I briefly discussed vertex operator algebras in my earlier post on generalized moonshine. While an ordinary commutative ring has a multiplication structure , a vertex operator algebra (or VOA) has a “meromorphic” version , and there is an integer grading on the underlying vector space that is compatible with the powers of z in a straightforward way.

I won’t say much about VOAs in general, but rather, I will consider those that satisfy some of the following nice properties:

Rational: Any V-module is a direct sum of irreducibles.

Holomorphic: Any V-module is a direct sum of copies of V.

cofinite: This is a rather technical-sounding condition that ends up being equivalent to a lot of natural representation-theoretic finiteness properties, like “every representation is a direct sum of generalized eigenspaces for the energy operator L(0)”.

It is conjectured that every rational VOA is cofinite.

As usual, when we have a collection of nice objects, we may want to classify them, or at least find ways of building new ones and discovering invariants and constraints.

Some basic invariants are the central charge c (a complex number), and the character of a module , given by the graded dimension , where the grading is given by the energy operator L(0), and we view the power series as a function on the complex upper half plane using . One of the first general results for “nice” VOAs is Zhu’s 1996 proof that if V is rational and cofinite, then the characters of irreducible V-modules form a vector-valued modular form for a finite dimensional representation of . Furthermore, he showed that in this case, the central charge c is a rational number, and if V is holomorphic, then c is a nonnegative integer divisible by 8.

Dong and Mason classified the holomorphic cofinite VOAs of central charge 8 and 16 – there is one isomorphism class for central charge 8, and 2 isomorphism classes for central charge 16. All three are given by a lattice VOA construction. In general, if you are given an even unimodular positive definite lattice (which only exists in dimension divisible by 8), you get a a holomorphic cofinite VOA from it, so the central charge 8 object comes from the lattice, and the central charge 16 objects come from the and lattices. Central charge 24 is at a sweet spot of difficulty, where Schellekens did a long calculation in 1993 and conjectured the existence of 71 isomorphism types. Central charge 32 is more or less impossible, since lattices alone give over types.

For central charge 24, because the L(0) eigenspace with eigenvalue 1 is naturally a Lie algebra, the proposed isomorphism types are labeled by finite dimensional Lie algebras. Schellekens’s list is basically

1. The monster VOA, with .

2. The Leech lattice VOA, with commutative of dimension 24.

3. 69 extensions of rational Kac-Moody VOAs by suitable modules (here the Lie algebras are products of simple Lie algebras and in particular noncommutative).

As far as existence is concerned, 23 of the 69 come from lattices, known as the Niemeier lattices. An additional 14 come from Z/2 orbifolds of lattices. Another 18 come from a “framed VOA” construction, given by adjoining modules to a tensor product of Ising models according to some codes (Lam, Shimakura, and Yamauchi are the main names here). The remaining 12 are more difficult, and after this recent paper, there are 2 that have not been constructed. There are only a few cases where uniqueness is known, such as the Leech lattice VOA. The case is wide open, and perhaps the worst for uniqueness, since there isn’t any Lie algebra structure to work with.

One of the results of van Ekeren, Möller, and Scheithauer was a reconstruction of Schellekens list, i.e., eliminating other choices of Lie algebras from possibility. This was desirable, since the original paper was quite sketchy in places and didn’t have proofs. A second result was a collection of new examples, in particular nearly filling out this list of 69. They did this by solving an old problem, namely the construction of holomorphic orbifolds. The idea is the following: Given a holomorphic cofinite VOA V, and a finite order automorphism g, take the fixed-point subalgebra , and take a direct sum with some -modules not in V to get something new. In fact, the desired -modules were more or less known – there is a notion of g-twisted V-module V(g), and one takes the submodules of all fixed by a suitable lift of g. To show that this even makes sense requires substantial development of the theory.

