8
votes
1answer
59 views

Parametrizing Walks on Sphere and Torus

This question is very underdeveloped, but I was wondering if there was a map from the sphere to the torus which preserves length of closed curves? I was just thinking about taking a walk on a ...
3
votes
1answer
76 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
1
vote
1answer
39 views

Intuition regarding the Whitney trick

I read here that a major ingredient in Whitney's strong embedding theorem and later Smale's celebrated h-cobordism theorem is the Whitney trick. Can someone give an intuitive description of the ...
4
votes
1answer
143 views

Is it possible to learn differential topology before analysis?

Currently I'm self studying for my own enjoyment topology and algebra (munkres and herstein). Since I start at the university next year everything I'm learning now is for my own enjoyment and I will ...
0
votes
1answer
67 views

Meaning of differentiability

Could anyone give an intuitive idea of the meaning of differentiability in general in any dimension and any space?
8
votes
1answer
253 views

Is the number 8 special in turning a sphere inside out?

So after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here? ...
3
votes
2answers
248 views

What is the difference between differential topology and calculus on manifolds?

I'm trying to teach myself one and bought a book on the other. It seems to me that they both cover about the same material. This leads to the question: What is the difference between differential ...
4
votes
0answers
221 views

Soft question: why are there non-smooth manifolds?

Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...
5
votes
3answers
257 views

Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?

I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology. I understand the ...
23
votes
3answers
1k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
6
votes
3answers
306 views

Why do people care about principal bundles?

I've started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought ...
2
votes
1answer
477 views

Prerequisite for Differential Topology and/or Geometric Topology

What are the prerequisites to learning both or one of the items? Consider that one will have done some of the "core" classes like Differential Geometry, Real Analysis, Abstract Algebra and POint-Set ...
2
votes
1answer
189 views

Why don't we have many differential topologist

I am interested in learning differential topology as Milnor, Guillemin, Pollack, Hirsch, Kosinski, etc.. did it. However, I am in University of Toronto as an undergraduate and none of colleges (or ...
1
vote
1answer
178 views

Taking Differential Topology concurrently with Analysis

So I'm trying to finalize my schedule for this semester. I can't decide whether I should enroll in a grad level Differential Topology (Milnor) class or just the undergrad general topology one. The ...
11
votes
5answers
1k views

Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
15
votes
2answers
536 views

Why study “curves” instead of 1-manifolds?

In most undergraduate differential geometry courses -- I am thinking of do Carmo's "Differential Geometry of Curves and Surfaces" -- the topic of study is curves and surfaces in $\mathbb{R}^3$. ...