Activities This Week

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.

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with Lagrangian boundary conditions and various constraints on boundary and interior marked points. The presence of boundary of real codimension 1 poses an obstacle to invariance. In a joint work with J. Solomon (2016-2017), we defined genus zero OGW invariants under cohomological conditions. The construction is rather abstract. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called open WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation. For the projective space, this PDE generates recursion relations that allow the computation of all invariants. Furthermore, the open WDVV can be reinterpreted as an associativity of a suitable version of a quantum product.

No prior knowledge of any of the above notions will be assumed.