I am looking for examples of scientific breakthroughs that have been made within the confines of a prison cell?
The French mathematician André Weil was imprisoned for failure to report for duty for a spell in 1940. It is said that "While in jail for six months at Rouen, he proved the Riemann hypothesis for curves over finite fields." Nowlan. He is said to have sent a 14 page letter to his sister from prison with some groundbreaking ideas on "Analogy in Mathematics". That letter is reprinted in the AMS article A 1940 Letter of André Weil on Analogy in Mathematics (trans. Krieger) with some introduction. Some more discussion on the contents and importance of this letter are here. Sources and stories abound online, and they largely seem to corroborate.
That seems to be a real solid example. I found several lesser examples of great scientific works, or accounts of Mathematicians and Scientists either writing something great or continuing with some great work. While these may not fit the request as straightforward as the Weil story, I believe they are worth noting nonetheless:
It is said about Galois that "During his imprisonment, he continued developing his mathematical ideas." Galois.
Galileo wrote what is said to be his greatest work Two New Sciences Galileo while in prison.
Bertrand Russell wrote Introduction to Mathematical Philosophy while in prison Russell.
Jean Leray is said to have done seminal work on topology while in prison.
Andre Weil's work on zeta functions of algebraic curves over finite fileds.
Andre Weil was one of the most famous mathematicians of the last century. He also was a pacifist, and when WWII approached, he left France to visit....Soviet Union:-) After Soviet Union, he visited Finland... just few days before Soviet Union attacked it. He was arrested by the Finns, as a Soviet spy with the nickname "Bourbaki". It did not help his case that he had a full suitcase of papers signed "Nicolas Bourbaki", and written in some misterious code. In view of the quick Russian offense they decided to execute him by shooting...
Fortunately for him, the great Finnish mathematician Nevanlinna (who was socializing with the highest Finnish officials) met the security chief on a party. The security chief told Nevanlinna that they caught a Russian spy whose name is Andre Weil, and this guy says that he knows Professor Nevanlinna personally! Nevanlinna confirmed that Weil is a harmless mathematician. And recommended to deport him to France.
So Weil was deported to his native country and put to prison as a deserter.... He spent his time in prison until the Germans defeated France and released him (as a French deserter). Probably they did not notice that he also was a Jew.
Anyway, after many adventures, Weil made his way to the US. In his memoirs he says that the work that he did in the French prison was his best. And that at a later age he even considered, instead of writing a grant applicaton, to ask to put him to a prison... perhaps he would do a work of similar quality:-)
All this is from Weil's memoirs "The apprenticeship of a mathematician", Translated from the 1991 French original by Jennifer Gage. Birkhäuser Verlag, Basel, 1992.
The work he did in prison was truly revolutionary.
Another famous and really unique case was Andre Bloch, another great French mathematician of 20-s century. He was drafted to WW I when he was a student (or maybe just graduated). He was severely wounded at his head (he fell from an artillery observation tower when it was hit). After a hospital and a honorable discharge he lived with his relatives. One morning he killed all of them during the breakfast (his uncle, aunt and several more relatives). It was decided that a war hero should not stand a trial during the war and he was confined in a mental asylum for the rest of his life. He died in his asylum in 1948.
During his stay in the asylum he worked and published many papers which revolutionized the subject of Complex Analysis. He is still remembered for the "Bloch's Principle", "Bloch Constant," "Bloch functions" and "Bloch's conjecture" (see Wikipedia).
I'm going to interpret "jail" and "prison" as including house arrest and concentration camps.
was kept under house arrest [by Fatimid Caliph al-Hakim] from 1011 until al-Hakim's death in 1021. During this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy.
After the Russian Revolutions of 1917, Trachtenberg fled to Germany where he became critical of Nazi policies. He was imprisoned in a Nazi concentration camp during World War II. He developed his system of mental arithmetic during his imprisonment.
Jean-Victor Poncelet was a prisoner of war in Russia in the years 1812–1814, and wrote Traité des propriétés projectives des figures there. This was an important work in projective geometry. More info on http://en.wikipedia.org/wiki/Jean-Victor_Poncelet