**What are time crystals?**

The paper is mathematical in nature. Shapere considers langrangians that are quartic functions of a phase speed. Crucially the term usually associated with kinetic energy (the square one) is negative. This 4th order potential leads to the swallowtail catastrophe of Thom. As a catastrophe it can make claims of generality and structural stability. Furthermore the swallowtail shape gives us several values that minimize the lagrangian, it is said to be 'multivalued'. This multi-valuedness in the potential means there are several states in the ground state. The system can oscillate between those states (since they have the same energy) and the ground state is thus dynamic and periodic. The periodicity is a bit of a 'leger de main' in the sense that Shapere waves his hands saying "if phi is a phase, the linear dependency in time makes it a periodic phenomenon".

**The Bruno controversy**

Shapere was asked about the controversy around his paper. Bruno, a french mathematician has raised the objection that the negative square part doesn't exist and considers instead a normal (positive) kinetic energy and claims to prove that time crystals are in fact impossible. Shapere acknowledged the paper but claims he is not looking at kinetic energy in the classic sense, simply lagrangian terms, expressed as the square of some phase velocity.

**Cold atoms link**

To emphasize the point during the presentation, Shapere established a link to cold atoms (where Georgia Tech is a leading expert). There in some phase as you expand apparently you will start seeing potentials as the one considered in the paper. I am not an expert but it seemed plausible.

**The swallow tail catastrophe**

The point of the Rene Thom catastrophes is that they are generic and structurally stable. Meaning that they will appear in many systems as sure as the caustics at the bottom of the pool (which are the first and second order catastrophes). The swallowtail is such a catastrophe, it is likely to appear. That these forms would appear in nature seems rather sound and the link to cold atoms a welcomed one.

I believe.

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