A numerical semigroup is a subset of the nonnegative integers that contains 0, is closed under addition, and contains all but finitely many of the integers. For example, the semigroup <3,5> consisting of all nonnegtive linear combinations of 3 and 5 is the set of integers {0,3,5,6,8,9,10,...}. For any element n of S, we may define the Apéry set Ap(S;n) to be the set of minimal elements of S in each equivalence class mod n. This can be interpreted as a point in R^n. The Kunz cone is the region of R^n in which these points lie. The faces of the Kunz cone are indexed by posets on Z/nZ, which correspond to the poset on Ap(S;n) induced by a precedes b if b-a lies in S.

The semigroup T=a*S1+b*S2 is considered a gluing of two numerical semigroups S1 and S2 if a and b are non-generator elements of S2 and S1, respectively, and gcd(a,b)=1. We determined the Apéry sets and posets of gluings when the modulus is one of their minimal generators. Building on previous work, we found an injection of faces that takes faces from lower-degree cones into faces from higher-degree cones. Further work needs to be done to investigate when this injection fills the corresponding face of the higher-dimensional cone.

This research was done with Isidora Bailly-Hall (Grinnell College) under the mentorship of Chris O'Neill and Emily O'Sullivan (SDSU).

At the 2021 Stanford University Research in Mathematics (SURIM) program, our team studied the mixing time of the Chung-Diaconis-Graham (CDG) random process and its generalizations, with particular emphasis on the Fibonacci Random Generator.

Generate a sequence of integers modulo m by successively multiplying by 2 and adding either -1, 0, or +1 with equal probability 1/3 (all arithmetic is done modulo m). This is the CDG Process. The Fibonacci Random Process replaces multiplying by 2 with the standard recursive Fibonacci relation.

It can be shown that for all odd m and all m, respectively, the probability distribution of later elements in the CDG and Fibonacci Processes approaches the uniform distribution over all possible values modulo m. The question studied effectively amounts to: As you increase m (the modulus), how many times must you iterate the process to achieve the same closeness to uniformity (specifically, such that the Total Variation distance between the value’s probability distribution and the uniform distribution is sufficiently small).

We tackled both problems using a bounding method originally applied by Chung, Diaconis, and Graham involving discrete Fourier analysis. In addition to attempting to improve the bound, we also attempted to determine the potential limits of the bounding strategy in generous cases. The distribution of Fibonacci-type sequences modulo m, at present a poorly-understood subject, appears to play a large part.