First, the existence and uniqueness of irreducible g-twisted V-modules V(g) was a nonconstructive theorem of Dong, Li, and Mason in 2000. Then, to get a multiplication operation on the component -modules, one first shows that irreducible -modules have a nice tensor structure (in particular, are simple currents), so that the space of suitable multiplication maps is highly constrained. This requires recent major theorems of Miyamoto ( is rational and cofinite – 2013), and Huang (if V is rational and cofinite, then Rep(V) is a modular tensor category and the Verlinde formula holds – 2008). By some clever applications of the Verlinde formula, van Ekeren, Möller, and Scheithauer showed that once we have simple currents with suitable L(0)-eigenvalues, the homological obstruction to a well-behaved multiplication vanishes, and one gets a holomorphic VOA.

The intermediate results that I found most useful for my own purposes were:

1. assembly of an abelian intertwining algebra (a generalization of VOA where the commutativity of multiplication is allowed some monodromy) from all irreducible -modules.

2. the explicit description of the action on the characters of irreducible -modules. This also solves a conjecture of Dong, Li, and Mason concerning the graded dimension of twisted modules.

In particular, if g has order n, then the simple currents are arranged into a central extension , where the kernel is given by an action of g, and the image is the twisting on modules. The group A is also equipped with a canonical -valued quadratic form. One obtains an A-graded abelian intertwining algebra with monodromy determined by the quadratic form (up to a certain coboundary), and the action is by the corresponding Weil representation (up to the c/24 correction).

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ALGECOM, the twice annual midwestern conference on algebra, geometry

and combinatorics on Saturday, October 24. We will feature four

speakers, namely,

Jonah Blasiak (Drexel University)

Laura Escobar (University of Illinois at Urbana-Champaign)

Joel Kamnitzer (University of Toronto)

Tri Lai (IMA and University of Minnesota)

as well as a poster session. If you would like to submit a poster, please e-mail (David Speyer) with a quick summary of your work by September 15.

A block of rooms has been reserved at the (Lamp Post Inn) under the name of ALGECOM.

This conference is supported by a conference grant form the NSF. Limited funds are available for graduate student travel to the conference. Please contact (David Speyer) to request support, and include a note from your adviser.

More information will be added to our website as it becomes available.

We hope to see you there!

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Suppose that a drug company wishes to create evidence that a drug is beneficial, when in fact its effect is completely random. To be concrete, we’ll say that the drug has either positive or negative effect for each patient, each with probability . The drug company commits in advance that they will state exactly what their procedure will be, including their procedure for when to stop tasks, and that they will release all of their data. Nonetheless, they can guarantee that a Bayesian analyst with a somewhat reasonable prior will come to hold a strong belief that the drug does some good. Below the fold, I’ll explain how they do this, and think about whether I care.

To be concrete, let’s suppose that the drug company knows that the analyst begins with a uniform prior on the drug’s efficacy: she thinks it is equally likely to be any real number between and . And the drug company’s goal is to get her to hold a greater than percent belief that the drug’s benefit is greater than .

The drug company chooses (and announces!) the following procedure: They will continue to run patients, one at a time, until a point where they have run patients and at least have benefited. This will eventually happen with probability . At this point, they stop the study and release all the data. If the analyst updates on this, she will believe that the drug has effectiveness with a probability that is roughly a bell curve around and standard deviation . (I didn’t check the constants here, but this is definitely the right form for the answer and, if the constants are wrong then just change to .) In particular, the analyst would be willing to bet at 19 to 1 odds that the drug does some good.

If we think that the key to this error is that the length of the experiment is allowed to be infinite, perversesheaf gives some practical numbers based on simulation, which I have also checked in my own simulations. If the experiment is cut off after patients, or when are helped, which ever comes first, then it is the latter situation about 30% of the time.

I mostly want to open this up for discussion, but here are some quick points I noticed:

The uniform prior isn’t important here. As long as the analyst starts out with some positive probability assigned to the whole interval for some , you get similar results.

As Reginald Reagan points out, the analyst rarely thinks the drug is very good.

