is to point out that these properties (and many more) follow straightforwardly from a natural action of the group , which we call , on the set of Ford circles. Even though the properties I described are proved with relatively short calculations, I think it doesn’t hurt to have a broader organizing principle in mind.

Recall that is made out of integer matrices satisfying . This is a group under matrix multiplication, and it has the notable property that its rows and columns are made out of coprime pairs of integers. It also acts on the complex upper half-plane by Möbius transformations: yields the transformation . One has the two distinguished generators , and . That is, any element of can be made by composing a word made from these two elements and their inverses. We can say that acts by Translation , while Spins the upper half-plane around by a distorted half-rotation: (this really is a half-rotation if you use the Cayley transformation to turn the half-plane to a disc).

Claim 1: The set of Ford circles has a transitive action of by Möbius transformations. In particular, given a matrix , the corresponding Möbius transformation takes the infinite Ford circle to the Ford circle .

Proof: There are many ways to prove the second sentence, and I will say more general things about transforming circles and lines at the end of this post. Here, it is probably easiest to verify directly: Apply the Möbius transformation to points to get , and check that the resulting points lie in . To show that this map from the line (plus the point at infinity) to the circle is a bijection, you can check that the derivative is nonvanishing, and note that image points approach the real axis as becomes large. To prove the first sentence, we note that by Euclid’s algorithm, any coprime pair of integers admits a pair such that . This implies all Ford circles lie in the -orbit of .
QED

By applying the claim to the transformation , we find that takes to . Note that the line is (setwise) stabilized by the infinite group , and the other circles are stabilized by conjugates.

In the proof of Claim 1, I gave a direct calculational basis for the fact that Möbius transformations take circles and lines to circles and lines. There are other explanations, for example using elementary inversive geometry, but I would be interested to see a solution that avoids calculation altogether. Another interesting question is: If instead of the direct definition we used, we were to define the Ford circles recursively by demanding that they are tangent to circles with smaller denominator, why should we expect their radii to depend only on the denominators? I only know how to motivate this using the group action.

Claim 2: The action of on the set of Ford circles induces an action on the set of ordered pairs of Ford circles, preserving .

Proof: The vectors and generate a subgroup of , and the corresponding quotient of has area equal to the index of the subgroup. We need to show that this area is preserved by the action and is equal to when finite. For the first part, we use the previous claim, where we saw that the action of on Ford circles induces an action on the corresponding integer row vectors by right multiplication, and the induced action on preserves area. The second part follows from the standard theory of cross-products. QED

Now we can prove the Ford circle versions of the claims:

1. We wish to show that for , the set of coprime integer pairs satisfying and is precisely the set of pairs satisfying . By radius considerations, all Ford circles tangent to have the form for some integer , and by Claim 1, there exists that takes to . Therefore, is tangent to if and only if it is the image of some under this transformation. By Claim 2, this holds if and only if . The absolute value sign can be removed, since we have chosen a suitable orientation.
2. We wish to show that if and are tangent to each other and to the real line, then is tangent to both of them. Since the conclusion is symmetric with respect to switching the circles and changing signs, we may assume that the corresponding fractions have positive denominator and that . Then produces the following maps . The claim then follows from the fact that is tangent to at and to at .

There are several useful corollaries to the use of group symmetry. For example, from the fact that , we can immediately conclude that if , then . We can also see that if , then this point of intersection lies on the semicircle whose diameter is the real interval , since the semicircle in question is the image of the positive imaginary ray under the transformation . There is also a connection to continued fractions: Given a real number , we can decree that a rational number is a good approximation of if . The set of good rational approximations to corresponds to the set of Ford circles that intersect the line nontrivially. The sequence of circles hit by the line as one approaches the from above correspond precisely to the convergents of the signed continued fraction expansion of . The signed continued fraction expansion of a convergent yields its expansion in terms of the generators .

Yesterday I had a very nice conversation about rhombus tilings with Lionel Levine, and I decided it was time to consolidate my knowledge and fill in the gaps. So I sat down and dumped everything I could think of into a question on MO. Note that this is a question and even a community wiki one — if you know of more results to add, please head over there and add them!

