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SUMMARY:Rouse Ball Lecture - Professor Richard Taylor\, Institute of Advan
 ced Study
DTSTART:20140619T103000Z
DTEND:20140619T113000Z
UID:TALK48633@talks.cam.ac.uk
CONTACT:virginia mullins
DESCRIPTION:*Counting solutions to congruences: reciprocity laws and densi
 ty theorems.*\nReciprocity laws provide a rule to count the number of solu
 tions of a fixed polynomial equation modulo a variable prime number. The r
 ule will involve very different objects: automorphic forms and discrete su
 bgroups of Lie groups. The prototypical example is Gauss' law of quadratic
  reciprocity\, which concerns a quadratic equation in one variable. Anothe
 r celebrated example is the Shimura-Taniyama conjecture which concerns a c
 ubic equation in two variables. I will start with Gauss' law and work my w
 ay up to somewhat more complicated examples. At the end of the talk I hope
  to indicate the current state of our knowledge. 
LOCATION:Room 3\, Mill Lane Lecture Rooms\, 8 Mill Lane\, Cambridge
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