$SU(2)$ has a lot of interesting mathematical and physical properties. In this post we study its irreducible representations in a mathematician's way.

Read moreWe develop a very straightforward way to compute the Fourier transform of $\exp(-cx^2)$, in the sense that any contour integration and the calculus of residues are not required at all. This cool approach enables us to think about these elementary concepts much deeper, so I highly recommend to study this approach as long as you are familiar with ODE of first order.

Read moreWe offer a detailed proof of the Riemann mapping theorem, which states that every proper simply connected region is conformally equivalent to the open unit disc.

Read moreIn this episode we focus on the rational field. What can we know about the Galois group of an irreducible polynomial with prime degree? There is a method by counting the number of nonreal roots. From this, we obtain an algorithm to compute the Galois group.

Read moreWe study the Galois group of a cubic polynomial over a field with characteristic not equal to 2 and 3.

Read moreWe try to prove the fundamental theorem of algebra, that the complex field is algebraically closed, using as little analysis as possible. In other words, the following proof will be *almost* algebraic.

Read moreIn this post we give several forms of Masher's theorem by studying group algebra, which eventually becomes a study of semisimple rings. One can consider this post a chaotic evil introduction to representation theory or something.

Read moreThis post is a continuation of a previous post about the ring of trigonometric polynomials over the real field. Now we have jumped into the complex field, and the extension is not a trivial matter.

Read moreIn this post, we study the concept of character, what it is about in abstract harmonic analysis and how to use it Galois theory.

Read moreThis blog post is intended to deliver a quick explanation of the algebra of Borel measures on \(\mathbb{R}^n\). It will be broken into pieces. All complex-valued complex Borel measures \(M(\mathbb{R}^n)\) clearly form a vector space over \(\mathbb{C}\). The main goal of this post is to show that this is a Banach space and also a Banach algebra.

In fact, the \(\mathbb{R}^n\) case can be generalised into any locally compact abelian group (see any abstract harmonic analysis books), this is because what really matters here is being locally compact and abelian. But at this moment we stick to Euclidean spaces. Note since \(\mathbb{R}^n\) is \(\sigma\)-compact, all Borel measures are regular.

To read this post you need to be familiar with some basic properties of Banach algebra, complex Borel measures, and the most important, Fubini's theorem.