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There is a beautiful article (a write-up of a talk, actually), by E.M. Purcell, Life at low Reynolds number, that explains how bacteria swim. Low Reynolds number is the technical way to phrase the sta …

In response to the second question (which I interpret as asking for math models of spider webs as they appear in Nature): There exist several distinct types of spider webs. The most common type, the o …

A 2013 issue of Interdisciplinary Science Reviews was entirely devoted to this topic. One viewpoint, by Jesper Lützen, struck me: When Wigner claimed that the effectiveness of mathematics in the natu …

If I understand the question correctly, the search is for a calculation of the asymptotic expansion of Gaussian integrals using concepts and techniques from category theory. Here is one such calculati …

To build intuition for the Planck constant $\hbar$, which I understand is the purpose of the OP, I would start by noting that $\hbar$ is not a dimensionless number: it has dimensions of energy $\times …

I would think that this presentation by Jaroslav Trnka, given here in Utrecht, goes at least some way towards a mathematical definition of the amplituhedron. To skip the physical motivation, start at …

• Concerning question 2, you might want to take a look at Simulation of cubical particle packing under mechanical vibration (2016). The precise effect mentioned in the 2017 paper is not considered in …

For an overview of applications of p-adic numbers in physics I would refer to the Wikipedia and Physics.stackexchange links, and to this nLab entry. Regarding the second question "What is the most con …

An application of the Ising model in social sciences is to voter models: The dynamics of the Ising model tries to align neighbouring spins, similarly, perhaps, to humans deciding on their political, r …

You could start with Michael Aizenman's list of a dozen specific problems from a variety of areas of mathematical physics. The list is two decades old, but most of these problems are still wide open. …

Dyson's A walk through Ramanujan's garden gives the background of this comment: He explains that the "seeds from Ramanujan's garden have been blowing on the wind and have been sprouting all over the l …

Michael Atiyah, On the Work of Edward Witten: In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to …

In integral form, this amounts to the Sokhotski-Plemelj theorem: $\lim_{\epsilon\rightarrow 0^{+}}\int_{-\infty}^{\infty}dx f(x) \frac{d}{dx}\log (x+i\epsilon)=-i\pi f(0)+{\cal P}\int_{-\infty}^{\in …

I would suggest John Barrett's essay on The Hyperbolic Theory of Special Relativity as a comprehensive answer. The principle of relativity corresponds to the hypothesis that the kinematic space is …

An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively. [For t …