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I usually have a more or less formed template of conversation to talk with people about mathematics, It's importance, methods, history, etc. I've been for some time interested in newer fields of mathematics that helped to solve or solved older problems of mathematics but my level of mathematical culture is still too low and I probably don't have good examples in mind.

Although the question may seem an enthusiastic atempt to spread the word of how mathematics is to the laymen, I'm also interested in examples that aren't so accessible for two reasons:

  1. I'm curious and perhaps I won't understand now but I have the future to get them.

  2. It could be useful/interesting for other members of the community that are way more advanced than the laymen.

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4 Answers

Probably the most famous (recent) example is the theory of elliptic curves which was used by Andrew Wiles to prove Fermat's Last Theorem, a conjecture about a particular class of diophantine equations. The problem was famously scrawled in the margin of Fermat's copy of Diophantus' Arithmetica with a claim that Fermat had found a proof. However, Fermat's claimed proof was never found and it took over $350$ years to find one using machinery that had not been invented at the time.

There is a lot to this story; I recommend the BBC documentary if you would like to know more

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  1. Fermat's Last Theorem exemplifying how the theory of elliptic curves solves a classical problem in number theory.

  2. The proof of the fundamental theorem of algebra using homotopy theory, showing how homotopy theory solves a problem in analysis.

  3. Robinson's formalism of nonstandard analysis, showing how logic settles a long standing debate about the existence of infinitesimals.

  4. Kolmogorov's axiomatization of probability theory, showing how measure theory is used to give a sound foundation for probability theory.

  5. Numerous results in group theory that are proved using Shelah's classification theory, showing how logic solves problems in algebra.

  6. Shannon's notion of entropy that created the whole field of information theory, shattering common beliefs in coding theory (e.g., Shannon's theorem on channel capacity).

I'm sure the list continues....

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Another problem was Seven Bridges of Königsberg was solved by Euler who invented graph theory to do so. This is still a 100-200 year gap –  Alizter Aug 4 '13 at 21:43
    
+ The proof of the upper bound conjecture using commutative algebra by Richard Stanley. –  some1.new4u Aug 4 '13 at 21:43
    
@Alizter You should post it as an answer. –  Igäria Mnagarka Aug 6 '13 at 5:35
    
@some1.new4u You should post it as an answer. –  Igäria Mnagarka Aug 6 '13 at 5:37
    
Can you be more specific about 5)? What is Shelah's classification theory and what specific results in group theory can be proven with it? –  Martin Brandenburg Aug 28 '13 at 12:15
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Distribution theory, invented about 1950, is now extremely important in the modern theory of linear partial differential equations, e.g. the Malgrange-Ehrenpreis theorem or regularity theorems for elliptic PDO's

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Galois theory

  1. proves the impossibility of classical construction problems with compass and straightedge. (squaring the circle, doubling the cube, angle trisection). Moreover it gives an exact criterion for the constructibility of regular polygons.

  2. characterizes the solvability of polynomial equations with radicals.

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protected by Alexander Gruber Aug 4 '13 at 23:01

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