To state the last point in a different manner, if the drug was even mildly harmful (say it helped 45% of patients and harmed 55%), this problem doesn’t occur. With those numbers, I ran a simulation and found that only 6 out of 100 analysts were fooled. Moreover, in the limit as the simulation goes to , the fraction of analysts who are fooled will stay finite: If a random walk is biased towards , the odds that it will be greater than , let alone greater than , drop off exponentially.

Normally, I’d like to think a bit more about the question before saying something, but I am getting tired and I want to put up this post for one very key reason: Tumblr is an absurd awful interface for conversations. So, I am hoping that if I get a conversation started here, maybe we will be able to actually talk about it usefully.

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These are excellent positions — available for up to 3 years, with no teaching requirements, and salaries in the AUD81-89k range.

Applications close at the end of January, and I hear Amnon is keen to hire as soon as possible.

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Dear Colleagues,We the undersigned announce that, as of today 15 September 2014, we’re starting an indefinite strike. We will decline all papers submitted to us at the Journal of K-Theory.Our demand is that, as promised in 2007-08, Bak’s family company (ISOPP) hand over the ownership of the journal to the K-Theory Foundation (KTF). The handover must be unconditional, free of charge and cover all the back issues.The remaining editors are cordially invited to join us.Yours Sincerely,Paul Balmer, Spencer Bloch, Gunnar Carlsson, Guillermo Cortinas, Eric Friedlander, Max Karoubi, Gennadi Kasparov, Alexander Merkurjev, Amnon Neeman, Jonathan Rosenberg, Marco Schlichting, Andrei Suslin, Vladimir Voevodsky, Charles Weibel, Guoliang Yu

More details to follow!

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The first interesting property of the Monster Lie Algebra has nothing to do with the monster simple group. Instead, the particular arrangement of generators illustrates a remarkable property of the modular J-function.

The more impressive property is a *particular* action of the monster that arises functorially from a string-theoretic construction of the Lie algebra. This action is useful in Borcherds’s proof of the Monstrous Moonshine conjecture, as I mentioned near the end of a previous post, and this usefulness is because the action satisfies a strong compatibility condition that relates the module structures of different root spaces.

The Monster Lie Algebra is a rank 2 Borcherds-Kac-Moody Lie algebra, and this implies it has a self-centralizing subalgebra of dimension 2, and decomposes under the adjoint action into a collection of eigenspaces, called root spaces. Here is the decomposition as a -graded vector space:

A brief examination reveals a bit of a mess in the upper left and lower right. Ignoring the mess for the moment, we have a in the middle, and two copies of the complex numbers in the upper left and lower right. This four dimensional vector space is a Lie subalgebra isomorphic to , i.e., we can view it as the Lie algebra of 2-by-2 matrices under conjugation, where the diagonal matrices form the 2-dimensional space in the middle.

The vector spaces in the mess have a symmetry between the degree piece and the degree piece, imposed by the action of the subalgebra . However, we can say more: the root multiplicities are determined by the coefficients of the normalized modular j-invariant . More precisely, when either m or n is nonzero, the degree subspace has dimension , the coefficient of the -coefficient of .

The simple roots of the Monster Lie Algebra span the degree spaces, i.e., those in the column, and they generate the “positive” subalgebra . That is, one has one simple root of degree , 198664 simple roots of degree , 21493760 simple roots of degree , etc. In the diagram below, the red line divides the Lie algebra into positive and negative subalgebras, and the semi-infinite red box contains the simple roots.

The degree simple root is called “real”, while the rest are “imaginary”, because the root space has an inner product where the vector has norm . To generate the full Monster Lie algebra from the simple roots, we follow the standard recipe for Borcherds-Kac-Moody Lie algebras, using quadratic relations, together with Serre’s relations for real roots. This makes the positive subalgebra into a Lie algebra “freely generated over ” by the imaginary simple roots.

The first miraculous fact about the Monster Lie algebra is that we started with simple roots whose multiplicities are coefficients of , and ended up with all roots having multiplicities given by coefficients of . This fact implies an infinite collection of identities relating the coefficients of . For example, an examination of the degree part reveals that , so .

