When is the Fusion inequality extremal?
Graph theory

When is the Fusion inequality extremal?

About the proof of the Fusion inequality

The spectral monotonicity mentioned that we see \sigma(L(U)) + \sigma(L(K)) \leq 2 \sigma(L(G)) in experiments but that we have not proven it. It actually follows. Here is an advantage of labeling eigenvalues \lambda_1 \leq \lambda_2 \leq \dots like differential geometers or analysis do and not the other way round, like often done in combinatorics. (The reason of course is that in the continuum, there is no maximal eigenvalue to begin with). If we write the k’th eigenvalue using the Courant-Fischer formula \lambda_k={\rm min}_{S} {\rm max}_{v \in S, |v|=1} ||Dv||^2 we can see that \lambda_k(U) \leq \lambda_k(G) and \lambda_k(K) \leq \lambda_k(G) and \lambda_k(L_{U,K}) \leq \lambda_k(L_U) + \lambda_k(L_K), where L_{U,K} = L_U \oplus L_K is the Hodge Laplacian of the non-interacting situation.

Theorem: b(K) + b(U) \geq b(G)

About the topic

The subject we have now pursued since January still seems to be new. I have not seen anything like that in the literature. It is a topic which could have been done in the 1930ies if somebody would have looked at it:

  1. finite abstract simplicial complexes were introduced in 1907
  2. the first minimax principle appeared by John von Neumann in 1928
  3. the Dirac operator was introduced in the continuum in 1928.
  4. The first ideas about Hodge theory appears around 1931 and 1944 in the discrete
  5. Finite topological spaces were studied first seriously by Alexandroff in 1937
  6. Mayer-Vietoris appeared in 1929 1930.

I’m very happy about this topic also because it a pretty much unexplored topic even so it is
very, very classical (old school” and also elementary. The topic should also be accessible to any student who has learned a semester of linear algebra. The Betti vectors are defined as nullity of concrete matrices

Schild’s ladder

There are many different ways how one can look at an open-closed complementary pair (U,K) in a simplicial complex. One way is to see K as a closed submanifold possibly with boundary and U as the complement. We started for example to look at the situation where K was a closed disk and U an open disk and G is a sphere. A statistical mechanics person would look at K as a spin up and U as spin down system but where we have some rules in that K is closed under taking subsets and U is closed under taking super sets. We can also look at the problem to extremize the excess cohomology as given in the Fusion inequality. We looked also at some Glauber type Markov dynamics where we switch randomly from open to close if we can. This is not simply a Markov process as we have some topological constraints in that an extension of U must remain open and an extension of K must remain closed. We have also voiced the picture of seeing K as a “laboratory system” and G as the world. This is typical in quantum mechanics and produces a lot of (in my opinion unnecessary puzzlement like the “wave collapse”). Of course, if we look at a small subsystem K of a quantum mechanical system, we neglect all the rest. Frame works like the collapse of a wave function are no paradoxa at all if we would look at the entire system. In our case, the cohomology of K alone is one thing but the cohomology of U also matters and then the interaction. Finally, we could see U as something which appears in Greg Egan’s novel “Schild’s ladder” .

Schild’s ladder deals with quantum graph theory. In the story the “Surampaet Rules were the laws of physics for 1000 years”. But there is a Baby universe U that steadily expands in G. The universe is eaten somehow by itself. The bubble U is more stable than ordinary vacuum and called “Novo vacuum” featuring an exotic geometry.

The novel starts with “In the beginning was a graph, more like diamond than graphite. … The edges had no length or shape, the nodes no position; the graph consisted only of the fact that some nodes were connected to others. This pattern of connections, repeated endlessly, was all there was. ” By the way, I was once using the novel in an exam problem (in 2021) and mentioned it in a paper of 2013 about discrete even dimensional manifolds.

What would be nice of course is to have a deterministic interesting evolution of a closed set K within G. Like in the theory of cellular automata, the problem is that one has too many possibilities to set up a rule. One could for example ask for a sort of mean curvature evolution or then use a probabilitistic evolution and take some quanttity like curvature to define a Glauber dynamics. Again, there are too many models one could consider but there could be some which produce something interesting. I was looking for random evolutions, where make the change if we increase the inequality gap. The hope was that we would reach an interesting situation when looking at maxima. It might however be not so interesting in that the maximum could be the number of locally maximal simplices. If U is the collection of these simplices, then we have excess cohomology in general. Take an orientable d-manifold G for example and pick a maximal simplex away. Then U will capture away the maximal Betti entry. Every time we take a maximal element away, we generated a harmonic pair (and it always will be an even and odd form in pairs which are generated).

I was looking for a picture of Alfred Schild and found one in an obituary in “physics Today” from 1978. Here is the myheritage animation shown in the youtube movie:

The picture to the left shows Alfred Shild. From an obituary of March 1978 in Physics Today: “Alfred E. Schild, one of the world’s leading research workers in the theory of gravitation and Ashbel Smith Professor of Physics at the University of Texas at Austin, died on 24 May. He was 56 years old. Schild clarified and enlarged general relativity through his studies of single- particle motion, quantization, special solutions and the conformal structure of space-time. His beautiful papers exhibiting the null structure of the world combined geometrical vision with analytical power. Much of his work on action principles and particles will be found inspiring by many physicists when read and understood. His style was direct and elegant. As far as I know, Schild never published a wrong formula. His expositions of tensor analysis and relativity are still among the best and clearest treatments of these subjects. Schild, an Austrian Jew, was born in Turkey. In 1939 he fled Nazi Austria, was interned in England and sent to Canada. He studied at the University of Toronto with John L. Synge and Leopold Infeld. In 1957 he came to the University of Texas at Austin. He created a center for relativity there and was instrumental ” 100 MHz counting speed Wide-range source compensation ; 6 operational modes Synchronous sampling & ” Real Time’ background subtraction Model 1140 Quantum Photometer n establishing centers for statistical mechanics and particle physics. His charm, warmth, vision and honesty recruited people from far away for the University of Texas, whose physics department now enjoys international distinction. He liked people; he was a humanist and fought for the rights of the individual with zest and compassion. Titles such as “chairman of the board of regents” or “President of the United States” did not impress him. His proposals for university reform were original; for example “Every newborn child should receive, with his birth certificate, a second document granting him a PhD …” He was a writer and lover of the arts with a tremendous joie de vivre and was completely unstuffy. When a Daily Texan interviewer asked him how he conducted his research, he answered, “I sit at my desk and think of girls, and sometimes I get a good idea.” Physicists all over the world have lost more than a distinguished colleague. One of the kindest, most amiable persons is dead. He was the most decent man I ever knew.” E. L. Schucking New York University

An extremal problem

Given a simplicial complex G, for which K do we get the largest \Pi(K) = \sum_k b_k(U)+b_k(K) -b_k(G)? If K consists of m different zero dimensional points, then \Pi(K) = 2|K|. An other case, where we know much is when G is an orientable manifold and if when U is the set of all facets, maximal simplices. Then \Pi(K) = 2 |U| -2. For example, if = C_m is the cyclic complex of length m, then the maximum is \Pi(K) =2m-2. What happens in general? Also interesting is the question when we have equality.