Calculus without limits

# McKean’s Themes

Last week, when looking up Henry McKean, I saw that he passed away on April 20, 2024. I don’t recall having met him in person but I have seen some of his work and met some of this students. Eugene Trubowitz was my Analysis 3 (complex analysis) and Analysis 4 (harmonic analysis) teacher at ETHZ. Trubowitz had just arrived at ETH in 1983 and the course I had from him must have been the first course he taught at ETH. Also Luigi Chierchia, who had collaborated at time with Alessandra Celletti had been a postdoc at ETHZ when I had been a senior undergraduate. An other student of McKean, is Victor Moll who I was lucky once to meet here at Harvard. Also Daniel Stroock (Harvard math graduate from 1962) had been giving courses at the Harvard Math department a few years ago. Also Pierre van Moerbeke who pioneered the integration of the Toda lattice I have never met but I had been exposed to his work as a graduate student, while working on Toda stuff. I had been especially fond of the Volterra lattice a(n)’ = a(n) (a(n+1)-a(n-1)) in Graduate school, first of all because it is simpler than the Toda system (which comes from a second order ODE) but because I learned then as a graduate student that the Volterra deformation is an isospectral deformation of the Dirac matrix D while the Toda system is the isospectral deformation of the Laplacian $D^2$. As a graduate student, I became obsessed already with the theme of factoring Laplacians $L=D^2+E$ and I’m still now, especially in the context of the Dirac matrix $d+d^*$ which is the square root of the standard Laplacian on differential forms. While looking what Trubowitz was doing, I already had got interested in spectral theory. Actually Trubowitz was covering a lot of Sturm-Liouville stuff in his undergraduate courses and told the class that he his busy learning more algebraic geometry. But McKean also turned up when I started to read the book of Cycon-Froese-Kirsch and Simon. This was more like an accident: Konrad Osterwalder was planning to do a seminar on Patodi’s proof of Gauss-Bonnet-Chern and I was asked as a graduate student to prepare for it, but somehow the topic must have scared the students too much (it is heavy!) but I read the chapter in the C-F-K-S book where there is also a nice treatment of the McKean-Singer symmetry which is now so important for me. Reading in that book also brought me closer to spectral theory of almost periodic operators and might have been one reason, I had been considered suited for my first postdoc position at Caltech (Apropos Random, McKean was a student of Willy Feller, and Fellers book was the side read I had when learning probability at the college). Sometimes, things are just random. McKean-Singer is a wonderful spectral symmetry showing that the non-zero eigenvalus of the Laplacian on even forms is the same than the spectrum of the non-zero eigenvalues on odd forms. The symmetry is given by the Dirac matrix. The importance of the Dirac matrix also had emerged at that time by Connes non-commutative geometry. To summarize, while working on isospectral deformations of the Dirac matrix in finite geometries, one is at the heart of not only one but several themes, which McKean was working on. Since I had not much to say otherwise, I was talking about this in the Youtube video below. This was a decision done in the early morning before the presentation. During the bike ride in, thought about out the rime “A theme of McKean” which then morphed into “McKean’s Themes” at the end of the ride because there is not only one theme, but several which intersect with what I’m thinking about at the moment.

Given a finite geometry (G,D,R), where G is a finite set of n elements, D is a $n \times n$ matrix and R a dimension function G to N producing a partition $G_k=R^{-1}(k)$ of G. The matrix D is assumed to be a Dirac matrix, meaning that it is block diagonal $D=d+d^* + m$ with $D^2=L=\oplus_{k=0}^p L_k$ is block diagonal. Especially $d^2=(d^*)^2=0$. The matrix D is in the finite operator algebra $mathcal{B}(l^2(G))$ which has an isospectral set which is a subgroup of $O(n)$. When we look at the isospectral set in the class of Dirac matrices, then this space becomes much smaller and is no more a group in general. What is this set? Is there an Abel-Jacobi map which linearizes the system in the similar way as the periodic Jacobi matrix situation has been solved? First of all one must point out that there is an isospectral flow. I pointed this out in June 2013 here and here https://arxiv.org/abs/1306.5597. The simplest deformation is $D'=[D^+-D^-,D]$. Note that unlike in the Toda case, where we have a discrete Schroedinger equation on a cyclic graph and which corresponds to the Hill situation related to operators $-d^2/dx^2 u + V(x) u$ occurring in Sturm-Liouville problems, we look in the geometric deformation of the Dirac operator at a situation which works for any compact Riemannian manifold with exterior derivative d and conjugate derivative $d^*$ defined by the metric and which works for any Dirac operator coming from a Delta set. This is very, very general, both in the continuum as in the discrete. In the discrete it covers the entire topos of delta set which contains all simplicial sets (a smaller topos containing more structure and axioms) and especially all simplicial complexes, (finite set of non-empty sets G closed under the operation of taking finite non-empty subsets). Now, this looks a bit like an other Toda story, but the behavior is much more spectacular! What happens is that if we start with a Dirac matrix $D=d+d^*$ coming for example from face maps on a delta set like in a simplicial complex, then almost all isospectral Dirac operators will have a diagonal part m. We write $D_t = d_t + d_t^* + m_t$, where we dubbed the diagonal part the dark matter part, the reason being that the $C_t=d_t+d_t^*$ can serve as a new geometry guiding electro-magnetic behavior for example while the part $m_t$ does not contribute to this. If we take the $C_t$ as a new Dirac operator, then by the Connes formula,space expands.

