Colloquium
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Posted October 21, 2019

3:30 pm - 4:20 pm TBD
Nathan Glatt-Holtz, Tulane University

TBD

Colloquium
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Posted September 24, 2019

Last modified November 2, 2019

John Voight, Dartmouth College

Heuristics for units in number rings

Units in number rings are gems of arithmetic, the most famous being the golden ratio and the integer solutions x,y to Pell's equation x^2 - D*y^2 = +/-1 for D > 0. Like gems, they are embedded deeply within. Refined questions about the structure of units remain difficult to answer, for example: how often does it happen that Pell's equation has a solution to the -1 equation? More generally, how often in a number ring is it that all totally positive units are squares? Absent theorems, we may still try to predict the answer to these questions. In this talk, we present heuristics (and some theorems!) for signatures of unit groups inspired by the Cohen-Lenstra heuristics for class groups, but involving an lustrous structure of number rings we call the 2-Selmer signature map. This is joint work with David S. Dummit and Richard Foote and with Ben Breen, Noam Elkies, and Ila Varma.

Colloquium
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Posted September 10, 2019

3:30 pm - 4:20 pm TBD
Leonid Berlyand, Department of Mathematics, Penn State University

TBD

Colloquium
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Posted September 13, 2019

3:30 pm - 4:20 pm TBD
Eric Rowell, Texas A&M

TBD