Algebraic geometry and its plot to take over the (analytic) world.

Recently, Peter Scholze posted a challenge to the Xena project to formalize a result in condensed mathematics. The motivation for this result is to provide a new foundation for analysis, not in terms of topology but using a different approach based off of algebraic geometry.

Mumford writes in Curves and their Jacobians: “[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.”

For some reason, this secret plot has so far stopped short of taking over analysis. The goal of this course is to launch a new attack, turning functional analysis into a branch of commutative algebra, and various types of analytic geometry (like manifolds) into algebraic geometry.

Peter Scholze

As an inhabitant of the analytic world, I’m curious about our would-be conquerors. Scholze’s work has revolutionized algebraic geometry, and I’m excited to what his insights into functional analysis are (though I almost certainly won’t understand them). However, I’m also a bit skeptical about the goal of replacing the foundations of functional analysis with an algebraic theory. I wanted to give my perspective about why category theory doesn’t yet play a major role in analysis and what some of the challenges are for any theory, algebraic or otherwise, to supplant analysis.

Let me start with the disclaimer that I am not an expert in functional analysis. Geometric analysis relies on functional analysis, but I mostly work on problems where standard elliptic or parabolic theory applies (or the hard analytic work has already been done). In this situation, it’s possible to black-box much of the underlying functional analysis, which is something I’m frequently guilty of doing.

I’m also a novice in terms of category theory and abstract algebra. In grad school I took a few courses that used categories, and we did cover some basic homological algebra. To be honest, I struggle with both abstraction and algebra, so it wasn’t really my cup of tea. I once knew what schemes and spectral sequences are, but that was long ago and I’ve happily forgotten by now. My impression is these topics are very useful/ubiquitous in algebraic branches of mathematics, as they really capture some essential structure and are useful for the sort of homological calculations that show up.

However, just because an approach works really well in one area, does not mean that it applies in completely different contexts. Category theory and homological algebra may be powerful and ubiquitous in many areas, but I don’t understand how they are equipped to handle day-to-day work in geometric analysis. In particular, it seems like difficult analytic questions are very different form the questions that algebraic geometry and higher category theory were developed to answer.

Difficult questions in geometric analysis tend to be rather specific, and not amenable to structural approaches. The interesting geometry often exists in edge cases where general functional analysis doesn’t apply and you have to take advantage of some specific feature of the problem (convexity of the domain, positive curvature, existence of a sub-solution, or something else) to make the analysis work.

In order to find an alternative approach for these sorts of questions, it would be necessary to find a way to encode the specific information into algebraic data, and develop a theory which can meaningfully use this data. One possibility to get started might be the theory of exterior differential systems. EDS is a very powerful theory so it’s likely that in some cases it could be made to work. However, in my opinion EDS has two major drawbacks compared to more conventional functional analysis. First, the standard existence theorems rely on analytic data, which is too rigid and restrictive for many applications (although might be fine for applications in condensed mathematics?). Second, the standard approach does not provide any sort of quantitative control over the behavior of the equation. It seems that Scholze has found a way of doing homological algebra while keeping track of quantitative estimates, which is promising in order to overcome this second challenge.

From an outside perspective, it might seem like partial differential equations is a disconnected field without a unifying theory. It’s natural to ask whether an analytic Grothendieck (who actually started his career in functional analysis) could find a coherent and general framework. However, the disjointed nature is a feature of PDE theory, not a bug. Fluids behave very differently from heat, so their theories are also very different.

An illustrative example

In keeping with the theme of specificity, I’ll discuss a particular example of a challenging analytic problem, which is the existence and uniqueness of Ricci flow. The Ricci flow deforms a Riemannian metric by its Ricci curvature and plays a central role in modern geometry, including Perelman’s proof of the Poincare conjecture. When the flow was first introduced by Hamilton, the original proof of existence and uniqueness was a tour de force involving the Nash-Moser inverse function theorem.

Intuitively, Ricci flow behaves like a heat flow for the geometry (with additional reaction terms which can cause spaces to violently tear themselves apart). However, it’s not possible to apply the standard theory for the heat equation to get existence and uniqueness for the Ricci flow. The issue is that solutions to the Ricci flow can be conjugated by diffeomorphism to get new solutions. As a result, the flow seems to think there are many solutions because it can see all the conjugations. From an analytic perspective, what this does is introduce degeneracy into the symbol of the flow. There is a standard and general theory that establishes existence and uniqueness for heat-type equations/systems, but this theory breaks down for degenerate cases. However, from a geometric perspective, conjugations by diffeomorphism are basically just coordinate changes, so it’s actually a good thing that these don’t make a whit of difference.

It’s probably not possible to find a general approach to abstractly deal with degenerate parabolic (or elliptic) equations. Degenerate equations are more pathological than their non-degenerate brethren (see, e.g., the regularity theory of optimal transport). By working specifically with the Ricci flow, Hamilton was able to find the right analytic approach to establish existence and uniqueness. Later on, Deturck found a proof for existence/uniqueness which takes advantage of the diffeomorphism invariance in a natural way.

