### Symplectic geometry of numbers

Ergodic Theory and Dynamical Systems Seminar

17th May 2018, 3:00 pm – 4:00 pm

Howard House, 4th Floor Seminar Room

Lattice point counting has a long history and deep connections to number theory and geometry. The set of all Euclidean lattices of a given co-volume forms a homogeneous probability space and classical results of Siegel and Rogers tell us about the moments of its lattice point counting function. In this talk we will introduce a new Rogers type formula due to the author and Jayadev Athreya for the space of symplectic lattices; this formula expresses the second moment of the lattice point counting function in terms of geometric and arithmetic data. Finally we will apply this formula to bound the discrepancy of random symplectic lattices in families of Borel sets.

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