JZdynamics

 Let $\Gamma$ be a finite connected graph (or multigraph) with first Betti number $r = \beta_1(\Gamma)$, and let $\Delta \subset \mathrm{Aut}(\Gamma)$ be a finite subgroup of automorphisms acting on $\pi_1(\Gamma)$ via pullback. Let $\mathcal{M}_\Delta := \mathrm{U}(1)^r / \Delta$ denote the moduli space of flat unitary representations of $\pi_1(\Gamma)$ up to $\Delta$-symmetry — equivalently, the moduli space of holonomy classes for $\mathrm{U}(1)$-bundles twisted by $\Delta$. Suppose further that there exists a stratified geometric object $\mathcal{I} \subseteq [-1,1]^3$, constructed from $\Gamma$ and $\Delta$, encoding topological, singular, or foliation-theoretic data derived from $\Gamma$ and its symmetries. For example, $\mathcal{I}$ may arise as a cone-singular surface with corners or as a compactification of a flow space determined by $\Gamma$. Spectral Moduli Duality Conjecture: There exists a natural equivalence of orbifolds (or derived stacks) $$\mathcal{M}_\Delta \cong...

Let $\Gamma$ be a finite connected graph (e.g., the 1-skeleton of a stratified space or foliated complex), and let $\mathcal{R}$ denote the stratified space of holonomy representations: $$\mathcal{R} = \bigsqcup_{G \subseteq GL_n(\mathbb{C})} \mathcal{R}_G,\quad \mathcal{R}_G := \left\{ \rho : \pi_1(\Gamma) \to G \right\}.$$ Each stratum $\mathcal{R}_G$ corresponds to a distinct choice of structure group $G$, such as $U(1)$, $SU(2)$, or   $GL_n(\mathbb{C})$. Define a stratified holonomy evolution governed by a sequence of generalized symmetry transformations $$\Psi_k : \mathcal{R}_{G_k} \longrightarrow \mathcal{R}_{G_{k+1}},$$ which may be continuous or discrete, invertible or not, and may preserve or enhance the structure group. The evolution of representations is given by the discrete recurrence: $$\rho_{k+1} = \Psi_k(\rho_k), \qquad \rho_k \in \mathcal{R}_{G_k},\; \rho_{k+1} \in \mathcal{R}_{G_{k+1}}.$$ This defines a dynamical system over the stratified moduli space $\mathcal{R...

 Consider a triple: $$(\mathcal I, z, \Sigma)$$ where $\mathcal I$ is the canonical stratified surface defined in previous posts (smooth away from a singular set) embedded in $[-1,1]^3$, $z$ is the vertical projection (interpreted as time), and $\Gamma=\Sigma_0$ is the critical level set. Define a movie that is given by: $$\Sigma_z :=\begin{cases}\{v_1, v_2, v_3, v_4\} & \text{if } z = \pm 1, \\\\\coprod_{i=1}^4 S^1_i & \text{if } z \in (-1, 0) \cup (0, 1), \\\\\Gamma & \text{if } z = 0,\end{cases}$$ where each $S^1_i$ denotes a topological circle, and $\Gamma$ is a finite connected 4-regular multi-graph formed by the merging of four disjoint circles into two overlapping ovals, with their four intersection points defining the graphs vertices.  For each $z \in (-1,0) \cup (0,1)$, the level set $\Sigma_z$ consists of four disjoint circles $S^1_1, \ldots, S^1_4$. Along each $S^1_i$, we define a real line field: $$V_i \subset T\mathbb{R}^3|_{S^1_i},$$ consisting of un...

We will construct a highly symmetric leaf (an interface leaf) as a special subset in some $\mathcal F$-completion, this time furnishing the leaf with additional geometric data including a metric and curvature. We start by asking the following optimization question: Fix $n=3$ and consider a surface of revolution $S$ and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with points $p,q$ elements of $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm {sup}~ \mathrm{dist}(p,q)=\sqrt{3}$.    What is $\rho_{\mathrm{max}}=\mathrm{max} \lbrace \mathrm{vol}(S) \rbrace_{p,q}$ assuming $S$ must remain a surface of revolution and have constant positive Gaussian curvature? An abstract surface of revolution with constant positive Gaussian curvature (to be embedded/optimized) within $X^3.$ In other words, what is the volume of the largest surface of revolution with constant Gaussian curvature that can be embedded in $X^3$ with a pair of antipodal corners as cone points? Let ...

