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{\Large Department of Mathematics, BGU}

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{\Huge BGU Probability and Ergodic Theory  (PET) seminar}\\[0.2\baselineskip]

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\textbf{On} \emph{Thursday, October 31, 2019}
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\textbf{At} \emph{11:10 -- 12:00}
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\textbf{In} \emph{-101}

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{\large\scshape Barak Weiss 
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  (Tel-Aviv University)
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will talk about
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{\Large\bfseries Geometric invariants of lattices and points close to a line, and their asymptotics\par}
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\textsc{Abstract:}
Given a lattice $\Lambda$ and a (perhaps long) vector $v \in \Lambda$, we consider two geometric quantities:
 - the projection $\Delta$ of $\Lambda$ along the line through $v$
 - the ``lift functional'' which encodes how one can recover $\Lambda$ from the projection $\Delta$
Fixing $\Lambda$ and taking some infinite sequences of vectors $v_n$, we identify the asymptotic distribution of these two quantities. For example, for a.e. line $L$, if $v_n$ is the sequence of $\epsilon$-approximants to $L$ then the sequence $\Delta(v_n)$ equidistributes according to Haar measure, and if $v'_n$ is the sequence of best approximants to $L$ then there is another measure which $\Delta(v'_n)$ equidistributes according to. The basic tool is a cross section for a diagonal flow on the space of lattices, and after some analysis of this cross section, the results follow from the Birkhoff pointwise ergodic theorem.

Joint work with Uri Shapira.








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