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Results tagged with big-list
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user 30684
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
16
votes
A search for theorems which appear to have very few, if any hypotheses
Any linear bounded operator on a Hilbert space can be written as linear combination of four unitary operators.
3
votes
Where can square roots come from when they are not distances?
Polar decomposition of square matrix $A$ involves square-root of $A^*A$
6
votes
Open problems in mathematical physics
The following contains several open problems (as of 2001, but most are still open I believe) in topological fluid dynamics by Moffatt :
Some Remarks on Topological Fluid Mechanics
4
votes
Practical applications of Sobolev spaces
Formulating optimization/control problems in Sobolev spaces often lead to better numerically conditioned problems, and more practically implementable solutions.
E.g: Consider the problem of devising …
- 2,281
10
votes
What are some deep theorems, and why are they considered deep?
Kolmogorov-Arnold-Moser (KAM) theorem in the study of perturbed Hamiltonian systems. This landmark theorem clarified a major point in the field, and answered in positive the question of existence of s …
34
votes
Publishing conjectures
Depending upon the content of the paper, you may look at the new Arnold Mathematical Journal (http://www.springer.com/mathematics/journal/40598).
From their site, "Problems, objectives, work in progr …
20
votes
Mathematical conjectures on which applications depend
Navier Stokes equations are believed to be well-posed.
3
votes
Mathematicians with both “very abstract” and “very applied” achievements
Jerrold Marsden made major contribution to symplectic geometry but also was a key contributor to problems in celestial mechanics and numerical methods.
2
votes
Online high quality colloquium talks
If applied math is allowed, here's one about invariant manifolds and interplanetary superhighway (Restricted three body problem)
http://www.podcast.ethz.ch/episodes/?id=1269
8
votes
Proofs where higher dimension or cardinality actually enabled much simpler proof?
Not a single theorem per se, but in dynamical systems, it is often very useful to translate questions about properties of a continuous system $\dot{x}=f(x)$ or discrete-time system $x_{k+1}=f(x_k)$, w …
9
votes
Open problems in PDEs, dynamical systems, mathematical physics
Dynamical systems is a huge field, with at least 3 (or more) subdisciplines which often interact with each other, but also have self-contained advances. Ergodic theory, topological dynamical systems, …
4
votes
Most helpful math resources on the web
http://www.proofwiki.org
It is a Wikipedia, for proofs.
57
votes
Every mathematician has only a few tricks
Integration by parts has allegedly earned some people big medals.