We can do a more systematic version of this examination using the Weyl (-Kac-Borcherds) denominator formula:

which arises from the Chevalley-Eilenberg resolution of the trivial representation of the positive subalgebra. Here, is the Weyl vector , and is an alternating sum of over finite orthogonal subsets of simple imaginary roots that add up to . The -th homology of the resulting complex is the subspace of the -th exterior power of supported in degree satisfying . For the Monster Lie algebra, the Weyl group has order 2 (as it is isomorphic to ), so the contributions to homology are easy to enumerate. If we set to be a basis vector in degree , we have , , , and for . We decategorify by evaluating the Hilbert-Poincaré series, and obtain the Koike-Norton-Zagier identity:

When considering characters, this identity is naturally an identity of formal power series, although we may turn this into a complex analytic identity near infinity by setting and . The product actually converges in the region where the product of the imaginary parts of and is greater than , since that is where is nonvanishing.

This identity has one quite remarkable property, namely the power series on the left is a sum of power series that are pure in and , while the power series on the right appears to be full of mixed terms containing both and . The vanishing of mixed terms on the left is what yields the identities between coefficients of that I mentioned before. In particular, the vanishing of the coefficient on the right is equivalent to .

The Koike-Norton-Zagier identity was proved independently during the 1980s by the three people named, but it seems that none of them bothered to write up a proof. An elementary argument can be given by multiplying both sides by and taking logs – the right side becomes a sum of where is the -th Hecke operator, while the left side is the sum of , where is the unique polynomial in of the form .

There is a higher-level argument using the theory of Borcherds products. Basically, the Borcherds-Harvey-Moore multiplicative theta lift sends to a function on that is invariant under , with zeroes of multiplicity one along the divisors for , and a cusp expansion as an infinite product whose exponents are coefficients of . It therefore suffices to examine the polar part at infinity to identify with , and the product formula yields the term .

I’d like to recapitulate what I said in the beginning about monster actions on the Monster Lie Algebra. Any linear action of a group on the simple root spaces extends naturally to an action by homogeneous Lie algebra automorphisms, so the bare fact that the monster acts is not so special.

However, Borcherds gave an alternative construction of the Monster Lie Algebra that produced a very well-behaved action. Instead of using generators and relations as above, he used a stringy quantization functor, which also goes by the name . This functor takes in a representation of the Virasoro algebra at central charge 26, and produces a vector space. If the representation has a product structure, in particular from a vertex algebra, then the output has a Lie algebra structure. The “cancellation of oscillators” theorem asserts that if the input has the form , where is a unitarizable representation of Virasoro with central charge 24, and is a Fock space for 2 free bosons with momentum , then the output is the weight part of (when ). This was first conjectured by Lovelace in 1971, and proved by Goddard and Thorn in 1972. We typically attach the name “no-ghost theorem” to this result, although the name refers to a somewhat different aspect of their theorem. In particular, the fact that the output space has no negative-norm states (known as ghosts) was a big deal in the early development of string theory.

Borcherds chose to input the tensor product of the Monster Vertex Algebra with the Lorentzian lattice vertex algebra . The lattice vertex algebra is -graded, and each graded piece is a rank 2 free boson. By cancellation of oscillators, we get an identification between graded pieces of the Lie algebra in degree away from and graded pieces of as monster modules. If we write , then the degree piece of the Monster Lie Algebra is identified with .

Here then is the distinguishing property of the monster action on the Monster Lie Algebra: for any , the root space of degree is a monster module whose isomorphism type depends only on the product . That is, the monster representation is constant along hyperbolas .

This property has the following impact: The identities we found relating the coefficients of , or equivalently the dimensions of root spaces, are promoted to relations between monster representations. For example, the identity of coefficients is promoted to a monster module isomorphism . This requires the identification of the monster action on the degree vector space with the action on the degree space, which does not hold for general monster actions. This additional information means that in addition to the ordinary Weyl denominator identity, we have a twisted denominator identity for each element of the monster, and one can use equivariant Hecke operators to organize the terms.

