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Results tagged with intuition
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user 391
Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a rough description of ones understanding of the subject at hand (on a technical level).
32
votes
Accepted
Why do Littlewood-Richardson coefficients describe the cohomology of the Grassmannian?
There are several rings-with-bases to get straight here. I'll explain that, then describe three serious connections (not just Ehresmann's Lesieur's proof as recounted in the OP).
The wrong one is $Rep …
- 35.5k
152
votes
Geometric interpretation of trace
Let's use $\det(\exp(tA)) = 1 + t\operatorname{Tr}(A) + O(t^2)$, and think about the vector ODE $\vec y' = A \vec y$, solved by $\vec y(t) = \exp(tA) \vec y(0)$. If we take a unit parallelepiped worth …
- 1,311
11
votes
What is a complex inner product space "really"?
This is a purely algebraic answer, so maybe not what you're looking for.
What would an inner product on a left $A$-module $M$ over a noncommutative algebra $A$ be? It should give a map $M \to M^*$ wh …
- 27.9k
4
votes
What is a Lagrangian submanifold intuitively?
Since you've already gotten lots of classical mechanical answers, I'll give my favorite source of Lagrangians. Let $\sigma$ be a holomorphic section of a Hermitian line bundle $\mathcal L$ with curvat …
- 27.9k
3
votes
Intuition/Heuristic behind I/I^2 definition of Kähler differentials
Start with $Spec(A)$ being a vector space $V$, and let $V'$ be the subspace $V\times 0$ complementary to the diagonal $V_\Delta$. Then the ideal $I$ is functions on $V\times V$ vanishing on the diagon …
- 27.9k
17
votes
Understanding moment maps and Lie brackets
I believe the following way (Kostant's, 1970) to be the best way to think about the Hamiltonian condition.
First, "why" is there a central extension $H^0(M; {\mathbb R}) \to C^\infty (M) \to symp(M)$ …
- 27.9k