Reflecting on the sociology of mathematics, it seems to me that we are seeing a growth in ways to do a quick and sloppy job publishing something. Fifteen years ago, this would have been a survey article that would have taken weeks for me to research and edit. Five years ago, this would have been a blog post written over several days. Now I’ve written something much less polished, but I was able to do it in an evening in between taking care of my baby. I’m not sure whether it’s good or bad, but it seems to have been the only way I could get this written at all.

This reminded me of one of my favorite ideas I’ve heard for an NSF policy change which would help career-life balance. Currently the MSPRF postdoc policy reads:

Changes in the host institution will be approved only under extremely unusual and compelling circumstances… Securing a position at an institution other than the proposed host institution is not considered an “extremely unusual and compelling circumstance.”

The suggestion is to change this by adding the line:

Nonetheless, if the fellow has a partner who is unable to procure a job near the sponsoring institution, and both the fellow and their partner have job offers in other city, that will be considered compelling circumstances.

For anyone who is going to be teaching about computing derivatives numerically, my students really enjoyed looking at the data in this paper. (I give them just a scan of table 1, and have them do the analysis themselves.)

Any other great data sources?

In this post I’ll state the main theorem, say a few words about the history, and then explain the main takeaway lesson we learned in this project.

Here’s the main theorem:

Any subfactor of index less than 5 has one of the following standard invariants:

• One of the ADE planar algebras, with index less than 4
• One of the affine ADE planar algebras, with index equal to 4
• An planar algebra, with index greater than 4
• The Haagerup planar algebra and its dual (index )
• The extended Haagerup planar algebra and its dual (index is a certain cubic integer)
• The Asaeda-Haagerup planar algebra and its dual (index )
• The 3311 Goodman-de la Harpe-Jones planar algebra and its dual (index )
• The 2221 Izumi planar algebra and its complex conjugate (index )

The history of this result is that the classification up to 4 was done in the 80′s (with Ocneanu being the main name), the classification up to was begun by Haagerup in 1994 and completed by work of Asaeda-Haagerup, Bisch, Asaeda-Yasuda, and Bigelow-Morrison-Peters-Snyder (which was the first PAPC paper back in 2009). In addition to the main series of papers (part 1 is Scott and I, part 2 is joint with Emily Peters and David Penneys, and part 4 is written by David Penneys and James Tener), the other key paper in extending the classification to index 5 was our paper with Frank Calegari.

The origin of this project was a conversation between Emily, Richard Burstein, and I in the Nashville airport after the Planar Algebras Shanks workshop. We were discussing the possibility of making an “atlas of subfactors” along the lines of Dror Bar-Natan and Scott’s “Knot Atlas” which would automate calculations about small subfactors. My original idea was that if we used global index instead of index, then the classification problem would become finite and thus automatable. As we learned more and more about the existing results, we ended up instead concentrating on two projects: automating and strengthening Haagerup’s approach for searching for small index subfactors (which lead to this project), and automating and improving techniques for constructing small index subfactors (which lead to the paper with Stephen Bigelow). Basically we would sit around in a house at Bodega Bay for a few days (generously lent to us by Vaughan) while I wrote code for the first project, Emily wrote code for the second project, and Scott ran back and forth getting us unstuck.

The most interesting “big picture” lesson to take from this project is the following. Haagerup’s initial classification result made it look like “exotic subfactors” were quite common. Not only did he find a brand new subfactor, to all appearances it looked like there was an infinite family of subfactors all with small index. Asaeda-Yasuda dashed the latter hope, showing that only the first two of this series were possible, but it still seemed likely that exceptional subfactors were quite common. In fact, we long thought that actually classifying all subfactors of index less than 5 was going to be too hard and that we should just search to find new subfactors in that range and construct them. But as it turns out, even though the number of combinatorial possibilities grows rapidly as you go from to 5, the only subfactors in that range were already known. Thus it now appears that exotic subfactors may be relatively rare. In fact, I would not be surprised if the full list of subfactors which are 4-supertransitive (this is a condition analogous to a group action being 4-transitive, but stronger) is just the ADE type graphs, Asaeda-Haagerup, and extended Haagerup.