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SUMMARY: Diophantine Equations and the Theory of Computation: A lecture se
 ries - Prof Minhyong Kim\, International Centre for Mathematical Sciences\
 , Edinburgh
DTSTART:20250521T130000Z
DTEND:20250521T143000Z
UID:TALK231100@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\nf(x_1\, x
 _2\, ... \, x_n) = 0\n\nwith integer coefficients whose solutions with rat
 ional or integer coordinates are the objects of interest. The computation 
 of such solution sets comprises some of the oldest problems in mathematics
 \, and any given case has a strong tendency to be surprisingly difficult e
 ven when approached with all the machinery that modern arithmetic geometry
  has at its disposal.\n\nThis series of lectures will focus on the computa
 tion of the full rational solution set for two-variable equations:\n\nf(x\
 , y) = 0.\n\nTo get 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 obvious solutions were the only ones. (Some 
 equations in three variables can be effectively reduced to the two-variabl
 e case.) Writing down a random equation like\n\ny^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 touc
 h on an incredibly broad range of mathematics\, including algebraic/differ
 ential geometry\, algebraic topology\, global analysis\, and mathematical 
 physics.\n\nEquations in two variables include elliptic curves (degree 3)\
 , out of whose study abstract and far-reaching frameworks for number theor
 y and algebraic geometry have emerged\, such as a conjectural category of 
 Grothendieck/Voevodsky motives\, with mysterious connections to automorphi
 c forms and the Langlands programme.\n\nThis represents a remarkable confl
 uence of theory and computation: about 60 years ago\, as modern scientific
  computation was first making its way into mathematics\, Birch and Swinner
 ton-Dyer started examining elliptic curves from an algorithmic viewpoint i
 n the Mathematical Laboratory at Cambridge and formulated their famous con
 jecture. The BSD conjecture is a powerful tool for computational arithmeti
 c geometry\, in that it can be used to compute solution sets even if the c
 onjecture itself is not known to be true.\n\nThese lectures will review so
 me of this history and the current status of the theory with an algorithmi
 c focus. Along the way\, if time allows\, some general reflections on the 
 interaction of number theory and computation will be presented\, including
  undecidability\, complexity\, quantum computation\, and the universality 
 of the theory of Diophantine equations.\n\nOne main goal will be to encour
 age interest in hyperbolic equations\, i.e.\, two-variable equations of de
 gree larger than 3\, including the so-called ‘non-abelian Chabauty theor
 y’ of algorithmic resolution. With the advent of machine learning\, I be
 lieve the time is right for this extension of the BSD philosophy to be bro
 ught to the attention of scientists with diverse backgrounds in the theory
  of computation. (In particular\, do not worry if some of the terms in thi
 s abstract are unfamiliar to you.)
LOCATION:The Computer Laboratory\, FW26 and Online
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