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1
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0answers
64 views

Geometric interpretation of Euler's identity for homogeneous functions [closed]

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is called homogeneous of degree $d \geq 0$ if $$f(\lambda x_1, \ldots, \lambda x_n ) = \lambda^d f(x_1, \ldots, x_n)$$ Differentiating both sides ...
25
votes
2answers
1k views

Geometric interpretation of the half-derivative?

For $f(x)=x$, the half-derivative of $f$ is $$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x = 2 \sqrt{\frac{x}{\pi}} \;.$$ Is there some geometric interpretation of (Q1) this specific derivative, and, ...
4
votes
0answers
178 views

Is there a picture I should have in my head of rational homotopy equivalence?

My understanding is that one thinks to rational homotopy theory for computational advantage. However, thinking about things in terms of localizations still lacks some amount of intuition for me. In ...
44
votes
5answers
3k views

Is there an intuitive reason for Zariski's main theorem?

Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version: Zariski's ...
7
votes
2answers
2k views

Geometric interpretation of Cartan's structure equations

Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^*M$ we can define the connection 1-forms $\omega^j_i$. We define the curvature 2-forms by ...
15
votes
5answers
1k views

What is the general geometric interpretation of modules in algebraic geometry?

Algebraic geometry is quite new for me, so this question may be too naive. therefore, I will also be happy to get answers explaining why this is a bad question. I understand that the basic philosophy ...
5
votes
1answer
762 views

Geometric meaning of torsion in homotopy groups

It is not too hard to understand the geometric meaning of torsion in homology groups of CW complexes. However, I thought it would be interesting to hear how people describe/think of the geometric ...
4
votes
1answer
433 views

Characterization of algebraic points on Shimura varieties?

Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points on Shimura varieties? The question of course does not always make sense for ${\bf{Q}}$-points: a theorem of Shimura shows ...
48
votes
11answers
3k views

How should one think about non-Hausdorff topologies?

In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" ...
0
votes
1answer
371 views

Piece of a sequence

Suppose we are given a representation of a finite series of natural numbers: $\sum_{i=0}^N{c_i x^i}$ The representation is essentially an expression that is a rational function of two polynomials. ...
19
votes
2answers
1k views

Geometric interpretation of group rings?

For a group $G$, is there an interpretation of $\mathbb C[G]$ as functions over some noncommutative space? If so, what does this space "look like"? What are its properties? How are they related to ...
19
votes
11answers
13k views

Why is the gradient normal?

This is a somewhat long discussion so bear with me. There is a theorem that I have always been curious about from an intuitive standpoint. This is an issue that has been glossed over in most ...
8
votes
7answers
1k views

Morphisms of (quasi-)projective varieties

This is another "homework help" question, which is still hopefully of at least pedagogical interest to working mathematicians. So, I'm currently taking an intro algebraic geometry class, and one ...