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SUMMARY: Diophantine Geometry and the Theory of Computation (Part II) - Pr
 of Minhyong Kim\, International Centre for Mathematical Sciences\, Edinbur
 gh
DTSTART:20250523T130000Z
DTEND:20250523T143000Z
UID:TALK232681@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                         ( x/z )^n + ( y/z )^n = 1 \, \n\nwhe
 re it took over 350 years to be sure that the obvious solutions were the o
 nly ones. (Some equations in three variables can be effectively reduced to
  the two-variable case.) Writing down a random equation like \n           
 \n                y^3 = x^6 + 23x^5 + 37x^4 + 691x^3 − 631204x^2 + 51693
 73941 \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 Dioph
 antine equations has a curious tendency to touch on an incredibly broad ra
 nge of mathematics\, including algebraic/differential geometry\, algebraic
  topology\, global analysis\, and mathematical physics. \nEquations in two
  variables include elliptic curves (degree 3)\, out of whose study abstrac
 t and far-reaching frameworks for number theory and algebraic geometry hav
 e emerged\, such as a conjectural category of Grothendieck/Voevodsky motiv
 es\, with mysterious connections to automorphic forms and the Langlands pr
 ogramme. This represents a remarkable confluence of theory and computation
 : about 60 years ago\, as modern scientific computation was first making i
 ts way into mathematics\, Birch and Swinnerton-Dyer started examining elli
 ptic curves from an algorithmic viewpoint in the Mathematical Laboratory a
 t Cambridge and formulated their famous conjecture. The BSD conjecture is 
 a powerful tool for computational arithmetic geometry\, in that it can be 
 used to compute solution sets even if the conjecture itself is not known t
 o be true. \n\nThese lectures will review some of this history and the cur
 rent status of the theory with an algorithmic focus. Along the way\, if ti
 me allows\, some general reflections on the interaction of number theory a
 nd computation will be presented\, including undecidability\, complexity\,
  quantum computation\, and the universality of the theory of Diophantine e
 quations. One main goal will be to encourage interest in hyperbolic equati
 ons\, i.e.\, two-variable equations of degree larger than 3\, including th
 e so-called ‘non-abelian Chabauty theory’ of algorithmic resolution. W
 ith the advent of machine learning\, I believe the time is right for this 
 extension of the BSD philosophy to be brought to the attention of scientis
 ts with diverse backgrounds in the theory of computation. (In particular\,
  do not worry if some of the terms in this abstract are unfamiliar to you.
 ) \n\nVenue: The Computer Laboratory\, FW26\nhttps://cl-cam-ac-uk.zoom.us/
 j/6590822098?pwd=VTBuUXRXN29qMDF4TGpaaEhFaytQQT09\n\nOnline: Meeting ID: 6
 59 082 2098\nPasscode: 1dYRka\n\n
LOCATION:Computer Lab\, FW26 and Online (link in abstract)
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