TL;DR
This paper discusses the growth rate of the number of numerical semigroups with a given genus, confirming Fibonacci-like growth and highlighting open problems and undergraduate contributions.
Contribution
It outlines Zhai's proof of the Fibonacci-like growth of the numerical semigroup count and discusses open questions in the area.
Findings
The sequence of numerical semigroups with genus g grows asymptotically like Fibonacci numbers.
It is still unknown whether this sequence is nondecreasing.
The paper highlights undergraduate contributions to the field.
Abstract
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to ? We outline Zhai's proof of a conjecture of Bras-Amor\'os that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems. We highlight the many contributions made by undergraduates to problems in this area.
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