numerical semigroups are combinatorial objects that lead to deep and subtle questions.  with tools from complex, harmonic, and functional analysis, probability theory, algebraic combinatorics, and computer-aided design, we answer virtually all asymptotic questions about factorization lengths in numerical semigroups.  our results yield uncannily accurate predictions, along with unexpected results about symmetric functions, trace polynomials, and the statistical properties of certain af c$^*$-algebras. 

partially supported by nsf grants dms-1800123 and dms-2054002.  joint work (in various combinations) with k.~aguilar, a.~b\"ottcher, \'a. ch\'avez, l.~fukshansky, m.~omar, c.~o'neill, j.~vol\v{c}i\v{c} and undergraduate 2022年亚洲世界杯预选赛 j.~hurley, g.~udell, t.~wesley, s.~yih.