SMRI Algebra and Geometry Online: Kim -- Recent progress on the effective Mordell problem

SMRI Algebra and Geometry Online 

’Recent progress on the effective Mordell problem’
Minhyong Kim (University of Warwick) 

Monday 7th December 8:00pm-9:30pm (AEDT) 

Online via Zoom 

Abstract: 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 

x^n+y^n=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.  

Register:
https://uni-sydney.zoom.us/meeting/register/tZwucOCtrj8oE9fa-wb15lPFkt8ENLNFwFgk 


The speakers blog is here: https://minhyongkimpublic.wordpress.com

Note: These seminars will be recorded (participants only when asking questions), and
uploaded to the SMRI YouTube Channel
https://www.youtube.com/channel/UCNrre_3ROSz7FqOka3lSMVA