פעילויות השבוע

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.

A singularity refers always to a special situation, something that is not true in general. The term ”singularity“ is often used in a philosophical sense to describe a frightening or catastrophically situation which is often unknown. Singularity theory in mathematics is a well defined discipline with the aim to tame the ”catastrophe“. I will give a general introduction to singularity theory with some examples from real life. Then I consider a special kind of taming a singularity, the normalization, and give an overview of classical and recent results on simultaneous normalization of families of algebraic and analytic varieties. I will also discuss some open problems.

ניתן להגדיר חבורות p-סילוב בכל חבורה שהיא. בדרך כלל משפטי סילוב המוכרים מעולם החבורות הסופיות לא פועלים. עם זאת משפט מקסים של עסאר (Asar) מראה שבכל זאת ניתן להציל משהו מתורת סילוב במקרים מסוימים.

For a polynomial $F(t,A_1,...,A_n)$ in $\mathbb{F}_p[t,A_1,...,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $F(t,a_1,...,a_n)$ in $\mathbb{F}_p[t]$ with $(a_1,...,a_n) \in S$, where $S=I_1\times\dots\times I_n\subset\mathbb{F}_{p^n}$ is a box, in the limit $p\rightarrow\infty$ and $deg(F)$ fixed. We show that under certain fairly general assumptions on $F$, and assuming that the box dimensions grow to infinity with one of them growing faster than $p^{1/2}$, the degrees of the irreducible factors of $F(t,a_1, \dots,a_n)$ are distributed like the cycle lengths of a random permutation in $S_n$.

This improves and generalizes previous results of Shparlinski and more recent results of Kurlberg-Rosenzweig, which in turn generalize the classical Polya-Vinogradov estimate of the number of quadratic residues in an interval.