Compute the Adams spectral sequence

I’m currently working on a program that computes the \(E_2\)-term of the Novikov spectral sequence at small primes, using a reinterpretation of Miller’s algebraic Novikov spectral sequence as the Eilenberg-Moore spectral sequence for the differential \(EBP\) Hopf algebroid.

I’ve been sitting on this idea for nearly 15 years but until recently didn’t dare to work on it seriously because I was discouraged by some program failures early on. I now believe I have a slow but working proof-of-concept implementation in Sage, and the next step is to translate this into streamlined code that can reach its full potential.

Question: once you have a program that efficiently computes the algebraic Novikov spectral sequence to large degrees, how much more difficult can it be to compute the real thing with possibly just some minor adjustments and tweaks? After all: isn’t the Adams spectral sequence just a non-commutative deformation of the Miller/Novikov one?

Right. Exactly. You’ve heard it here first. Watch this space.

Just don’t hold your breath quite yet.

Invent non-commutative elliptic cohomology theories

This pipe dream comes from reading too much of my own produce, specifically my paper on non-commutative complex-oriented ring spectra (a.k.a. formal groups over non-commutative rings). One suggestion from that paper is that interesting non-commutative ring spectra should exist at every height \(\ge 2\). The first reflex is to look at elliptic cohomology theories. So what exactly is a “non-commutative elliptic curve”?

You can actually put this question to Google and get good answers: scrolling through the search results long enough will lead you to Manin’s real multiplication program which asks for the arithmetic significance of non-commutative \(2\)-tori (in the sense of non-commutative geometry). This is fascinating stuff, and if there is any chance of connecting this with \(M\xi\), I’ll take it.

Making sense of Rosenfeld's projective planes

I first read about Rosenfeld planes in John Baez’s This Week’s Finds in Mathematical Physics (Week 106). Rosenfeld proposed that the Lie group \(E_8\) can best be understood as the isometry group of the exotic “projective plane” \(P^2 ({\mathbb O}\otimes{\mathbb O})\) - and that the other groups in the Freudenthal magic square arise similarly as symmetries of the planes \(P^2 ({\mathbb A}\otimes {\mathbb B})\) with \({\mathbb A}, {\mathbb B} = {\mathbb R}, {\mathbb C}, {\mathbb H}, {\mathbb O}\).

I love simple explanations like this, so I really want this to be true. The big question is whether one can in fact give a construction of these spaces that admits a meaningful interpretation as a projective plane. Many experts seem to suggest “no”.

My paper on the Bruhat decomposition of Grassmannians was a by-product of an attempt to learn about these spaces. The constructions there give a CW decomposition of the projective line \(P^1 ({\mathbb O}\otimes{\mathbb O})\) (which shows up as \(PG_8^{\rm or}({\mathbb R}^{16})\) in that paper).