A Real World Application of $\mathcal F$-completions

-


The following diagram is a high explosive lens mold from the Manhattan project, drawn in secret by David Greenglass, to pass on to the Soviets:


Below is a 3D symmetrical extension of this high explosive lens mold:


This is precisely $\mathcal I$ which is a special subset of $CX_V$ in dimension $n=3$. You can see the outer colored loops that wrap around $\mathcal I$. This is a subset of $\Gamma$. This also gives a clean example of the coloring function $\phi: E \to \mathcal C$. 

The flat local system $\mathcal L$ lives on these colored loops and it is inherited directly from the geometry of $\mathcal I$.

Thus, we can form the enrichment $\mathcal X=(\phi,\mathcal L, \rho).$ For more details on this see the previous post: https://jzdynamics.blogspot.com/2025/04/the-geometry-of-delta-groupoid.html. 

Without going into too much detail on the physics - the right thing to do is to take geodesic flows along the white strands and add (i.e. direct sum) the force vectors restricted to the collision interfaces. This encodes the force vectors into a flat bundle or local system giving the net force bundle. We can now interpret $\rho$ (the holonomy representation) in terms of a flat connection (i.e. no curvature is detected). This is because any parallel transport along any colored loop path comprising a subset of $\Gamma$ yields no angle defect.


 





Comments

Post a Comment

Popular posts from this blog

An introduction to the $\mathcal F$-completion, the Interface construction and the Gamma set

-
Image
I'll start off with a definition that might initially sound pretty abstract and maybe confusing. The point in defining things as I do is to make our lives easier later on. The goal is to focus on a symmetrical subspace of what I'll call the $\mathcal F$-completion. This subspace is called $\mathcal I$. So let's begin: Let $X$ be a smooth, open, geodesically convex $n$-dimensional manifold with regular polytope boundary, and let $V = \{v_i\}$ be a finite set of vertices in $\partial X$. A block is defined as $\mathcal{B} := X \cup \partial X$, where $\partial X$ denotes the boundary. A foliational completion (or $\mathcal{F}$-completion) of a block $\mathcal{B}$ is a structure obtained by extending foliations to accumulate at specific vertices in the boundary. Formally, for each pair of distinct vertices $(v_i, v_j) \in V \times V$, there exists a foliation $\mathcal{F}_{v_{ij}}$, whose leaves accumulate at $v_i$ and $v_j$. The $\mathcal{F}$-completion is the union of these ...
Read more

Automorphisms and zeta space

-
Image
Here I will operate under the assumption that $\phi_s$ is a unique and well-posed solution to the following PDE. This means whatever conditions are needed for that to occur, will be imposed. $$s\frac{\partial^2}{\partial s^2}\phi_s(x)=\mp x\frac{\partial}{\partial x}\phi_s(x) $$ for which we have our pre-assigned unique solution: $$\mathcal \phi_{s}:=\bigg\lbrace e^{\frac{\pm s}{\log x}}:s \in \Bbb R \bigg\rbrace$$ We have singularities at $s=0$ and $x=0,1$ and this will be called a "cross singularity" or  $CS$. A natural idea would be to intialize some distribution on this $CS$ and run time forward giving a wellposed analytic solution for all time. So we choose our pre-assigned solution $\phi_s$ and run the solution back in time where we obtain what we wanted, at least qualitatively. We can throw out our solution for $s<0$ because it doesn't directly apply to the situation, and it does not rapidly decay. Meanwhile, for $s>0$ it is Schwartz i.e. has a rapid decay cr...
Read more

Grothendieck Site over the Spectral Tower

-
Sheaf Stack over the Dimensional Stratification Site We define a stratified site $\mathcal{S}$ indexed by dimension $n \in \mathbb{Z}_{\geq 2}$, whose objects are geometric-combinatorial models $\mathcal{I}^{(n)}$ associated to dimension-$n$ F-completions. Morphisms in this site correspond to boundary inclusions, coordinate projections, and degenerations between dimensions: $$\mathcal{I}^{(n)} \to \mathcal{I}^{(n-1)} \quad \text{via coordinate slice or foliation boundary}.$$ Let $\mathcal{M}_\Delta^{(n)}$ denote the moduli space of $\Delta$-twisted holonomy representations in dimension $n$, with $\mathcal{M}_\Delta^{(2)} = \mathrm{U}(1)^r / \Delta$ the classical representation torus quotient. We define a sheaf stack of zeta data over $\mathcal{S}$: $$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks}$$ such that: For each $n$, $\mathscr{Z}(\mathcal{I}^{(n)})$ is a stack (e.g. derived or topological) encoding the family of zeta sheaves over $\mathcal{M}_\Delta^{(n)}...
Read more