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SUMMARY: Diophantine Geometry and the Theory of Computation (Part III) - P
 rof Minhyong Kim\, International Centre for Mathematical Sciences\, Edinbu
 rgh
DTSTART:20250526T130000Z
DTEND:20250526T143000Z
UID:TALK232684@talks.cam.ac.uk
CONTACT:Challenger Mishra
DESCRIPTION:A Diophantine equation\, named after the Egyptian mathematicia
 n Diophantus of Alexandria (c. 2C)\, is a polynomial equation \n\n        
                  f(x_1\,x_2\,...\,x_n) = 0 \n\nwith integer coefficients w
 hose solutions with rational or integer coordinates are the objects of int
 erest. The computation of such solution sets comprises some of the oldest 
 problems in mathematics\, and any given case has a strong tendency to be s
 urprisingly difficult even when approached with all the machinery that mod
 ern arithmetic geometry has at its disposal. \nThis series of lectures wil
 l focus on the computation of the full rational solution set for two-varia
 ble equations: \n\n                                f(x\,y) = 0.
\n\nTo g
 et a sense of why this might be difficult\, consider the case of Fermat’
 s equation \n                         \n                       ( x/z )^n +
  ( y/z )^n = 1\, \n\nwhere it took over 350 years to be sure that the obvi
 ous solutions were the only ones. (Some equations in three variables can b
 e effectively reduced to the two-variable case.) Writing down a random equ
 ation like \n           \n                y^3 = x^6 + 23x^5 + 37x^4 + 691x
 ^3 − 631204x^2 + 5169373941 \n\nand noting that it has the solution (1\,
  1729)\, it is very hard to know if it has any other solutions. \n\nAs of 
 now\, the study of Diophantine equations has a curious tendency to touch o
 n an incredibly broad range of mathematics\, including algebraic/different
 ial geometry\, algebraic topology\, global analysis\, and mathematical phy
 sics. \nEquations in two variables include elliptic curves (degree 3)\, ou
 t of whose study abstract and far-reaching frameworks for number theory an
 d algebraic geometry have emerged\, such as a conjectural category of Grot
 hendieck/Voevodsky motives\, with mysterious connections to automorphic fo
 rms and the Langlands programme. This represents a remarkable confluence o
 f theory and computation: about 60 years ago\, as modern scientific comput
 ation was first making its way into mathematics\, Birch and Swinnerton-Dye
 r started examining elliptic curves from an algorithmic viewpoint in the M
 athematical Laboratory at Cambridge and formulated their famous conjecture
 . The BSD conjecture is a powerful tool for computational arithmetic geome
 try\, in that it can be used to compute solution sets even if the conjectu
 re itself is not known to be true. \n\nThese lectures will review some of 
 this history and the current status of the theory with an algorithmic focu
 s. Along the way\, if time allows\, some general reflections on the intera
 ction of number theory and computation will be presented\, including undec
 idability\, complexity\, quantum computation\, and the universality of the
  theory of Diophantine equations. One main goal will be to encourage inter
 est in hyperbolic equations\, i.e.\, two-variable equations of degree larg
 er than 3\, including the so-called ‘non-abelian Chabauty theory’ of a
 lgorithmic resolution. With the advent of machine learning\, I believe the
  time is right for this extension of the BSD philosophy to be brought to t
 he attention of scientists with diverse backgrounds in the theory of compu
 tation. (In particular\, do not worry if some of the terms in this abstrac
 t are unfamiliar to you.) \n\nVenue: The Computer Laboratory\, FW26 https:
 //cl-cam-ac-uk.zoom.us/j/6590822098?pwd=VTBuUXRXN29qMDF4TGpaaEhFaytQQT09\n
 \nOnline: Meeting ID: 659 082 2098 Passcode: 1dYRka
LOCATION:Computer Lab\, FW26 and Online (link in abstract)
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