The application to moonshine is the following: if we want to understand the character of an element in the monster acting on , we may use the twisted denominator identities to obtain recursion relations between the traces on graded pieces. For example, the previous monster module isomorphism yields for any element in the monster simple group. The general form of these recursion relations is known as complete replicability, and Koike showed in unpublished work that the candidate moonshine functions listed by Conway and Norton are completely replicable. Once these recursions were in place for characters of the monster action on , Borcherds proved the Monstrous Moonshine conjecture by comparing the graded pieces of the monster vertex algebra to Conway and Norton’s candidate representations for .

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I don’t yet have any of the juicy numbers revealing what libraries are paying for their Elsevier subscriptions (as Timothy Gowers has been doing in the UK; if you haven’t read his post do that first!). Nevertheless there are some interesting details.

Essentially all the Australian universities, excepting a few tiny private institutes, subscribe to the Freedom collection (this is the same bundle that nearly everyone is forced into subscribing to). The contracts are negotiated by CAUL (the Council of Australian University Librarians).

My librarian was very frank about Article Processing Charges (APCs) constituting double-dipping, whatever it is that Elsevier and the other publishers say. The pricing of journal bundles is so opaque, and to the extent we understand it primarily based on the historical contingencies of print subscription levels more than a decade ago, that in practice the fraction of articles in a subscription bundle for which APCs have been paid has no meaningful effect on the prices libraries pay for their bundles.

I think this point needs wider dissemination amongst mathematicians — whatever our complaints about APCs inhibiting access to journals for mathematicians without substantial funding, we are just plain and simple being ripped off. **Gold open access hybrid journals are a scam.**

Now, on to some details about contracts. First, my librarian confirmed the impression from Gowers’ investigations in the UK — bundle pricing is based largely on historical spending on print subscriptions, with annual price increases. Adding some interesting context on the numbers we’re now seeing out of the UK, she told me that the UK is widely perceived as having received a (relatively) great deal from Elsevier, in terms of annual price increases. If the UK numbers scared you, be aware that here in Australia we may well have it worse. A curious anecdote about historical pricing of subscriptions is that one division of CSIRO happened to have cancelled most of their print journals the year before they took out an electronic subscription with a commercial publisher, and as a result got an excellent deal. The Australian universities have apparently mostly signed confidentiality agreements regarding their journal subscription costs (as we expect, by now), but my understanding of the conversation was that the ANU in particular had not.

Finally, my librarian pointed out that doing what I hope to do next, namely use the FOI act to obtain detailed information on Elsevier subscription costs, may be counterproductive, as the most likely result of unusual discrepancies in pricing being revealed is some libraries simply having budgets cut, rather than actually giving the negotiators any more power in the future. I got the impression she’d talked to other Australian librarians about this, and there was some amount of nervousness.

I’ve been told I should go talk to Andrew Wells, the librarian at UNSW, and after posting this I’m going to get in touch with him!

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In my experience, students in algebraic geometry tend to pick up the rough idea but remain hazy on the details, most likely because there are many different ways to make these details precise. I decided to try and write down all the basic results I could think of along these lines.

Let be some parameter space such as the space of pairs of two conics. Let be some space of solutions, such as the space of triples where is a point on . Let be a map, such as projection onto the components. We want theorems which will discuss the size of the fibers of , in terms of some global degree of the map .

We work over some field . For simplicity of presentation, we’ll assume that is affine, meaning that it is a subset of defined by polynomial equations

We’ll write for the ring .

It would be silly to ask for any such results if were disconnected. A very basic observation of algebraic geometry is that is connected if and only if has no nontrivial idempotents. In fact, we will ask for something stronger: That is an integral domain. The terminology for this is that is **irreducible**. From now on, we will make:

**Assumption** is irreducible. ( is an integral domain.)

If is also affine, with corresponding ring , then is an module. We define the **degree** of in this case to be the dimension of as a vector space. Degree can be defined in much greater generality; we will feel free to refer to it in greater generality without giving the definition. We will denote the degree of by . Roughly, we want theorems which say that the fibers of have size .

