Seminar
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Location:  60 Evans Hall 
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Lindenstrauss 2915
A theorem of Gauss and Legendre states that every integer n that is not congruent to 0,4 or 7 mod 8 can be presented as a sum of three relatively prime squares. In fact, we know (ineffectively) that under these conditions there are n^{1/2+o(1)} such representations of n.
Rescaling the points to the unit sphere, we get as n increases more and more points on this sphere, and some 50 years ago Linnik, in a book with the intriguing name "Ergodic properties of number fields", used ergodic theory to study the distribution of these points, at least for sequences of n satisfying an auxiliary congruence condition  for instance if n is in addition assumed to be congruent to 2 (mod 3).
An alternative route, using bounds on Fourier coefficients of certain automorphic forms, was given by Duke and Iwaniec. This alternative route has several advantages, e.g. does not need a congruence condition and gives much more quantitative information. But the highly original work of Linnik on the subject is still very relevant, and had motivated much recent work.
I will explain why and how ergodic theory  more specifically, homogeneous dynamics  can be used to study integer points on the sphere, and time permitting present some recent work in this direction.
Lindenstrauss 2915
H.264 Video 
Lindenstrauss_2915_final.mp4

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