Dynamical Dirac

Let (G,T) be a finite dynamical system. This means that G is a finite abstract simplicial complex and T is an invertible simplicial map (a permutation of G coming from a permutation of the vertex set V or alternatively, a homeomorphism of the topological space which origins from a permutation of the vertex set V. So, (G,T) is a topological dynamical system on a finite set but unlike what one usually assumes in topological dynamics, the space G is not a metric space. Every finite metric space is automatically discrete. We need a non-Hausdorff topological space to have a reasonable topology. Indeed the topological dimension of the Alexandrov topology agrees with the maximal dimension (an observation of Jennifer Gao from 2024).

Non-Hausdorff topologies have been demonized a bit maybe at first by Hausdorff himself who convinced Alexandrov and Hopf to make Hausdorff the standard assumption in their book. Traditionally, non-Hausdorff spaces have had a hard time. In finite topologies they are unavoidable however! If our world should be finite as some “Neo Demokrits” speculated over the last centuries, then we would live in a non-Hausdorff space as finite Hausdorff spaces are boring and would never support interesting structures necessary for life to emerge: in a finite non-Hausdorff topological space, every point {x} is both open and closed. It is clopen. (Even the worst dictators of the last millenium knew that this is nonsense. Parodies using footage of the iconic “Downfall” 2004 movie have become “memes.” The topology one from 2010 is still the best I have seen.)

P.S. Even in algebraic geometry, the Zariski topology is a bit snuffed at as being “too weak” as it fails to separate points. Much of mathematics is cultural of course. Maybe as an allegory to the distain about the Zariski topology, one can add that we had in the math department a bust of Zariski in the common room. [P.P.S In a math 21a project of 2001 (a time when our math courses were much less structured and teachers could pretty autonomously also add creative innovative aspects into the courses), I once wrote an X-windows program (in C) which allowed students by hand to match points in different pictures in order to do a “structure from motion” reconstruction. I worked with Jose Ramirez in 2008/2009 more on Structure from motion, especially in omni-directional vision mathematics. One can find the C source code here all written from scratch. ] The “missing bust of Zariki” is probably at the same location where the missing math department models are now. Maybe these missing parts are at the same place than the lost arc appearing in the Indiana Jones movie. There are certainly top men working on it. “Top Men!” Of course, the US storage facility hinted at in “Raiders of the Lost Arc” must be at Harvard, where else?The missing “Zariski bust” also illustrates that “fame is ephemere” (Cole prize, National medal of science, Wolf and Steele prize does not prevent you to disappear in a secret storage facility. Zariski seems have had a healthy influence on his students: Hironaka, Mumford, Hartshorne, Kleiman or Artin are all fine.) Back to the Zariski topology: when I learned algebraic geometry (see my notes of the course given by Knus), it was all commutative algebra = ring theory. I don’t think a single algebraic curve was ever drawn …. The Zariski topology appeared in the first 10 minutes of that course. It started as follows: “Let R be a commutative ring with 1. There exist maximal ideals and so prime ideals in R. The spectrum X of R is the set of non-empty prime ideals. If S in an arbitrary subset of R, then V(S) is the set of prime ideals containing S. This defines a topology on X, the Zariski topology …”.

Not only the connection matrix or Green function matrix can be defined “dynamically”. One can also define dynamically. If is the usual exterior derivative, then define . It turns out that while depends on T, the Hodge Laplacian does not. So, there is no change of cohomology when using the deformed Dirac matrix. This is like with the isospectral deformation of the Dirac operator, which I worked on last summer a bit more showing that one can get the time t map of this differential equation using a simple QR decomposition. I talk a bit about this and also say something more about the Laeuchli paraodox. It was Prompted by the last Wednesday talk of Michael Freedman on the proof of the Poincare conjecture. It was a nice and generally accessible talk. There will be 6 more talks about the remaining conjectures. I’m especially looking forward to the Yang-Mills one in October, as I believe this is a terrible choice of problem. Maybe I will change my mind after the talk. As for now, I have the impression that the Yang-Mills gap problem is a political problem in the sense that the decision whether a given “solution” is probably going to be partisan and disputed. Is the mass gap for connection matrices in a Yang-Mills setup a valid approach? Of course not. Physics is a tricky business. Even in very basic frame works like statistical mechanics or quantum mechanics, there is much disagreement. Whether “a non-trivial quantum Yang-Mills theory exists in very much depends on what one considers to be a non-abelian quantum field theory. What kind of mathematics is allowed? The criteria of whether one has a solution is required to “standard rigor in mathematical physics and in particular constructive quantum field theory”. Chatterjee is a student of Diakonis and a probabilist. Is a probabilistic approach to the Yang-Mills setup allowed? Or does it have to be in the axiomatic quantum field frame work which has been tried for decades by Mathematicians like Osterwalder (or my own PhD advisor Lanford)? The vagueness of the set-up is rather annoying in my current opinion. And it is a bit telling that many mathematical physicists who had worked in that subject have changed subject.