Title: Regularity lemmas as a measure of complexity
Speaker: Caroline Terry
Abstract: A major theme in combinatorics is understanding the structure of graphs with forbidden subgraphs. This can be phrased by asking, given some local combinatorial restriction in a graph, what are the global implications? Are there special local restrictions which yield very strong information about global structure? These kinds of questions are also studied in model theory, but with a focus on the infinite setting.
Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemerédi’s regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon-Fischer-Newman, Lovász-Szegedy, and Malliaris-Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies always have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn Semester. Pre-candidacy PhD students can sign up for this lecture series by registering for one or two credit hours of Math 6193 with Professor Nimish Shah.
