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        "combined_title":"GEOMETRY SEMINAR:  Roots of cubic congruences and dynamics on SL(3,R): Matthew Welsh (U Maryland)",
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        "description":"Roots \u00b5 mod m  of polynomial congruences F(X) \u2243 0 mod m are fundamental objects in number theory.\r\nStatistical information about the normalized roots  \u00b5(m) \u2208 R \/ Z for m up to large x is used, for example, to prove that n^2 + 1 is infinitely often a product of at most two primes.\r\nOriginally, this statistical information for quadratic roots was obtained from algebraic geometry (the Weil bound) and later from the spectral theory of automorphic forms.\r\nRecently in joint work with Jens Marklof and Zonglin Li, we studied quadratic roots using techniques from homogeneous dynamics on SL(2, R) that were similar to techniques developed to study Farey fractions.\r\nIn this talk, I present work in progress on extending these techniques to roots of cubic congruences and  SL(3,R)..",
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