Questions tagged [geometric-intuition]

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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, ...
Martin Hurtado's user avatar
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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, \...
charlie katerba's user avatar
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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 ...
Exterior's user avatar
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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 ...
Joseph Victor's user avatar
2 votes
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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 ...
Arrow's user avatar
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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 ...
Amrit Awasthi's user avatar
1 vote
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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 ...
Arrow's user avatar
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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 ...
Cézar Bezerr's user avatar
1 vote
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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 ...
amakelov's user avatar
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