Linked Questions
11 questions linked to/from Results true in a dimension and false for higher dimensions
2
votes
1
answer
130
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Difference in essential spectrum between Schrodinger operators
I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
189
votes
79
answers
42k
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Which math paper maximizes the ratio (importance)/(length)?
My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...
76
votes
34
answers
7k
views
Dimension leaps
Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" ...
63
votes
52
answers
23k
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Colloquial catchy statements encoding serious mathematics
As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...
5
votes
5
answers
269
views
Peculiarities in low dimensions or low order or etc
I have been pondering about certain conjectures and theorems viewed as either low vs high dimensions, or smaller vs larger primes, or anything of the sort "low vs high order". Let me mention a couple ...
0
votes
0
answers
145
views
Stability of Schrodinger operators on bounded domains
Consider a Schrodinger operator $H=H_0+V$, where $H_0$ is a $L_2$ realization of the negative Laplace operator $-\Delta$ with homogeneous Neumann boundary condition on a bounded, smooth domain $\Omega\...
9
votes
3
answers
707
views
When are $\mathbb{R}^n$ and $\mathbb{R}^m$ essentially similar?
Here is a rather vague and subjective question: for which $n$ and $m$ are
$\mathbb{R}^n$ and $\mathbb{R}^m$ ``essentially similar''? The answer depends partly on what type of
mathematician is ...
3
votes
1
answer
199
views
Bounded solutions for Schrödinger equation at the edge of the essential spectrum
Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
18
votes
1
answer
654
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Higher dimensional generalization of: Any quadrilateral tiles the plane?
Any (non-self-intersecting) quadrilateral tiles the plane.
(MathWorld image.)
Q. What is the strongest known generalization of this statement to higher dimensions?
I.e., $\mathbb{R}^d$ ...
4
votes
1
answer
722
views
binary intersection property in finite dimension
I have recently discovered a survey article called "The Hahn-Banach Theorem: The Life and Times" by Narici and Beckenstein and I have read it with great amusement:
I was particularly fascinated by ...
14
votes
2
answers
1k
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Exotic spheres and stable homotopy in all large dimensions?
Given that the kervaire invariant one problem has been solved in (almost) all dimensions....my question is whether there exists an exotic sphere in all sufficently lagre dimensions? Given the Kervaire-...