An analogy with information geometry

I don’t want to give the impression that I’m just bashing category theory, so let me talk about a situation where differential geometry has been proposed as a foundation for a more applied field. Information geometry uses ideas from differential geometry to understand parametrized families of probability distributions. I’ve heard it suggested that this should revolutionize statistics, but I (and the statisticians I have discussed IG with) are skeptical.

Most statisticians don’t care about affine connections or Riemannian metrics for their own sake, so to motivate them to learn information geometry, it needs to be useful for what they care about. However, the difficult questions that working statisticians have can’t be answered using foundational approaches. For instance, differential geometry on its own probably isn’t going to help answer the question “how sensitive is my hierarchical Bayesian model to the choice of hyperpriors?”

That’s not to say that information geometry has nothing to offer. For instance, the notion of natural gradients is really natural from a geometric perspective and does speed up convergence of gradient descent. There are also some fascinating insights that statistical manifolds have for geometers. But information geometry works best as a collaboration between two fields, not as an attempt to rewrite statistics from the ground up. Put bluntly, you should be extremely skeptical of anyone who says that what’s needed to model the spread of infectious diseases is a deep understanding of Legendre duality.

[The] teaching of statistics cannot be done appreciably better by mathematicians ignorant of the subject than by psychologists or agricultural experimenters ignorant of the subject.

Harold Hotelling

Welcome to our algebraic overlords

All this being said, it’s definitely a bad idea to bet against Peter Scholze. And I, for one, welcome our new algebraic overlords. Here are some places where I think that a more structural framework to analysis might be useful.

  1. There are a class of functional inequalities that are applied as standard tools. The two most well-known examples are the Sobolev and Morrey inequalities, but there are quite a few related results. Having a general framework to understand these inequalities seems like a natural place where structural insights can be helpful.
  2. Similarly, there is a “standard theory of elliptic equations,” which often is a just a polite way of saying “proved somewhere in Gilbarg-Trudinger.” It might be possible to find a unified approach for Schauder estimates and whatnot and perhaps algebra will have some important insights for this.
  3. One project that is much more ambitious would be to find a truly algebraic proof of the Calabi conjecture or something similar (i.e. a proof that either doesn’t require or abstracts away the a priori estimates for complex Monge-Ampere equations). That’s a lot to ask for a fledgling theory, but the problem seems closely enough related to algebraic geometry to be a test case for the theory.

Of course, it’s likely that the insights from algebraic geometry will be completely different. I look forward to seeing what new advances they bring even if I dread the day when I’ll be expected to teach analysis in terms of universal properties.

Addendum: some remarks by Peter Scholze

Thanks a lot for your blogpost! I hope it is OK if I make a comment here…

As I hope was clear from the phrasing, the opening paragraphs from my
course were not at all serious (I mean, Mumford was joking, so was I). I
have no ambitions to overtake analysis, I just want to be able to talk
about those parts of analysis that I care most about in a way that
combines seamlessly with how arguments work in arithmetic/algebraic
geometry. The hope that this could be done resulted from having had to
do quite a bit of p-adic analysis, and this never felt much like
analysis to me.

So yes, I just hope that it may be a helpful alternative language for
some analytic questions close to algebraic geometry. Your question
whether it may give a more “algebraic” proof of the Calabi conjecture is
very much appreciated! In fact, I don’t really know what good test
questions for the formalism would be, but this is obviously a central
question in the overlap between algebraic and differential geometry. The
Simpson correspondence would be another such thing.

Some things Clausen and I have already thought about in terms of this
formalism:

— it gives formal proofs that coherent cohomology groups on compact
complex manifolds are finite-dimensional, and satisfy Serre duality.
Usually, this is done by invoking ellipticity of
\overline{\partial}-operators; our proof stays purely in the realm of
holomorphic things, and reduces everything to a simple computation for
the open unit disc.

— it gives a generalization of Serre duality to open complex manifolds,
even in cases where the relevant complexes do not have the property that
differentials have closed image (so the cohomology groups are
pathological as topological vector spaces)

— hopefully, it can similarly lead to nice proofs of
Grothendieck-Hirzebruch-Riemann-Roch (for general compact complex
manifolds) or Atiyah-Signer; but we currently have some confusions about
the “correct” way to do K-theory in this setting.

So these points are in the spirit of blackboxing some things usually
done with or closely related to elliptic operators.

Another thing that came up in some discussions once: You might want to
model the “rational homotopy type” of a smooth manifold M by its
commutative differential graded algebra of differential forms
\Omega^\bullet(M). Then you would like to formulate a Künneth theorem
that for a product manifold M\times N, one gets the tensor product
\Omega^\bullet(M\times N)\cong \Omega^\bullet(M) \otimes \Omega^\bullet(N) of those commutative differential graded algebras.
This forces you to take a certain completed tensor product, and this may
or may not interact well with whatever homological algebra you also
intend to do. If you apply the formalism of liquid vector spaces, it is
simply the liquid tensor product, and it interacts perfectly well with
whatever homological algebra you intend to do. For any given
application, I’m sure there’s a more classical way of proceeding, but
the liquid formalism handles such questions gracefully.

Upshot: If you are an actual analyst, you probably have no reason to
care about whatever it is that I’m doing. If however you are an
algebraist that is forced to use some analysis, there is a nonzero
chance that these “liquid vector spaces” can be useful.

Peter Scholze, December 13 2020

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