Here is our first result.

**Theorem** (Shafarevich, II.6.3, Theorem 4) If has characteristic zero and is algebraically closed then for almost all in . More precisely, there is some polynomial , not identically zero on , so that implies .

**Warning** This isn’t true if is not algebraically closed: Consider the map from .

**Warning** This isn’t true in characteristic : Consider .

We now want results which let us say something, not just about almost all , but about all .

We will at first focus on counting the size of in a naive sense: We think of as sitting in (or in ) and we literally count points of the fiber. We can’t hope for the fibers to always be of full size because even the nicest map, , has fiber of size , not , over the point . So, using the naive size, we can only hope for upper bounds.

There are two additional problems. The first one is if we have something like projecting onto the coordinate. In this case, the degree is but the fiber over has size . When is affine, with corresponding ring , we can fix this by requiring that is torsion free as an -module. In general, the right condition is that no irreducible component of maps to a proper subvariety of .

More subtly, suppose that is a nodal curve, such as , and is its desingularization. (In this case, the line with as the map .) Then the degree of the map is , but the fiber over is , of size . The hypothesis to rule this out is that is integrally closed in its fraction field. By definition, this is the same as saying that is **normal**.

Once we rule out these possibilities, we have

**Theorem** (Shafarevich, II.6.3, Theorem 3) If is normal, and no irreducible component of maps to a proper subvariety of , then every fiber of has naive size .

I can’t resist mentioning a result which far harder than these:

**Theorem** (A consequence of Zariski’s Main Theorem) Let be normal and let have degree . Assume that no irreducible component of maps to a proper subvariety of . For any in , the number of connected components of is at most $d$.

We now consider counting size in a less naive way. Again, for simplicity, suppose that is affine, with corresponding ring . Let be a point of , so there is a map of rings by . Consider the ring , where acts on by the above map. The maps from this ring to are the point in . Thus, is an upper bound for the number of points of above . We will call this dimension the **scheme theoretic size** of the fiber. Once again, it can be defined when is not affine as well.

We have the following cautionary example: Let mapping onto the coordinate. Then the degree is , but the fiber above has size , either scheme theoretically or naively. To rule this out, we impose that is **finite** over . By definition, this means that is affine, and is a finitely generated module.

You might worry about how we could ever prove that is affine if it is not given to us as a closed subset of . Fortunately, we have:

**Theorem** (Hartshorne, Exercise III.11.2) If is projective with finite fibers, then it is a finite map. Here projective means that is a closed subset of , projecting onto . (This is not the morally right definition of a projective map, but if you are ready for the right definition, then you should be working with “proper” rather than “projective” anyway.)

We then have

**Theorem** (Hartshorne, Exercise II.5.8) If is finite over , and no irreducible component of maps to a proper subvariety of , then every fiber of has scheme theoretic size .

**Theorem** Let be a finite map. Then all fibers have scheme theoretic size if and only if is **flat** over .

Unfortunately, flat is a rather technical condition. The first thing to understand is that some nice looking maps can fail to be flat:

**Warning** Let be , let and let the map be . This is a finite map. (We can alternately describe as .) This map is degree , but the fiber over has scheme theoretic size (and naive size ).

If your eye is well enough trained that this doesn’t look nice to you, try the examples here.

There are two good conditions that imply flatness:

**Theorem** (Hartshorne III.9.7) If is normal and one dimensional, and no irreducible component of maps to a proper subvariety of , then is flat over .

**Theorem** (The miracle flatness theorem) If is Cohen-Macaulay, is smooth of the same dimension as , and is finite, then is flat.

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We still have room for a number more applicants, so we would like to encourage more of you to apply. Please note that the application deadline of March 1 is firm.

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