[Paper Review] Diagonalizing the genome II: toward possible applications

Satyan L. Devadoss, Jack Morava|arXiv (Cornell University)|Sep 25, 2012

Genomics and Phylogenetic Studies11 references4 citations

TL;DR

This paper proposes a topological resolution of the space of real symmetric matrices (quadratic forms) using an orbihedral stack ${\widetilde{\sf Q}}^{*}_{n}$, modeled on the moduli space of stable genus-zero curves with marked points. By relating eigenvalue configurations to configuration spaces and leveraging associahedral tessellations, the construction yields a simply-connected, stratified space with finite isotropy, suggesting applications in modeling coupled oscillators in genomics and evolutionary biology.

ABSTRACT

In a previous paper, we showed that the orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. In this draft of a sequel, we construct a related (stacky) resolution of a space of real quadratic forms, and suggest, perhaps without much justification, that systems of oscillators parametrized by such objects may may provide useful models in genomics.

Motivation & Objective

• To develop a refined, geometrically structured resolution of the space of real symmetric $n \times n$ matrices, addressing its singular stratification due to eigenvalue multiplicities.
• To establish a topological groupoid framework connecting the moduli space of real genus-zero curves with marked points $\overline{\mathcal{M}}_{0,n+1}^{\rm or}(\mathbb{R})$ to the space of normalized quadratic forms.
• To explore the potential of systems of coupled oscillators, parameterized by eigenvalues of symmetric matrices, as models for genomic and evolutionary dynamics.
• To construct a stack $\widetilde{\sf Q}^{*}_{n}$ that resolves the singularities of the quotient space $\mathcal{Q}^{*}_{n}/{\rm O}(n)$, with finite isotropy and controlled homotopy type.
• To suggest that this resolution may provide a geometric foundation for modeling evolutionary transitions as phase changes in high-dimensional stratified spaces.

Proposed method

• Constructs a stack $\widetilde{\sf Q}^{*}_{n}$ as a pullback over the orientable moduli space $\widetilde{\sf M}_{n+1}$, which resolves the singularities of the space of real quadratic forms.
• Uses the braid hyperplane arrangement and the action of the symmetric group $\mathbb{S}_n$ to stratify the space of symmetric matrices by eigenvalue multiplicities.
• Applies a duality between cubical and associahedral tessellations of hyperbolic manifolds to define a compact, aspherical resolution of the configuration space of points on the real line.
• Defines a topological groupoid structure on $\widetilde{\sf Q}^{*}_{n}$ with isotropy groups isomorphic to wreath products $\mathbb{S}_{n_i} \wr \mathbb{Z}_2$, reflecting the symmetries of repeated eigenvalues.
• Establishes a functor between the moduli space $\overline{\mathcal{M}}_{0,n+1}^{\rm or}(\mathbb{R})$ and the space of normalized quadratic forms, preserving geometric and groupoid structure.
• Applies van Kampen's theorem to compute the fundamental group of $\widetilde{\sf Q}^{*}_{n}$, showing it is trivial for $n > 2$, implying simple connectivity.

Experimental results

Research questions

• RQ1 Can the singular stratification of the space of real symmetric matrices be resolved via a topological stack with finite isotropy and controlled homotopy type?
• RQ2 How can the configuration space of eigenvalues of symmetric matrices be related to the moduli space of stable genus-zero curves with marked points?
• RQ3 What is the topological and geometric significance of the orbihedral stack $\widetilde{\sf Q}^{*}_{n}$ in modeling systems of coupled oscillators in genomics?
• RQ4 Can the resolution of the space of quadratic forms via $\widetilde{\sf Q}^{*}_{n}$ provide a geometric framework for understanding evolutionary transitions as phase changes?
• RQ5 What role do the braid group and cactus group play in the homotopy and cohomology of the resolved space $\widetilde{\sf Q}^{*}_{n}$?

Key findings

• The orbihedral stack $\widetilde{\sf Q}^{*}_{n}$ is a topological stack of constant relative dimension $\frac{1}{2}n(n-1)$ over its quotient space, resolving the singularities of the space of real symmetric matrices.
• The fundamental group $\pi_1(\widetilde{\sf Q}^{*}_{n})$ is trivial for $n > 2$, indicating that the resolution is simply-connected.
• The isotropy groups of $\widetilde{\sf Q}^{*}_{n}$ are finite, isomorphic to $\prod N(n_i) = \prod \mathbb{S}_{n_i} \wr \mathbb{Z}_2$, reflecting the symmetries of repeated eigenvalues.
• The space $\widetilde{\sf Q}^{*}_{n}$ admits an action of ${\rm O}(n) \rtimes J^{*}_{n}$ by orbifold automorphisms, suggesting rich symmetry structures.
• The construction provides a geometric model for systems of coupled oscillators with labeled eigenvalues, potentially relevant to genomic systems such as those in mitochondria or *E. coli*.
• The resolution links the moduli space $\overline{\mathcal{M}}_{0,n+1}^{\rm or}(\mathbb{R})$ to the space of quadratic forms via a functorial groupoid structure, enabling topological analysis of eigenvalue configurations.