Taxicabs, cubic curves, and faking irrationality -- Florian (Ian) Sprung, March 1, 2005

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This talk will mainly focus on cubic curves, i.e. solutions to a 
polynomial f(x,y) of degree three. A remarkable theorem by Siegel 
states that there can only be finitely many points (x,y) on these 
curves where both x and y are integers. The known proofs are a little 
difficult, so we won't bother. But we can consider easier cases of this 
- e.g. by Thue. At some point, we will approximate ("fake") an 
irrational number by rational ones, and furthermore see that there are 
only finitely many rationals that fake it well. This will lead us to a 
brief sketch of a theorem from "diophantine approximation theory", and 
how it was improved for 100 years, before being perfected by Roth (by 
very elementary methods, and he won a fields medal for it!). Some crazy 
results at the end, as usual.

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