Consider a family of functions $$f_t(x) := e^{\frac{t}{\log x}}, \quad x \in (0,1), \quad t \in [1/2, 2],$$ each of which can be viewed as a smooth curve in the $(x, f)$-plane for fixed $t$. As $t$ varies, this defines a continuous family of curves, or equivalently, a surface embedded in the extended space $(x, f, t)$. We now define a topological gluing operation by identifying the boundary curves corresponding to $t = 1/2$ and $t = 2$ in the $(x, f)$-plane. The identification is performed along straight lines of slope $+1$, i.e., lines of the form $$f = x + b, \quad b \in [-1, 1].$$ Explicitly, we declare the point $(x, f_{1/2}(x))$ to be equivalent to $(x', f_2(x'))$ whenever both points lie on the same line of slope $+1$; that is, whenever $$f_{1/2}(x) - x = f_2(x') - x',$$ so that both have the same intercept $b \in [-1, 1]$. This identification defines an equivalence relation that glues the endpoints of the parameter domain $t \in [1/2, 2]$ together along a $1$-par...

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 restricte...

This post serves to marry the last three posts into one cohesive framework. The connection to foliational completions is also made clear. Enriched Graphs and Delta Groupoid Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph, embedded in $\mathbb{R}^3$. Define an enrichment of $\Gamma$ to be a triple: $$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$ where: - $\phi: E \to \mathcal{C}$ is a coloring or stratification into a finite set of colors $\mathcal{C}$, - $\mathcal{L}$ is a local system (functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$), assumed flat, - $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation. Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments. Define the Delta groupoid $\mathcal{D}_\Gamma$ as follows: - Objects: enrichments $\mathcal{X} \in \mathcal{E}_\Gamma$. - Morphisms: invertible maps   $$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')...

 Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph embedded in $\mathbf R^3$. Define an enrichment of $\Gamma$ to be a triple: $$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$ where, $\phi: E \to \mathcal{C}$ is a coloring or stratification of the edge set into a finite set of colors $\mathcal{C}$. $\mathcal{L}$ is a local system (i.e., a functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$). It is assumed a priori that connections are flat. $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation. Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments. The Delta groupoid $\mathcal{D}_\Gamma$ is defined as follows: Objects are enrichments $\mathcal{X} = (\phi, \mathcal{L}, \rho) \in \mathcal{E}_\Gamma$. Morphisms are invertible maps $$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')$$ that satisfy the following properties: The underlying graph $\Gamma$ is fixed, thus t...

Let $\Gamma = (V, E)$ be a finite, planar labeled, undirected, tailless, topological multigraph ($4$-regular) arising from a $z=0$ slice of a stratified foliated manifold. The graph admits a decomposition into two overlapping subgraphs ("ovals") $\mathcal{O}_1, \mathcal{O}_2 \subseteq \Gamma$, such that their union reconstructs $\Gamma$: $$\Gamma = \mathcal{O}_1 \cup \mathcal{O}_2, \quad \text{with } \mathcal{O}_1 \cap \mathcal{O}_2 \neq \varnothing.$$ We define a coloring function $$\phi: E \to \mathcal{C} = \{c_1, c_2\}$$ that assigns each edge a domain label corresponding to one of the two ovals. This coloring encodes the stratified foliation structure by partitioning edge interactions into two local holonomy domains. Each edge $e \in E$ carries a local holonomy map $\tau_e \in \mathrm{Aut}(\mathcal{F})$, where $\mathcal{F}$ is a coefficient system (e.g., a local system or vector bundle). For any based cycle $\gamma = (v_0 \to e_1 \to \cdots \to e_k \to v_0)$, the holonomy...

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 ...