PIMS Network Wide Colloquium: Jordan Ellenberg

The array

3     -3    4     -4    5     -5    0     0

4     -4    -3    3     0     0     5     -5

-5    5     0     0     -3    3     -4    4

has an interesting feature: its columns are each vectors satisfying the linear relation 3x + 4y + 5z = 0, and its rows are permutations of each other. Are there arrays of integers like this for other simple linear relations? This question was raised by Chris Smyth in 1986 in connection with an audacious conjecture in algebraic number theory about linear relations between Galois conjugates. We explain how to prove Smyth's conjecture, and what it has to do with eigenvalues of sums of permutation matrices, weightings on hypergraphs, and (depending on time and audience indulgence) a local-to-global principle for equations in random variables. 

Joint work with Will Hardt.