Speaker: Jiyoung Han(Tata Institute)
3 December, 2021 14:30 UTC
Abstract: The Siegel transform is one of the main tools when we consider problems related to counting lattice points using homogeneous dynamics. It is revealed by many mathematicians that Siegel’s integral formula and Rogers’ second moment formula are very useful to solve various quantitative and effective variants of classical problems in the geometry of numbers, such as the Gauss circle problem (generalized to convex sets) and Oppenheim conjecture. Furthermore, Rogers’ higher moment formulas, together with the method of moments, give us information about limit distributions related to these problems. In this talk, we revisit Rogers’ higher moment formulas with a new approach, and introduce higher moment formulas for Siegel transforms on the space of affine unimodular lattices and the space of unimodular lattices with a congruence condition. Using these formulas, we obtain the results of limit distributions, which are generalizations of the work of Rogers (1956), Södergren (2011), and Strömbergsson and Södergren (2019). This is joint work with Mahbub Alam and Anish Ghosh.