Georgia Institute of Technology College of Sciences

We present a novel example of a Lorentzian manifold-with-boundary featuring a dramatic degeneracy in its deterministic and causal properties known as “causal bubbles” along its boundary. These issues arise because the regularity of the Lorentzian metric is below Lipschitz and fit within the larger framework of low regularity Lorentzian geometry. Although manifolds with causal bubbles were recently introduced in 2012 as a mathematical curiosity, our example comes from studying the fundamental equations of fluid mechanics and shock singularities which arise therein. No prior knowledge of Lorentzian geometry or fluid mechanics will be assumed for this talk.

A recent advance by Smith establishes a quantitative converse (conjectured by Smyth and Serre) to Fekete's celebrated theorem for compact subsets of $\mathbb{R}$. Answering a basic question raised by Smith, we formulate and prove a quantitative converse of Fekete for general symmetric compact subsets of $\mathbb{C}$. We highlight and exploit the algorithmic nature of our approach to give concrete applications to abelian varieties over finite fields and to extremal problems in algebraic number theory.