Effective July 2021, Professor Bridget Tenner has been awarded a 3-year $175,000 research grant from the National Science Foundation's Division of Mathematical Sciences to investigate the subject of Higher-Order Pattern Containment. Congratulations Professor Tenner!
From the abstract of Professor Tenner's award:
Permutations are orderings of a set of objects, typically positive integers. They can model aspects of many different problems and scenarios, and they are used throughout mathematics. A permutation pattern is a subset of a permutation that appears in a particular order. For example, it might have an increasing subsequence of some length, or it might avoid having a large number followed by two smaller numbers. The presence or absence of particular patterns can be critical to features of the data that the permutation is modeling, which has led to great interest in the analysis of permutation patterns for the past several decades. The goal of this project is to establish a broader context to the traditional study of permutation patterns. Historically, research in this area has been almost entirely in terms of a binary question: does a permutation contain a given pattern or does the permutation avoid it? Recent evidence has shown that the specific number of times that a pattern appears is of great importance, and consequently a more granular notion of pattern containment should be studied. Specifically, the question should instead be: how many times does the permutation contain the pattern? This project will address that higher-order question in several ways, with implications for combinatorics and its connections to algebra, topology, and computer science.