Activities This Week

קרה לכם שהסתכלתם על נוסחה ואמרתם לעצמכם - חבל שאין לה הסבר בעל פה? אציג סיפורים קומבינטוריים וגיאומטריים מאחורי כמה זהויות בינומיות.

In their paper, Frączyk and Gelander prove a conjecture by Margulis: for a higher-rank simple Lie group $G$, a discrete subgroup has an infinite injectivity radius if and only if it has infinite covolume. The novel methods used to resolve this rely on ergodic theory—specifically, analyzing random walks on the space of discrete subgroups, alongside new stiffness and rigidity results for these stationary measures. In this talk, I will introduce the foundational definitions and present an outline of the main arguments used to prove the conjecture.

Link to paper

The covering radius of a shift space is a quantity of interest for information-theoretic applications of data transmission over noisy channels. In this talk we will explain what is the covering radius of a sofic shift and why people care about it. We will outline a proof that the covering radius of a primitive sofic shift is always a rational number, and outline an algorithm to compute the covering radius from a labeled graph presentation. We will also briefly explain how these results relate to dynamics, to a certain zero-sum two-player game and to an old meta-conjecture about typical ground states in statistical mechanics. The notions will be defined, no specific background assumed. Based on joint work with Aidan Young as in https://arxiv.org/abs/2603.21449, and previous joint work with Dor Elimelech and Moshe Schwartz as in https://ieeexplore.ieee.org/document/10360152