Miscellanea I

While researching for this blog or idling browsing, I come across a number of interesting results that I don't really know the wider context for, or especially clear and convincing expositions of various specific theorems from areas in which I am definitely not an expert. I really want to share these, but there is not always a reblog function and it seems rather pointless to reproduce the arguments verbatim, lovely as they are. The compromise is that I will occasionally write posts like this, where I just list a smorgasbord of miscellaneous results that I hope become better known. Also, if I write posts as opposed to creating a separate page, they can be indexed by the tag function.

• Theorem (Chebotarev): Let ${\displaystyle \mathrm{\Omega }}$$\omega_n$ be a primitive $n^{th}$ root of unity and $M_n$ be the $n \times n$ matrix with entries ${\displaystyle a_{ij}={\omega }^{ij},1\le i,j\le n \$}$${\displaystyle \omega =e^{2\mathrm{i}\pi /n}}$$a_{ij}=\omega_n^{ij}$. If $n$ is prime then any minor of ${\displaystyle \mathrm{\Omega } }$$M_n$ is non-zero. 
  
  Apparently this less known theorem of Chebotarev has been rediscovered many times, and even appeared as a problem at the famous Miklos Schweitzer competition. Frenkel and Tao have both given great proofs of this result. In particular, Tao gives a proof of the Cauchy-Davenport theorem as well as an uncertainty principle for functions on the cyclic group $\mathbb{Z}/p\mathbb{Z}$.
• A couple of natural questions on integer and integer-valued polynomials bounded by $e^n$ with pretty nice solutions. 
• I found some nice notes on differential graded Lie algebras and why one might care about derived algebraic geometry/functors.  Perhaps sources for where to read about such motivation exist and are known to the experts/people in the field, but as an outsider I find it difficult to get excited about such things because expositions accessible to me are hard to find and this does a good job of that.
• The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle induces a stable framing of the stable tangent bundle, so stably framed manifolds represent elements of the stable homotopy groups of spheres. It turns out all elements are represented by an "honestly" framed manifold. 
• The relation between stable homotopy groups of spheres and manifolds is a close one, so it is very satisfying that $\pi_3^{st}=\mathbb{Z}/24$ has a geometric explanation, and there is an octonionic analogue!

• (Catching a ray of light) I came across a very natural and interesting question: given a smooth curve $\gamma$ of radius of curvature at most 1, suppose at each point there is a disc that is orthogonal to $\gamma$. This sweeps out a tube which one can view as a fibre optic cable, and then ask whether shining a light in through one end always results in the light coming back out.