Subhash Khot is giving a talk at Current Developments in Math this year and his talk has the intriguing phrase “Grassmann graph” in it so I thought I’d look up what it is and what he and his collaborators prove about it, and indeed it’s interesting! I’m just going to jot down something I learned from “Pseudorandom Sets in Grassmann Graph have Near-Perfect Expansion,” by Khot, Dor Minzer, and Muli Safra, in a way that makes it sound like something algebraic geometers might be interested in, which, indeed, I think they might be!

Suppose you have a sheaf F on a space, and the space has a covering U_1, .. U_N. The sheaf axiom says that if we have a family of sections s_i of F(U_i) such that s_i and s_j agree on for all i,j, then there is actually a global section s in F(X) which restricts to each s_i.

What if we only have an *approximate* section? That is: what if we have a family of s_i such that: if I select i and j uniformly at random, the probability that s_i and s_j agree on is bounded below by some p > 0. Call such a family a “p-section.” (You should take the view that this is really a family of problems with X changing and N growing, so that when I say p > 0 the content is that p is bounded away from some 0 uniformly in X,N.)

The question is then: **Is an approximate section approximately a section?**

(This is meant to recall the principle from additive number theory that *an approximate subgroup is approximately a subgroup*, as in e.g. Freiman-Rusza.)

That is: if s_1, .. s_N from a p-section, is there some actual section s in F(X) such that, for i chosen uniformly at random,

for some p’ depending only on p?

The case which turns out to be relevant to complexity theory is the Grassmann graph, which we can describe as follows: X is a k-dimensional vector space over F_2 and the U_i are the l-dimensional vector spaces for some integer l. But we do something slightly weird (which is what makes it the Grassmann graph, not the Grassmann simplicial complex) and declare that the only nonempty intersections are those where has dimension l-1. The sheaf is the one whose sections on U_i are the linear functions from U_i to F_2.

Speculation 1.7 in the linked paper is that an approximate section is approximately a section. This turns out not to be true! Because there are large sets of U_i whose intersection with the rest of X is smaller than you might expect. This makes sense: if X is a space which is connected but which is “almost a disjoint union of X_1 and X_2,” i.e. and $\latex X_1 \cap X_2$ involves very few of the U_i, then by choosing a section of F(X_1) and a section of F(X_2) independently you can get an approximate section which is unlikely to be approximated by any actual global section.

But the good news is that, in the case at hand, that ends up being the *only* problem. Khot-Minzer-Safra classify the “approximately disconnected” chunks of X (they are collections of l-dimensional subspaces containing a fixed subspace of small dimension and contained in a fixed subspace of small codimension) and show that any approximate section of F is approximated by a section on some such chunk; this is all that is needed to prove the “2-to-2 games conjecture” in complexity theory, which is their ultimate goal.

So I found all this quite striking! Do questions about approximate global sections being approximated by global sections appear elsewhere? (The question as phrased here is already a bit weird from an algebraic geometry point of view, since it seems to require that you have or impose a probability measure on your set of open patches, but maybe that’s natural in some cases?)

I guess the connection with expansion in graphs can be made very direct: given a graph X, you can take the U_i to be the vertices and the U_i cap U_j to be the edge ij, when it’s there, or 0 otherwise, and F the space of continuous functions on the graph. Then an approximate section would be a function on Vert(X) which with very high probability takes the same value on two randomly chosen adjacent vertices. The question of whether there exists such a function which isn’t constant on a large proportion of Vert(X) is basically asking whether the graph has a good minimum cut.

Instead of imposing a probability measure, you can equivalently ask: over how large a subset of the base space do you have a section? Jamie Tucker-Foltz (then an undergraduate!) and I elaborated this viewpoint here: https://arxiv.org/abs/1803.06800. Many constructions around the Unique Games Conjecture turn out to have a natural topological interpretation.

Small correction: it’s Subhash Khot, not Subhosh.

Oops, thanks! Now fixed!

Not an expert, but perhaps the statistical mechanics community has thought about these questions. For instance, one can ask to what extent a low temperature state of the Ising model (where most of the sites have matching spins) is close to a zero temperature state (where all the spins are aligned), i.e. when is the system magnetised. As you point out it will depend on how highly connected the graph is, what the sparsest cut is, etc..