Blog

Latest Posts

• Gauss Sums And Kronecker Weber

  Recently I was a counselor at the Ross math program where quadratic reciprocity was one of the main things we built up to. (as a side note, I seriously recommend this job, some of the most fun I have had.) Quadratic reciprocity has always been mysterious to me. It is a simple enough statement: knowing when \(p\) is a QR mod \(q\) can be determined if we know when \(q\) is a QR mod \(p\). However this has led itself to 246 published proofs somehow, which always suggested to me something deep should be embedded into the statement. However before last month I could not really convince myself of this being a deep statement. A lot of proofs of it is often just e.g. counting or doing a computation and it just comes out.

• Stone Čech Compactification And Multiplier Algebras

  There is an isomorphism theorem of Gelfand, detailed in the next paragraph. It ends up giving an (anti)-equivalence of categories between \(C^\ast\)-algebras and the category of locally compact haussdorf spaces. It gives rise to the philosophy that \(C^\ast\) algebras are really the study of "non-commutative topology". One of the ideas here is that phenomena in topology should have their counterparts in \(C^\ast\)-algebras. One of the most important examples of this is the extension of \(K\)-theory to \(C^\ast\)-algebras. Today, we do something simpler but still very much striking. We talk about compactifications and their analogues, and how we can use this interplay between topology and \(C^\ast\)-algebras to learn more about compactifications.

• The Inverse Galois Problem Over C(t)

  Galois theory in its vanilla form gives us a connection between field theory and group theory. Specifically, for nice field extensions (so called Galois extensions), you can associate to it the group of its automorphisms (the Galois group). Galois theory claims that sub-extensions correspond to subgroups of the Galois group. Now a natural question is, can we say that each finite group \( G\) is the galois group over some extension of a given field? For \( \mathbb{Q}\) , the problem is still open. However for the field \( \mathbb{C}(t)\) , the answer is positive! The proof requires some Riemann surface theory and some covering space theory. The former is to be expected, as we are talking about \( \mathbb{C}(t)\) which has inherent connections to complex analysis. The latter comes in because there is an analogue of Galois theory for covering spaces. One nice thing about this problem is that it directly translates the covering Galois theory to the vanilla field one for the context. This post will assume the prerequisites of Galois theory and some basic pointset topology. Some algebraic topology will be used, but the reader can safely blackbox those. I will explain briefly the concepts used and the reader wont need to know those beyond the details I provided for this blog.

• Proof Of Cayley Hamilton Using The Zariski Topology

  Cayley-Hamilton is a well known theorem typically introduced in first year linear algebra classes. The proof in these classes are usually some variant of using the Jordan normal form, whose construction is a bit disturbing and its really not interesting. Even historically it is a bit messed up: Hamilton initially proved it for only linear functions on his quaternions, this is the 4 by 4 case of Cayley-Hamilton over R. Later Cayley went and proved it for the 2 by 2 and 3 by 3 cases by literally computing by hand(he really only published the 2 by 2 case and asked the reader to believe he verified the 3 by 3 case). However it was Frobenius who actually proved this theorem, in a 63 page article on it. Unfortunately the theorem is not named after Frobenius who really deserves the credit. This post will present a more interesting and less tedious approach to proving Cayley-Hamilton.