Final ACO Doctoral Examination and Defense of Dissertation

Title: Graphs and geometry: an interplay between local and global views

Jing Yu

ACO PhD student, School of Mathematics

Date: 7/19/2024

Time: 1-3pm

Location: Skiles 005

Zoom: https://gatech.zoom.us/j/2722564420

Advisor: Dr. Anton Bernshteyn, School of Mathematics, Georgia Institute of Technology

Committee:

Dr. Anton Bernshteyn, School of Mathematics, Georgia Institute of Technology (advisor)

Dr. Rose McCarty, School of Mathematics and School of Computer Science, Georgia Institute of Technology

Dr. Will Perkins, School of Computer Science, Georgia Institute of Technology

Dr. Anush Tserunyan, Mathematics & Statistics Department, McGill University

Dr. Xingxing Yu, School of Mathematics, Georgia Institute of Technology

Reader: Dr. Anush Tserunyan, Mathematics & Statistics Department, McGill University

Thesis draft: Dissertation.pdf

Abstract: In this dissertation, we explore problems related to graphs and geometry. This work consists of two projects, and they are independent and utilize distinct proof techniques. However, they share a common underlying philosophy: we alternate between local and global perspectives as required.

In Project I, we investigate the large-scale geometry of Borel graphs of polynomial growth. Krauthgamer and Lee showed that every connected graph of polynomial growth admits an injective contraction mapping to $(\mathbb{Z}^n, \|\cdot\|_\infty)$ for some $n \in \mathbb{N}$. We strengthen and generalize this result in a number of ways. In particular, answering a question of Papasoglu, we construct coarse embeddings from graphs of polynomial growth to $\mathbb{Z}^n$. Furthermore, we extend these results to Borel graphs. Namely, we show that graphs generated by free Borel actions of $\mathbb{Z}^n$ are in a certain sense universal for the class of Borel graphs of polynomial growth. This provides a general method for extending results about $\mathbb{Z}^n$-actions to all Borel graphs of polynomial growth. For example, an immediate consequence of our main result is that all Borel graphs of polynomial growth are hyperfinite, which answers a well-known question in the area. Additionally, our results yield nice applications in graph minor theory.

In Project II, we investigate outerplanar graphs with positive Lin–Lu–Yau curvature. we show that all simple outerplanar graphs with minimum degree at least 2 and positive Lin-Lu-Yau curvature on every edge have maximum degree at most 9. Furthermore, if G is maximally outerplanar, then G has at most 10 vertices. Both upper bounds are sharp.