A brief update. I’ve been at Cambridge for the last week or so now, and lectures have finally started. I am, tentatively, taking the following Part II classes:
- Riemann Surfaces
Topics in AnalysisProbability and Measure
- Graph Theory
- Linear Analysis (Functional Analysis)
- Logic and Set Theory
I will also attempt to sit in on Part III Algebraic Number Theory, and I will also be self-studying Part II Number Theory and Galois Theory for the Tripos.
As far as this blog goes, my current plan is to blog about interesting topics which come up in my lectures and self-study, partly as a study tool and partly because there are a few big theorems I’d like to get around to understanding this year and some of the material in my lectures will be useful for those theorems.
Today I’d like to blog about something completely different. Here is a fun trick the first half of which I learned somewhere on MO. Recall that the Abel-Ruffini theorem states that the roots of a general quintic polynomial cannot in general be written down in terms of radicals. However, it is known that it is possible to solve general quintics if in addition to radicals one allows Bring radicals. To state this result in a form which will be particularly convenient for the following post, this is equivalent to being able to solve a quintic of the form
for in terms of . It just so happens that a particular branch of the above function has a particularly nice Taylor series; in fact, the branch analytic in a neighborhood of the origin is given by
This should remind you of the well-known fact that the generating function for the Catalan numbers satisfies . In fact, there is a really nice combinatorial proof of the following general fact: the generating function satisfies