'But I don’t want to go among mad people," Alice remarked.
"Oh, you can’t help that," said the Cat: "we’re all mad here. I’m mad. You’re mad."
"How do you know I’m mad?" said Alice.
"You must be," said the Cat, "or you wouldn’t have come here."' (Lewis Carroll, Alice in Wonderland)
This blog post gives an in depth discussion on the mathematical background leading up to Einstein's famous Field Equations. In order for the reader to understand all the material covered, it is strongly suggested that all of my previous blog posts be read prior to this one.
Topics such as Ricci curvature, vacuum solutions, Schwarzschild De Sitter metrics, and more are covered!
This blog post gives an in depth discussion on intermediate examples in Riemannian geometry, such as Christoffel symbols, the covariant derivative, and geodesics. Furthermore, we discuss one of the first known examples of a non-Euclidean geometry on the Poincare half-plane, which breaks Euclid's fifth parallel postulate.
This blog post goes more in depth regarding the topic of Lie groups and Lie algebras, as introduced in Riemannian Manifolds: Part I. Once preliminary definitions and theorems are covered, we show how Lie theory can be used to give a much more elegant structure to many common matrix groups such as the unitary and special orthogonal groups. Lastly, this blog post goes into a brief description of classical mechanics and how Lie theory ties into the study of symplectic structures.
This blog post gives an introduction to my second series on higher-level geometry through the perspective of Riemannian manifolds. Riemannian manifolds offer just enough of an abstract structure to define concepts such as curvature and torsion. We will start off by giving a bit of an introduction on Lie groups and Lie algebras, and then show how generalizations of the directional derivative (called affine connection) allow the structure of a Lie algebra to be carried to another tangent plane — this information will ultimately be used to define geometric concepts such as the Gaussian curvature. We will wrap up the article by proving Gauss' Theorema Egregium which shows that the Gaussian curvature of a shape is intrinsic.
Using everything we've learned about manifolds from the past three posts, this post delves into an alternate approach to electromagnetism and Maxwell's famous equations. Moreover, this post covers all background material on Minkowski spacetime and the Hodge dual, so that we have a good understanding of the intersection between electromagnetism and theoretical physics.
Using our vector bundles discussed in the last post, we look at a few additional mechanics that should have been covered earlier (my mistake) and then make way into the daunting realm of cohomological algebra. Surprisingly enough, this leads to generalizations of well-known theorems such as Green's theorem and Stokes' theorem, as well as some of my favorite theorems in cohomological algebra (i.e. the Snake theorem).
With topology covered in the last blog post, this article goes into the details of the two primary fiber bundles over a smooth manifold: the tangent bundle and the cotangent bundle. I will additionally provide the reader with a fresh new perspective on vector fields and extend the analogy of k-forms!
In all honesty, this blog post is a bit more mathematically intensive than the last. In order to fully understand the mechanics of what's going on, I would suggest a minimum of linear algebra. However, I encourage the everyday reader to take a look at it and see if anything rustles your jimmies.
Over a series of blog posts, I plan to introduce the theory of manifolds which are used heavily in theoretical mathematics and physics today. Though a bit of a complicated subject, I'll try my best to reduce concepts to layman terms, so that anyone who has gone through calculus can understand.
Before I can go into the details of manifolds, fiber bundles, Lie Groups, etc. I must first introduce elementary topology. Topology provides a bridge between set theory and analysis, so that fields like geometry may be studied rigorously. Though not super mathematically-rigorous on the reader, this post does introduce a fair amount of new terminology.
My name's Chris and I'm a mathematics Ph.D. student at University of California, Santa Barbara (UCSB). I recieved my masters in mathematics from Virginia Tech studying Morse Theory and geometry, and did an undergrad in computer science engineering as well as an undergrad in math. Over the past few years, I've become a huge enthusiast of Hellenistic and Roman culture, as well as their respective languages, but also enjoy freediving and pyrography. Thanks for visiting!