Activities This Week

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.

Recently Uri Gabor refuted an old conjecture stating that any finitary factor of an i.i.d process is finitarly isomorphic to an i.i.d process. Complementing Gabor’s result, in this talk, which is based on work in progress with Yinon Spinka, we will prove that any countable-valued process which is admits a finitary a coding by some i.i.d process furthermore admits an $\epsilon$-efficient finitary coding, for any positive $\epsilon$. Here an ‘’$\epsilon$-efficient coding’’ means that the entropy increase of the coding i.i.d process compared to the (mean) entropy of the coded process is at most $\epsilon$. For processes having finite entropy this in particular implies a finitary i.i.d coding by finite valued processes. As an application we give an affirmative answer to an old question about the existence of finite valued finitary coding of the critical Ising model, posed by van den Berg and Steif in their 1999 paper ‘‘On the Existence and Nonexistence of Finitary Codings for a Class of Random Fields’’.