Ciro Santilli @cirosantilli 35

The BSD conjecture states that if your name is long enough, it will always count as two letters on a famous conjecture.

Maybe also insert a joke about BSD Operating Systems if you're into that kind of stuff.

The conjecture states that Equation 1. "BSD conjecture" holds for every elliptic curve over the rational numbers (which is defined by its constants $a$ and $b$)

The conjecture, if true, provides a (possibly inefficient) way to calculate the rank of an elliptic curve over the rational numbers, since we can calculate the number of elements of an elliptic curve over a finite field by Schoof's algorithm in polynomial time. So it is just a matter of calculating $N_p$ like that up to some point at which we are quite certain about $r$.

The Wikipedia page of the this conecture is the perfect example of why it is not possible to teach natural sciences on Wikipedia. A million dollar problem, and the page is thoroughly incomprehensible unless you already know everything!

Mechanics before quantum mechanics and special relativity.

Written mostly by Eric W. Weisstein.

Ciro once saw a printed version of the CRC "concise" encyclopedia of mathematics. It is about 12 cm thick. Imagine if it wasn't concise!!!

Infinite Napkin is the one-person open source replacemente we needed for it! And OurBigBook.com will be the final multi-person replacement.

If a Big Company makes a product that Does Something, they just call it Big Company Does Something.

If a product is called "Big Company Catchy Name Does Something", then it came from an acquisition, and they wanted to keep the name due to its prestige and to not confuse users.

TODO what's the point of it.

Bibliography:

By the Open University. "Open" I mean.

Some/all courses expire in 4 weeks: www.futurelearn.com/courses/intro-to-quantum-computing. Ludicrous.

Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".

Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.

Another important way to think about Lie algebras, is as infinitesimal generators.

Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.

To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.

As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.

Bibliography:

Recommended from Physics from Symmetry by Jakob Schwichtenberg (2015) page 92: