“The pursuit of mathematics is a divine madness of the human spirit.”
Alfred North Whitehead
Yesterday I gave a talk on OGMC (Ontario Graduate Math Conference) at Waterloo on rigid analytic geometry. Here are the slides. I had a lot of fun at OGMC, and went to lots of great talks. The talks I liked the most was one on algebraic K-theory, and another one on categorical logic (eventhough I can’t say I understood most of them). They seem to be very interesting topics.
“God exists since mathematics is consistent, and the Devil exists since we cannot prove it.”
André Weil
I’ve recently finished my final project for my algebraic number theory course on adeles and ideles, which can be found here. Adeles is an object in algebraic number theory that solves the technical problem of doing analysis over \(\mathbb Q\) so to speak. It lets us work over all completions of a global field simultaneously. One of the applications of it is that certain compactness theorems on it proves the Dirichlet unit theorem and class number theorem. If I had more time, I would have elaborated further at the last part in my project but i unfortunately did not have enough time.
Recently, I gave another talk on the Short Attention Span Math Seminars which is a friendly introduction to category theory. Here are my slides, and here are the slides for the first five minutes. In my talk, I first talked about the idea of abstractions, then I talked about the categorical way of thing and along the way talked about many ideas like universal properties, functors, natural transformations. In the end, this culminated in the Yoneda lemma or the fundamental theorem of category theory. I also discussed a little an application of it to algebraic geometry.
It is a well known fact that there is no general formula for quintic equations or algebraic equations of any higher degree, and a typical proof of this fact uses heavy machinery from Galois theory. However, there is a far more elementary but much less well known proof by V.I. Arnold using nothing more than basic knowledge of complex numbers and topology. Last week, I gave a talk on the Short Attention Span Math Seminars organized by the Pure Math Club at University of Waterloo. In my talk, I explained the main idea of his proof: moving the coefficients along loops to induce permutations of the roots. I also talked about how it could be turned into a rigorous proof using Riemann surfaces of algebraic functions and their monodromy groups, as well as its connections to Galois theory. In the end, I also discussed briefly some ideas related to this proof. Here are my slides.
I’m in a good mood today because it’s my birthday 🎉 and I finally finished recording my final presentation for PMATH 965! For my final presentation, I did an exposition on Galois categories and étale fundamental groups. Here’s the Youtube video of my presentation, and you can download the slides here.
“In mathematics the art of proposing a question must be held of higher value than solving it.”
George Cantor
This week I finally finished my final project for my PMATH 499 reading course in arithmetic geometry. My project is called “Galois cohomology and weak Mordell–Weil theorem”, which is available here. Galois cohomology is the application of group cohomology to Galois groups. In particular, we know that for a perfect field \(k\) with an algebraic closure \(K\), the Galois group \(\mathrm{Gal}(K/k)\) is isomorphic to an inverse limit of topological Galois groups \(\mathrm{Gal}(L/k)\) ranging over the finite Galois extensions \(L/k\) with the natural projections. This fact makes \(G\) a profinite group, and the Krull topology, the topology of \(G\) endowed by the inverse limit process by viewing each finite Galois group as a topological group with the discrete topology, is a topology where a basis for the neighborhood at the identity is the collection of normal subgroups having finite index in \(G\). This is a motivation to use homological algebra in such situation.
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