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Results tagged with big-list
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user 3709
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.
11
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
Sard's theorem provides such an example. Given a random smooth map between two manifolds (lets say compact and of the same dimension), there is no canonical way of constructing a regular value. But, S …
7
votes
What is your favorite isomorphism?
The De Rham Isomorphism.
6
votes
Where is number theory used in the rest of mathematics?
If Arakelov geometry counts as number theory, then, http://arxiv.org/pdf/math/0401029v1.pdf demonstrates the computation of the Analytic torsion (a purely analytic object involving the product of dete …
5
votes
What's your favorite equation, formula, identity or inequality?
$(A-\lambda _1) (A-\lambda _2) \ldots = 0$, the Cayley-Hamilton theorem.
5
votes
Why were matrix determinants once such a big deal?
1) The Chern-Weil theory of characteristic classes is built upon determinants of functions of curvature forms of vector bundles.
2) Feynman path integrals require determinants (but typically in infini …
5
votes
Proofs that require fundamentally new ways of thinking
Proving that subgroups of free groups are free requires the knowledge of topology, a completely different field which a priori does not have anything to do with groups.
4
votes
Tweetable Mathematics
Every non-constant complex polynomial has a complex root : If not the inverse is bounded analytic. Use Liouville. #FundamentalTheoremOfAlgebra
3
votes
Why do we need random variables?
Practically everything we measure in real life (for instance the time taken for an apple to fall on Newton's head) is "random" in the sense that if we perform the experiment again, we will not get the …
- 3,383
3
votes
What should be learned in a first serious schemes course?
Being a differential geometer, it might be nice if you can point out analogies (perhaps even make them rigorous ?) to differential geometry. Like a scheme being flat over another is perhaps akin to a …
3
votes
A good book of functional analysis
I second Reed and Simon's methods of mathematical physics. However, if you are interested primarily in the applications of functional analysis to PDE, for the most part a couple of appendices of Evans …
1
vote
Tweetable Mathematics
There is no smooth surjection from $S^5$ to $S^6$. #Sard