פעילויות השבוע

A circle packing is a collection of circles in the plane with disjoint interiors. An accumulation point of the circle packing is a point with infinitely many circles in any neighborhood of it. A site percolation with parameter p on the circle packing means retaining each circle with probability p and deleting it with probability 1-p, independently between circles. We will explain the proof of the following result: There exists p>0 satisfying that for any circle packing with finitely many accumulation points, after a site percolation with parameter p there is no infinite connected component of retained circles, almost surely. This implies, in particular, that the site percolation threshold of any planar recurrent graph is at least p. It is conjectured that the same should hold with p=1/2. The result gives a partial answer to a question of Benjamini, who conjectured that square packings of the unit square admit long crossings after site percolation with parameter p=1/2 and asked also about other values of p. Time permitting, we will discuss an application of the result to the existence of macroscopic loops in the loop O(n) model on the hexagonal lattice. Portions joint with Nick Crawford, Alexandar Glazman and Matan Harel.

A union closed family F is a family of sets, so that for any two sets A,B in F, A$\cup$B is also on F. Frankl conjectured in 1979 that for any union-closed family F of subsets of [n], there is some element i $\in$ [n] that appears in at least half the members of F.

The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system. No prior knowledge on quantum mechanics or representation theory will be assumed.

Tensor categories are abelian k-linear monoidal categories modeled on the representation categories of affine (super)group schemes over k. Deligne gave very succinct intrinsic criteria for a tensor category to be equivalent to such a representation category, over fields k of characteristic zero. These descriptions are known to fail badly in prime characteristics. In this talk, I will present analogues in prime characteristic of these intrinsic criteria. Time permitting, I will comment on the link with a recent conjecture of V. Ostrik which aims to extend Deligne’s work in a different direction.