Questions tagged [geometric-intuition]

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Topological intuition for the cancellation property of separated maps w.r.t a class of properties of continuous maps

Recall a continuous map is separated if its diagonal is closed. This is equivalent to the fibers being relatively Hausdorff in the total space. Proposition. Suppose $\mathrm P$ is a class of ...
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37 votes
3 answers
3k views

What is so geometric about symplectic geometry?

Symplectic geometry is often motivated by the Hamilton's equation which in turn are a reformulation of Newton's third law. But the subject itself is of independent mathematical interest. What I don't ...
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57 votes
9 answers
4k views

Examples of back of envelope calculations leading to good intuition?

Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from ...
3 votes
3 answers
232 views

Can you measure the degree of uniformity of a 2d shape?

Is there a calculation that could take the points that make of the outline of a 2 dimensional shape and provide a numeric evaluation representative of the uniformity or symmetry of the shape. Such as ...
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3 votes
1 answer
533 views

Geometric interpretation of sections $H^0(\Theta_X, X)$ of the Tangent sheaf over curve

I'm reading Mumford's & Oda's Algebraic Geometry II and I'm confused about explanations on geometric intuition of sections $H^0(\Theta_X, X)$ of the tangent sheaf on page 287: Let $X$ a ...
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1 vote
0 answers
40 views

Geometric interpretation for uniformly elliptic pde of 2 second order

Let $\Omega \subset \mathbb{R}^{2}$ a domain,let $u \in C^{2}(\Omega)$, the operator $Lu= tr(A.D^{2}u) + <\nabla u,b> +cu$ where $A$ is a symmetric matrix, $b$ is a vector field continuous ...
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7 votes
0 answers
1k views

Visualization of an algebraic stack

As the visuallization of an algebraic stack is virtually impossible I warn about this is a soft question. I am interested in thinking visually about algebraic stacks (also higher and derived stacks, ...
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1 vote
0 answers
117 views

Intuition for analysis of basic gradient descent variants

I'm currently learning the basic variants of gradient descent for minimizing convex functions under various assumptions, such as Lipschitz, smooth, strongly-convex, ... . I've found various sources ...
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  • 967
7 votes
0 answers
382 views

Geometric interpretation of minimal number of generators of a module

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, \...
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  • 223
2 votes
0 answers
201 views

Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question. I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will trivially ...
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  • 9,753
15 votes
1 answer
1k views

(really) basic intuition for $\mathbb A^1$-homotopy theory

Apologies in advance if this question is inappropriate for MO. I'm trying to read here and there about $\mathbb A^1$-homotopy theory in algebraic geometry. I understand some abstract machinery is ...
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17 votes
1 answer
823 views

Axiom of choice as zero dimensionality

In the paper Quantifiers and Sheaves by Lawvere, at the bottom of the second page, the author writes: "... the condition that every epi splits, which geometrically we would call 0-dimensionality ...
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7 votes
0 answers
291 views

Grothendieck - A group as a sheaf over simplicial complexes

In this blog post, Terence Tao gives the following definition of a group. Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
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  • 845
31 votes
6 answers
4k views

How to explain the concentration-of-measure phenomenon intuitively?

One way to phrase the "concentration-of-measure" phenomenon is that, for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$, "most of the mass is close to the equator, for any equator."1 Q. ...
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6 votes
1 answer
249 views

Geometric Intuition of $P^+$ in Modular Tensor Categories

I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...
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46 votes
2 answers
6k 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, (...
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4 votes
0 answers
337 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 ...
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17 votes
1 answer
3k views

Geometric interpretations of matrix inverses

$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (...
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  • 673
6 votes
3 answers
8k views

Proof without words for surface area of a sphere [closed]

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...
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68 votes
5 answers
7k 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 main ...
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22 votes
2 answers
10k 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 $\Omega_i^...
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35 votes
5 answers
4k 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 ...
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6 votes
1 answer
1k 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 ...
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5 votes
1 answer
725 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 ...
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2 votes
0 answers
3k views

What is the geometric meaning of the third derivative of a function at a point? [closed]

What is the geometric meaning of the third derivative of a function at a point? This question is now asked on the sister site: https://math.stackexchange.com/questions/14841/what-is-the-meaning-of-...
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65 votes
11 answers
10k 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" (sequences/...
0 votes
1 answer
392 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. ...
20 votes
2 answers
2k 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 ...
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12 votes
2 answers
992 views

Geometrically interpreting the answer to a vector calculus question involving tangent line segments to ellipses.

Let E be an ellipse centered at the origin on the x, y plane with major radius b and minor radius a. The length of the shortest line segment tangent to E that begins on the x-axis and ends on the y-...
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80 votes
12 answers
77k views

Why is the gradient normal?

This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I ...
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  • 3,433
14 votes
7 answers
4k 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 ...
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