# Critical sequences of rank-to-rank embeddings and a tower of finite left distributive algebras

## Sheila Miller

### City Tech - CUNY

In the early 1990’s Richard Laver discovered a deep and striking correspondence between critical sequences of rank-to-rank embeddings and finite left distributive algebras on integers. Each $A_n$ in the tower of finite algebras (commonly called the Laver Tables) can be defined purely algebraically, with no reference to the elementary embeddings, and yet there are facts about the Laver tables that have only been proven from a large cardinal assumption. We present here some of Laver’s foundational work on the algebra of critical sequences of rank-to-rank embeddings and related algebraic work of Laver’s and the author’s, describe how the finite algebras arise from the large cardinal embeddings, and mention several related open problems.

Sheila Miller is an Assistant Professor at the New York City College of Technology, in CUNY. She received her Ph.D. from the University of Colorado at Boulder in 2007, with a thesis entitled “Free Left-Distributive Algebras,” written under the supervision of Richard Laver, and subsequently held a postdoctoral position at the USMA in West Point. In addition to set theory and distributive algebras, she studies mathematical biology, with ongoing research into populations of sea turtles.