# Questions tagged [geometric-intuition]

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### Identity relating floor function and hexagonal numbers

While playing around with squares, I wondered about the sum of square roots of all natural numbers between two perfect squares(both inclusive). After taking the floor value of the expression for first ...
1k views

### Tensor product of vector bundles

The Whitney sum (where fibre dimensions add) of two real, or two complex, vector bundles $\pi : E \to X$ and $\pi' : E' \to X$ over a topological space $X$ is not hard to get an intuitive grasp of. ...
258 views

### How does Zariski's main theorem factor $A[x]\to A[x]/\langle x(x^2-1)\rangle$?

Over the reals, consider the projection from the graph of $x\mapsto x(x^2-1)$ to the $y$-axis. This is a bundle over the real line whose total space looks an elongated reverse 'S': over $(-1,1)$ in ...
3k views

### How can Machine Learning help “see” in higher dimensions?

The news that DeepMind had helped mathematicians in research (one in representation theory, and one in knot theory) certainly got many thinking, what other projects could AI help us with? See MO ...
1 vote
57 views

### 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 ...
4k 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 ...
5k 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 ...
387 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 ...
761 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 ...
1 vote
44 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 ...
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, ...
1 vote
127 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 ...
401 views

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### 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 ...
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 ...
774 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 ...
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-...
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/...
394 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. ...
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 ...
1k 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-...