# Questions tagged [geometric-intuition]

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3answers
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### 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 ...
9answers
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 ...
3answers
152 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 ...
1answer
369 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 ...
0answers
38 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 ...
0answers
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, ...
0answers
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 ...
0answers
368 views

5answers
3k 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 ...
1answer
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 ...
1answer
653 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 ...
0answers
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-...
11answers
9k 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/...
1answer
386 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. ...
2answers
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 ...
2answers
983 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-...
12answers
66k 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 ...
7answers
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 ...