Here, I solve Problem $5-5$ from John Lee’s book on Riemannian Manifolds, which demonstrates the non-flatness of the ($2$-)sphere. This problem is particularly interesting because it serves as the motivating example for a later chapter in the book on curvature. As a by-product, we will put to rest the concerns of flat-Earthers. Connections First, I go over the technical tools needed to state (let alone solve) the problem. A connection $\nabla$ on a smooth manifold $M$ is a way of differentiating vector fields (and more generally, tensor fields) along curves in $M$: ...

A Lie group $G$ is a group that is also a (continuous, differentiable) topological space. An example to keep in mind is $G=\mathbb R^n$ which is a group under vector addition and has well-defined notions of continuity and differentiation. To measure lengths and volumes (and relatedly, to define and integrate probability densities) we need to endow $G$ with additional structure so that it is not merely a manifold, but a Riemannian manifold. Luckily for us, we only need to define an inner product for the Lie algebra, after which there is a natural definition of length and volume that can be made for the entire group manifold. I say that the resulting choice of volume (called the Haar measure) is natural because it is compatible with the group structure of $G$ as well as its differential structure as a manifold. This can be compared to how the standard notion of volume for $\mathbb R^n$, the Lebesgue measure, is compatible with vector addition; we have for a (measurable) set $A\subseteq \mathbb R^n$ and for every $\mathbf x_0 \in \mathbb R^n$, ...

There are multiple ways to construct new groups from old ones. For instance, the semidirect product $SO(3) \ltimes \mathbb R^3$ is the Special Euclidean group $SE(3)$, which is composed of all the rigid transformations of $\mathbb R^3$ (minus reflections). Here, I provide an intuition for how these constructions work. I will also go over some of the additional structures that can be imposed on Lie groups, paving the path towards doing differential geometry and calculus on Lie groups. While the word geometry implies the presence of an inner product/Riemannian metric, the word calculus hints at the possibility of differentiation and integration on Lie groups. ...

A topological group is a set of elements $G$ that has both a group operation $\odot$ and a topology . The group operation satisfies the usual axioms (same as those of finite groups ), and the presence of a topology lets us say things like ’the group is connected’ and ’the group operation is continuous’. $G$ is called a Lie group if it is also a smooth manifold. The smooth structure of the manifold must be compatible with the group operation in the following sense: $\odot$ is differentiable with respect to either of its arguments 1. The compatibility of its constituent structures is what makes a Lie group so special, enabling it to capture the essence of a continuous symmetry . ...

Over the past year, I have struggled to pin down what the scope of my blog should be. There is plenty of exposition out there on just about every aspect of modern mathematics, but especially on exterior calculus and differential geometry due to their situation at the intersection of several areas in theoretical and applied mathematics. (As a case in point, the two main references that I’ve been using to self-learn differential forms were the creations of a theoretical physicist and a computer scientist , respectively). So then what is the scope of my blog? Maybe it is for me to catalog the process of self-learning mathematics as an engineering major who lacks a curricular background in modern mathematics. Maybe it is to assure others like me (who are also privileged enough to learn mathematics in isolation of such material concerns as its ‘job prospects’) that it can be done. This post will do a bit of both; it serves in part the purpose of organizing my own thoughts on these matters, and in part the purpose of providing a roadmap for others who are interested in embarking on a similar journey. ...

The Fourier transform takes a (absolutely integrable) function $f:\mathbb R \rightarrow \mathbb R$ and outputs a different (possibly complex-valued) function. If the first is interpreted as a signal (e.g., the waveform of an audio that is parameterized by time), then its Fourier transform has its ‘peaks’ at the dominant frequencies of the signal. I will not expound too much on the Fourier transform itself, but its computation looks something like this1: ...

In this post, I want to bridge the gap between abstract vector spaces (which are the mathematical foundation of linear algebra) and matrix multiplication (which is the linear algebra most of us are familiar with). To do this, we will restrict ourselves to a specific example of a vector space – the Euclidean space. Unlike the typical 101 course in linear algebra, I will avoid talking about solving systems of equations in this post. While solving systems of equations served as the historical precedent1 for mathematicians to begin work on linear algebra, it is today an application, and not the foundation of linear algebra. ...

A running gag in engineering colleges is that a lot of instructors begin their first class of the semester with this question: “What is a vector?”. I used to find this ritual almost pointless because to me, every answer to this question felt either like a non-answer or a matter of context. I mean it depends, right? A structural engineer should have a different answer to this question than, say, a data scientist. To a structural engineer, a vector is a physical measurement that has a magnitude and a direction, whereas a data scientist may not necessarily think of a vector as having a direction. Indeed, in its full generality, a vector does not need to have geometric notions such as directions and angles associated with it. Today, I no longer think that this is a matter of context. By virtue of how we phrase the question “What is a vector?”, we may be asking (without ambiguity) the question: “If I call some mathematical object a vector, what does that tell you about it?” ...

We talked about why sparsity plays an important role in many of the inverse problems that we encounter in engineering. To actually find the sparse solutions to these problems, we add ‘sparsity-promoting’ terms to our optimization problems; the machine learning community calls this approach regularization. ...

The so called curse of dimensionality in machine learning is the observation that neural networks with many parameters can be impossibly difficult to train due to the vastness of its parameter space. Another issue that arises in practice is that most of the neural network does not do anything, as a lot of its weights turn out to be redundant. This is because many (if not all) of the problems we’re interested in solving as engineers have some inherent sparsity. Steve Brunton has an excellent video explaining why this is so. ...