SMRI Algebra and Geometry Online Seminar
Recent progress on the effective Mordell problem
Minhyong Kim (University of Warwick)
Monday, 07 December, 20:00–21:30 AEDT (Sydney)
Other time zones: Vancouver, Mon 01:00; Toronto, Mon 04:00; London, Mon 09:00; Cape Town, Mon 11:00; Mumbai, Mon 14:30; Beijing, Mon 17:00
In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of genus at least two have only finitely many rational points. This can be understood as the statement that most polynomial equations (in a precise sense)
f(x,y) = 0
of degree at least 4 have at most finitely many solutions. However, the effective version of this problem, that of constructing an algorithm for listing all rational solutions, is still unresolved. To get a sense of the difficulty, recall how long it took to prove that there are no solutions to
xⁿ + yⁿ = 1
other than the obvious ones. In this talk, I will survey some of the recent progress on an approach to this problem that proceeds by encoding rational solutions into arithmetic principal bundles and studying their moduli in a manner reminiscent of geometric gauge theory.
Please register to attend.
The seminar will be held online via Zoom. You will be sent a confirmation email with connection details.
This seminar will be recorded and made available on the SMRI Youtube channel.
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