Boltzmann machine, a predecessor of modern machine learning models, is based on a spin-glass model of Sherrington-Kirkpatrick's stochastic Ising Model.
A pair of homologous cycles in a cellular sheaf has effective resistance as defined via the minimal-norm bounding chain. Effective resistance acts as a probability on graph cycles, thus allowing for random sampling to compress a sheaf with control of the eigenspectrum of the sheaf Laplacian. Cellular sheaves thus have natural variational approximations.
Behavior Space manifold of model parameters as coordinates has simpler models as its boundaries with parameters $\theta_\nu = 0, \infty, \theta_\mu$ , the model manifold is usually a hyperribon.
The loss function landscape pertaining to can be thought of thermodynamically as an energy function over thermal degrees of freedom? Stochastic gradient descent then acts as a dissipative process introducing self-generated disorder. Furthermore a phase transition occurs at singularity in free energy.
Any real function on a cotangent bundle can be interpreted to be a Hamiltonian, so real-valued cotangent bundle is a phase space.
A more general form of Arrow's impossibility theorem can be shown to be true as a consequence of homotopy.
A recent model called rapid freeze neuroplasticity, a phenomenon similar to frozen random patterns in coupled map lattices pertaining to self-organized quenched disorder, was proposed as an explanation for neurodynamics of conspiracy theories?
Philip J. Anderson's treated emergence centrally through generalized rigidity, to be understood as a consequence of broken symmetry and the order parameter.
Nerves, a construct of Čech cohomology theory, can be used to track down attractors in spike train activity in computational neuroscience?

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August 5th 2023: Added some more navboxes, as if we did not have enough of those yet.

August 4th 2023: Added theorem include box:

[[
include inc:theorem |
title=Poincaré duality|
body=The [[$\mathbb{F}_2$]]-homology of a compact [[$n$]]-dimensional manifold [[$M$]] is symmetric in its grading. That is for all [[$k,n$]], [[$0 \leq k \leq n$]]: [[$$H_k(M; \mathbb{F}_2) \cong H_{n-k}(M; \mathbb{F}_2)$$]]|
]]

will produce:

Poincaré duality

Theorem. The $\mathbb{F}_2$ -homology of a compact $$n$$ -dimensional manifold $$M$$ is symmetric in its grading. That is for all $$k,n$$ , $0 \leq k \leq n$ : $H_k(M; \mathbb{F}_2) \cong H_{n-k}(M; \mathbb{F}_2)$

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