A good way for motivating young students in undergraduate level is telling them that you can do great works!
Question. What are good examples of research level mathematical discoveries done by undergraduate students?
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3$\begingroup$ For the case of high school students, I already asked this question in Mathematics Stack Exchange (math.stackexchange.com/q/174009/18398). $\endgroup$– JRNCommented Mar 25, 2014 at 6:34
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$\begingroup$ @JoelReyesNoche Ah! Interesting. Thanks for the link. I will edit the question. $\endgroup$– user230Commented Mar 25, 2014 at 6:43
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2$\begingroup$ Well, a 22 year old Ike Newton discovered the generalized binomial theorem and a good chunk of what would become calculus in his final year. Though I suppose this example could be more intimidating than inspiring. Use with caution. ;) $\endgroup$– David HCommented Mar 25, 2014 at 11:20
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1$\begingroup$ One example that sticks in my mind is the proof of the double bubble conjecture in two dimensions by a group of undergraduates. $\endgroup$– Jim BelkCommented Mar 25, 2014 at 14:33
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1$\begingroup$ Thank you. I would have edited it myself, but was not completely sure which interpretation you meant. $\endgroup$– zyxCommented Mar 25, 2014 at 17:35
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4 Answers
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You can point them to Involve, a journal all of whose articles involve substantial contributions by undergraduates. [Disclaimer: I am on its editorial board]
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2$\begingroup$ Very interesting! I am happy to have heard about this. $\endgroup$ Commented Mar 25, 2014 at 15:09
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An example from 2002 is the polynomial-time algorithm to decide primality. Neeraj Kayal and Nitin Saxena were two of the three authors, and had just gotten their undergraduate degrees that year.
An example from 1956 is the construction of a set of intermediate degree, by Richard Friedberg in his senior thesis at Harvard, and also by a not-much-older Albert Mucnik in Russia. This is a set which is neither recursive nor strong enough to allow the halting problem to be computed recursively from it.
These were both influential results in logic.
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One certain source can be found by looking through winners of the Morgan Prize, which is awarded for mathematical research by undergraduate students. Besides the wikipage linked above, there is also an AMS page and MAA page.
Although you can find some very "strong" mathematics and mathematicians among award recipients, I feel it is worth noting which institutions show up repeatedly (listed alongside winners and runners-up on the wikipage). If the goal is "motivating young students" (per the OP) then it might be worth stepping back to wonder why the same names of certain universities keep showing up; I imagine it could be de-motivating for students at other schools to see this list and perhaps, mistakenly, believe that quality undergraduate research is outside of their own reach by virtue of the university that they attend.
answered Jun 2, 2018 at 17:31
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This story is very motivating. It is not direct answer to your specification of UG research (was a grad student). But it meets the spirit. And serves the motivation rationale:
https://www.snopes.com/fact-check/the-unsolvable-math-problem/
