Lecture 1: Topology (International Winter School on Gravity and Light 2015)

What people said

Strong rec symplectomorphic · 2016-12 · Mathematics Stack Exchange · on “What is topology?”

You can watch plenty of people do this on Youtube, e.g. this wonderful introduction with amazing boardwork [this video] by Frederic Schuller, pitched to an audience interested in general relativity. Unfortunately Munkres's lectures at MIT were never recorded for OCW.

Strong rec Daniel Adams · 2021-01 · Mathematics Stack Exchange · on “Topological understanding of real Analysis”

The way I see it you have two options : Forget about the word topology for the time being. Understand open and closed sets in $\mathbb{R}$ (or $\mathbb{R}^d$ ) via the definition of open balls, that is a set $A\subset\mathbb{R}^d$ is open if for all $x\in \mathbb{R}^d$ there exists $r>0$ such that $B_r(x) \subset A$ , where $B_r(x)$ is the open ball of radius $r$ around $x$ $$ B_r(x):=\{y\in\mathbb{R}^d~:~|x-y|<r\}. $$ Take the definition of a continuous function as the " $\epsilon,\delta$ " definition which you will of learnt in your Analysis course. Continue along using these definitions / concepts, until you realise you need topology, then take a course in it. If you have some spare time

Recommends Arthur · 2018-02 · Mathematics Stack Exchange · on “How can a undergraduate student start learning geometry?”

Differential geometry was where I started (Spivak vol 1). If you want to get into general relativity (or just learn an awesome new tool for differential geometry), then tensor calculus is something that I've found very rewarding. Here is one series of lectures on the subject [LINK] . Or if you want to skip that and dive straight into general relaivity instead then this lecture series [this video] really helped me when I took the GR course.

Recommends Nicolas Lussier-Clément · 2018-03 · Mathematics Stack Exchange · on “Manifolds and open sets in them with different dimensions”

Just to warn, I'm fairly new to this discipline so you might verify further. From Lecture 1 of Gravity and light [this video] and books, I understood that a topological space is a set $M$ equip with a topology $\mathscr{O}_M)$ that is a subset of the power set of $M$, $(M,\mathscr{O}_M)|\mathscr{O}_M\subset\mathscr{P}(M)$. The topology $\mathscr{O}_M)$ must satisfy 3 additional axiom: $\varnothing, M \in\mathscr{O}_M$ $U,V\in \mathscr{O}_M \Rightarrow U \cap V \in \mathscr{O}_M$ $U_\alpha\in \mathscr{O}_M \Rightarrow \left( \bigcup\limits_{\alpha \in A} U_\alpha \right) \in \mathscr{O}_M$ Meaning that the topological space does not carry dimensions and in that sense you might be able to define su

Recommends Steffen Plunder · 2018-05 · Mathematics Stack Exchange · on “reference for special relativity”

I never took a course on relativity, but I enjoyed these video lectures on YouTube: Frederic P Schuller: A thorough introduction to the theory of general relativity [this video] Having 13 lectures mainly devoted to geometry, it is maybe not suited as a quick introduction to the topic and feels more like a math course than a physics lecture. But there are many side remarks on how differential geometry and physics are connected and it is very rigorous from a mathematical point of view.

Recommends marc · 2018-05 · Mathematics Stack Exchange · on “d-dimensional topological manifold example”

I am having trouble understanding the concepts of manifolds, so I recently watched these two videos on the topic topology [this video] manifolds [LINK] . During watching I came up with the following example, for which I will briefly denote the required definitions I am trying to grasp first. The definition in question given for a $d$-dimensional topological manifold is: Definition : A topological space $(M, \mathcal{O})$ is called a d-dimensional topological manifold , if $$ \forall p \in M : \exists \mathcal{U} \in \mathcal{O} : \exists x : \mathcal{U} \to x(\mathcal{U}) \subseteq \mathbb{R}^d,$$ such that $x$ is invertible: $x^{-1} : x(\mathcal{U}) \to \mathcal{U}$ $x$ is continuous $x^{-1}$ is

Recommends peek-a-boo · 2021-02 · Mathematics Stack Exchange · on “How to define the differential on a manifold/How to proove a function is differentiable over a manifold?”

Given two smooth manifolds $(M,\mathcal{A}_M)$ and $(N,\mathcal{A}_N)$ of dimensions $m$ and $n$ respectively, a function $f:M\to N$ is defined to be of class $C^r$ ( $r \geq 0$ any integer, usually we consider $C^{\infty}$ ), if for every chart $(U,\alpha)\in\mathcal{A}_M$ and $(V,\beta)\in\mathcal{A}_N$ with $f(U)\subset V$ , we have that the composite map \begin{align} \beta\circ f\circ \alpha^{-1}:\alpha[U]\subset \Bbb{R}^m\to \Bbb{R}^n \end{align} is of class $C^r$ in the usual sense that one learns in multivariable calculus. If this is your first time studying smooth manifolds, this can all be very abstract, so I highly recommend you watch this lecture series [this video] by Frederic Schull

Recommends Markus Scheuer · 2021-03 · Mathematics Stack Exchange · on “Are there any famous/notable mathematicians who have their own YouTube channel?”

I appreciate the central lecture course from The WE-Heraeus International Winter School on Gravity and Light [this video] by Frederic P. Schuller [LINK] . He formidably presents in $28$ lectures, each roughly one and a half hour long the mathematical foundations necessary to understand general relativity. The lectures are built around the definition of the term spacetime , defined as Spacetime is a four-dimensional topological manifold with a smooth atlas carrying a torsion-free connection compatible with a Lorentzian metric and a time orientation satisfying the Einstein equations.

Recommends Clemens Bartholdy · 2021-05 · Mathematics Stack Exchange · on “Relating co-variant basis of different coordinate systems”

I'll try example with a concrete example. Suppose we had a function, $$ f(r,\theta) = r \sin \theta$$ Then, in Cartesian we could write, $$ f(x,y) = \sqrt{x^2 + y^2} \sin \tan^{-1}( \frac{y}{x} )$$ But, in the above function if we substituted $x=r \cos \theta$ and $y= r \sin \theta$ , we'd have: $$ f \big(x(r,\theta), y(r,\theta) \big) = r \sin \theta$$ Meaning, $$ f(r,\theta) := f \big(x(r,\theta),y(r,\theta) \big)$$ The actual topic is about Chart transition maps, I found this playlist [this video] to be of aid, it's about GR but about lecture-5 or 6, transition maps are discussed.

Recommends Ben H · 2022-12 · Mathematics Stack Exchange · on “Could you give me advice on self-study for topology, measure theory and functional analysis?”

Check out the lectures by Frederic Schuller. You can watch the first two in his GR lecture series, on Topology [this video] and on Topological Manifolds [LINK] , to see if you like his style, and then try Lectures on Quantum Theory [LINK] , which are highly mathematical, following von Neumann's functional analysis approach (and have a standalone lecture on Measure Theory [LINK] ). Paired with appropriate textbooks for further reference, they're perfect.