Activities This Week

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’’.

The classification of hypersurface singularities aims at writing down a normal form of the defining power series with respect to some equivalence relation, and to give list of normal forms for a distinguished class of singularities. Arnold’s famous ADE-classification of singularities over the complex numbers had an enormous influence on singularity theory and beyond. I will report on some of the impact of his work to other disciplines and to some real-life applications of the classification. Stimulated by Arnold’s work, the classification has been carried on to singularities over fields of positive characteristic, partly with surprising differences. I will report on recent results about this classification and about related problems.