Joe Shipman recently posted an interesting e-mail on the Foundations of Math e-mail list:

I propose the thesis “any mathematics result more than a century old is suitable for undergraduate math majors”.

Note that the original proofs may be too difficult for undergraduates, I am only requiring that today a “boiled-down” proof (which may be embedded in a much larger theory than existed at the time of the original proof) could be taught.

So far I have only found one significant counterexample, Dirichlet’s theorem (which, in its logically simplest form, states that if a is prime to b, there exists a prime congruent to a mod b).

Can anyone think of better counterexamples? Does anyone know of a proof of Dirichlet’s theorem that does not require prerequisites beyond the standard undergraduate curriculum?

(Two other possible counterexamples, the Prime Number Theorem and the Transcendence of Pi, are proven sufficiently easily at the following links that they would, in my opinion, be appropriate for a senior seminar:

Click to access Newman.pdf

Click to access pitrans.pdf

).

Another version of the thesis is “any mathematics result more than 200 years old is suitable for freshmen” (note that most high schools offer a full year of Calculus). Results that were merely conjectured more than 200 years ago but not really proved until later don’t count.

— JS

I’ve sometimes considered something like this. Can anyone else think of potential counterexamples? I wonder if there were some results known on solutions of differential equations in the 18th century that would be too advanced for first-years. And probably some particular calculations done in the 19th century that are just too large for an undergraduate to properly manage. I think it’s also possible that some of Cantor’s results on the possible Borel structures of the sets of discontinuities of real-valued functions might be too advanced, but it’s also possible that advanced seniors can manage them. Or perhaps the Riemann-Roch theorem? (I don’t actually even know enough to state that theorem myself.)

Another interesting corollary to this discussion – what’s the earliest result of the 20th century that is beyond the reach of an advanced undergraduate?

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Ars Mathematica » Blog Archive » Short Proof of Prime Number Theorem(21:49:25) :[…] Via Antimeta. […]

Noah Snyder(17:26:31) :I am highly skeptical that the prime number theorem is accesible when Dirichlet’s theorem on primes in arithmetic progressions is not. The level of difficulty is essentially the same, and every undergrad class that I’ve had or taught that covered one also covered the other.

Kenny(00:34:49) :Timothy Chow just made that same point on the FOM e-mail list (and also brought up Riemann-Roch). He also made a further comment that I think brings up a lot of the real issues motivating this discussion. He brings up some good examples of how some areas of math just aren’t considered of much interest any more, as well as some other points of how the history of math doesn’t just lead nicely to the present situation as we might like.

Phil(13:18:40) :If she is willing to sacrifice some breadth, I guess any advanced, motivated undergraduate can at least “learn about” pretty much any subject. Of course, it is arguable whether knowing a specific subject and ignoring it’s connections and applications to other fields has any use. For what it’s worth, one of my friends (a junior undergrad) spent the summer studying Floer homology. He gave talks about it at a PhD seminar, and everyone was impressed. On the other hand, his knowledge of analysis outside of what is strictly required for what he does is quite limited for an “advanced undergraduate”.