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SUMMARY: Diophantine Equations and the Theory of Computation (Part I) - Pr
 of Minhyong Kim\, International Centre for Mathematical Sciences\, Edinbur
 gh
DTSTART:20250521T130000Z
DTEND:20250521T143000Z
UID:TALK231103@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\nf(x_1\, x_2
 \, ... \, x_n) = 0\nwith integer coefficients whose solutions with rationa
 l or integer coordinates are the objects of interest. The computation of s
 uch solution sets comprises some of the oldest problems in mathematics\, a
 nd any given case has a strong tendency to be surprisingly difficult even 
 when approached with all the machinery that modern arithmetic geometry has
  at its disposal.\n\nThis series of lectures will focus on the computation
  of the full rational solution set for two-variable equations: f(x\, y) = 
 0.\n\nTo get a sense of why this might be difficult\, consider the case of
  Fermat’s equation (x/z)^n+(y/z)^n =1\n\nwhere it took over 350 years to
  be sure that the obvious solutions were the only ones. (Some equations in
  three variables can be effectively reduced to the two-variable case.) Wri
 ting down a random equation like y^3  =  x^6 + 23 x^5 + 37 x^4 + 691 x^3 -
  631204 x^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 on an
  incredibly broad range of mathematics\, including algebraic/differential 
 geometry\, algebraic topology\, global analysis\, and mathematical physics
 .\n\nEquations in two variables include elliptic curves (degree 3)\, out o
 f whose study abstract and far-reaching frameworks for number theory and a
 lgebraic geometry have emerged\, such as a conjectural category of Grothen
 dieck/Voevodsky motives\, with mysterious connections to automorphic forms
  and the Langlands programme.\n\nThis 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 t
 his history and the current status of the theory with an algorithmic focus
 . Along the way\, if time allows\, some general reflections on the interac
 tion of number theory and computation will be presented\, including undeci
 dability\, complexity\, quantum computation\, and the universality of the 
 theory of Diophantine equations.\n\nOne main goal will be to encourage int
 erest in hyperbolic equations\, i.e.\, two-variable equations of degree la
 rger than 3\, including the so-called ‘non-abelian Chabauty theory’ of
  algorithmic resolution. With the advent of machine learning\, I believe t
 he time is right for this extension of the BSD philosophy to be brought to
  the attention of scientists with diverse backgrounds in the theory of com
 putation. (In particular\, do not worry if some of the terms in this abstr
 act are unfamiliar to you.)
LOCATION:The Computer Laboratory\, FW26 and Online
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