Department of

Mathematics


Seminar Calendar
for events the day of Thursday, May 7, 2015.

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Thursday, May 7, 2015

11:00 am in 241 Altgeld Hall,Thursday, May 7, 2015

The Apollonian structure of imaginary quadratic fields

Kate Stange (University of Colorado)

Abstract: Let $K$ be an imaginary quadratic field with ring of integers OK. The Schmidt arrangement of $K$ is the orbit of the extended real line in the extended complex plane under the Bianchi group $PSL(2,OK)$ (realised as Mobius transformations). The arrangement takes the form of a dense collection of intricately nested circles. I'll explain how the number theory of $K$ influences the arrangement, and I'll use these arrangements to generalise Apollonian circle packings and define a new collection of thin groups of arithmetic interest.

1:00 pm in Altgeld Hall 243,Thursday, May 7, 2015

Simultaneous dense and nondense orbits

Jimmy Tseng (University of Bristol)

Abstract: We consider pairs of maps on (usually) the same phase space and, in particular, examine pairs for which many points have drastically different orbit structures. Our main example is a pair of commuting automorphisms of the d-torus, for which the set of points with dense orbit under one map and nondense orbit under the other has full Hausdorff dimension. Two other examples that we only very briefly mention are two linearly independent elements of the Cartan action on compact higher rank homogeneous spaces and the multiplication-by-n map on the circle and the geodesic flow under the induced map on the circle corresponding to the expanding horospherical subgroup. The last result is an example for which the phase spaces are not the same (because the geodesic flow acts on the space of unimodular lattices) but, nevertheless, it allows us to obtain a counterpart to a classical result of R. Kaufman in Diophantine approximation. This talk is based on my joint work with V. Bergelson and M. Einsiedler and my other recent joint work with R. Shi.