This survey for mathematicians summarizes several works by the author on protein geometry and protein function with applications to viral glycoproteins in general and the spike glycoprotein of the SARS-CoV-2 virus in particular. Background biology and biophysics are sketched. This body of work culminates in a postulate that protein secondary structure regulates mutation, with backbone hydrogen bonds materializing in critical regions to avoid mutation, and disappearing from other regions to enable it.

We prove that the angle defect minus the area of a super hyperbolic triangle is not identically zero and explicitly compute the purely fermionic difference. This disproves the Angle Defect Theorem for N=1 super hyperbolic geometry and provides a novel non-trivial additive function of super triangles. The proof techniques involve the real orthosymplectic group OS(1|2) in its action on the real super Minkowski space of dimension 2,1|2 and brute-force computation.

In 2017, D. Horsman, C. Heunen, M. Pusey, J. Barrett, and R. Spekkens proved that there is no physically reasonable assignment that takes a quantum channel and an initial state and produces a joint state on the tensor product of the input and output spaces. The interpretation was that there is a clear distinction between space and time in the quantum setting that is not visible classically, where in the latter, one can freely use Bayes' theorem to go between joint states and marginals with noisy channels. In this paper, we prove that there actually is such a physically reasonable assignment, bypassing the no-go result of Horsman et al., and we illustrate that this is achievable by restricting the domain of their assignment to a domain which represents the given data more faithfully. This answers an open question at the end of their work, thus indicating the possibility that such a symmetry between space and time may exist in the quantum setting.