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Department of Mathematical Sciences, University of Liverpool
Mathematical Sciences Building, Liverpool L69 7ZL, United Kingdom
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Selected Topics in Mathematics
Speakers
22 October (Tuesday, 5pm) and 29 October (Tuesday, 5pm)
Oleg Karpenkov, University of Liverpool

Mean value property for nonharmonic functions

Abstract. In this talk we show how to extend the mean value property for harmonic functions to the nonharmonic analytic case. In order to get the value of the function at the center of a sphere one should integrate a certain power series with respect to Laplace operator over the sphere. We write explicitly such series in the Euclidean case and in the case of infinite homogeneous trees.
22 October (Tuesday, 5pm) and 29 October (Tuesday, 5pm)
Dr. James Tseng, Bristol University

Spiraling of lattice approximates and spherical averages of Siegel transforms

Abstract. A classical theorem of Dirichlet from the theory of Diophantine approximation, a subfield of number theory, says that, for every real d-vector x, there exists infinitely many pairs of (p,q) satisfying ||qx - p|| < |q|^(-1/d) where p is a d-vector with integer components, q is a nonzero integer, and ||*|| is the sup norm.
One can regard these pairs (p,q) as lattice points in a thinning region of a unimodular lattice associated to x. Take the orthogonal projection of these lattice points in this region. For each resulting d-vector, further project radially onto the unit d-1-sphere (which lies in the orthogonal projection) and consider the distribution of such vectors. One can now ask for this distribution for an analogous thinning region for any unimodular lattice. We show that, on average, the distribution is uniform--i.e. on average, the directions of such orthogonally projected lattice points are uniformly distributed.
On the other hand, we construct specific examples for which the distribution is nonuniform.
In showing our equidistribution theorem, we use the Siegel transform, an important, classical tool in the geometry of numbers. In particular, we need a spherical average result for these transforms, which we show by adapting a very recent proof of Marklof and Strombergsson. Our techniques are elementary and hinges on counting lattice points in balls or, equivalently, generalizations of the Gauss circle problem. Results like ours date back to the work of Eskin-Margulis-Mozes and have wide-ranging applications.
This is joint work with J. Athreya and A. Ghosh.
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