New in this story in the last two weeks was the realization that like in the Toda case, where William Symes has noticed that QR decompositions allow to compute Toda flows, we can also here in a general geometric frame work compute the isospectral deformation using QR decomposition. It might be a bit more difficult to flesh this out in a Riemannian manifold setting. What I did in my thesis, when generalizing the Toda dynamical systems from periodic Jacobi matrices to Random Jacobi matrices $L u(\omega) = a(\omega) u(T(\omega) + a(T^{-1}(\omega)) u(T^{-1}\omega) + b(\omega) u(\omega)$ where $T: (\Omega,\mathcal{A},P) \to (\Omega,\mathcal{A},P)$ is an automorphism of a probability space $(\Omega,\mathcal{A},P)$. Random Jacobi matrices are non-commutative random variables, there does not need to be anything random to it. $\Omega$ can be a finite set or $T: \Omega \to \Omega$ could be a irrational rotation $T(\omega) = \omega+\alpha$ leading to almost periodic Jacobi matrices. The QR decomposition story also works in the geometric setting but it is not clear yet how it relates. What we have is that there are two ways to do isospectral deformation. We can take a function h and look at $D' = [h(D)^+-h(D)^-,D]$ which produces isospectral $D_t$. The second possibility is to look at $e^{t g(D)} = Q_t R_t$ which produces the isospectral deformation $D-t = Q_t^* D Q_t$. What happens is that $Q_t$ is still block tri-diagonal and $D_t$ remains a Dirac matrix. Now this is not only great that we have not to have to use numerical integration schemes. An other interesting featue is that there are deformations which are local in the sense that the time one map value of the Dirac matrix at some cell x only is affected by the immediate neigborhood of x. As pointed out some time ago, the linear wave solution like $u_t = \cos(t D) u_0$ solving $d^2/dt^2 u = - L u =-D^2 u$ does not honor locality. Only in a continuum limit like on a Riemannian manifold, the wave equation has finite propagation speed. In short, a quantized space forces quantized time, if we insist on finite propagation speed of signals.

Lets look at a classical example which is well studied and is now almost 50 years old. Assume $L$ is a periodic Jacobi matrix $Lu(k) = a(k) u(k+1) + a(k-1) u(k-1) + b(k) u(k)$ with with $a(k)>0$, where the isospectral set is a torus, the Jacobi variety of the hyperelliptic curve. The inverse problem is related to Toda flows which are translations on this torus. The solution has been solved in 1975 by Mark Kac and Pierre van Moerbeke and the algebraic connection fleshed out more by Henry McKean and Moerbeke in 1980 (Hill and Toda curves). The picture is that L defines a hyperelliptic curve of some genus g (the number of lacunae= spectral gaps) of the matrix L considered as a periodic matrix on $l^2(\mathbb{Z})$ rather than a $n \times n$ matrix. The auxililary spectrum obtained by imposing Dirichlet boundary conditions on one of the vertices produces by the interlace theorem eigenvalues in the lacunae. If the matrix L moves in the isospectral set, like driven by a Toda flow $L'=[B,L]$, then these divisors move on a straight line on the hyperelliptic curve. The Abel Jacobi map then linearises the map by produces from the divisor class a point on the Jacobi variety. The Toda lattice $a'(k)= a(k)(b(k+1)-b(k)), b'(k) = 2a(k)^2-2a(k-1)^2$ had been introduced in 1967 in solid state physics, it is equivalent to a periodic system of particles on the circle with exponential potential $q(k)'' = e^{-(q(k)-q(k-1))} - e^{-(q(k+1)-q(k))}$. Morikazu Toda lived from 1917 to 2010. Relevant to the work done now is the realization of William Symes from 1982, that if we do QR decomposition of $exp(t L) = Q R$, then the orthogonal path $Q_t$ satisfies $Q_t' = B_t Q_t$ which is equivalent to the Toda system $L_t' =[B_t,L]$