Georgia Institute of Technology College of Sciences

In the last years, connections between graphs and Riemann surfaces have been
discovered on several different levels. In particular, graphs are closely related
to singular Riemann surfaces and the boundary in the Deligne–Mumford com-
pactification of moduli spaces. Moreover, in both settings there is a notion of a
canonical measure (the Arakelov–Bergman and Zhang measures) which reflects
crucial geometric information.
In this talk, we focus on the following question: what is the limit of the canon-
ical measures along a family of Riemann surfaces? Combining the canonical
measures on Riemann surfaces and metric graphs, we obtain a full description
and a new compactification of the moduli space of Riemann surfaces in terms
of hybrid curves.

Based on joint work with Omid Amini (École polytechnique).

BlueJeans link: https://bluejeans.com/476849994

Over recent years there has been much interest in both Turán and Ramsey properties of vertex ordered graphs (i.e., graphs equipped with an ordering of their vertex set). In a recent paper, József Balogh, Lina Li and I initiated the study of embedding spanning structures into vertex ordered graphs. In particular, we introduced a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. In this talk I will discuss this work, in particular emphasizing how we adapt the regularity and absorbing methods to be applicable in the ordered setting.

The notion of weak saturation was introduced by Bollobás in 1968. A graph $G$ on $n$ vertices is weakly $F$-saturated if the edges of $E(K_n) \setminus  E(G)$ can be added to $G$, one edge at a time, in such a way that every added edge creates a new copy of $F$. The minimum size of a weakly $F$-saturated graph $G$ of order $n$ is denoted by $\mathrm{wsat}(n, F)$. In this talk, we discuss the weak saturation number of complete bipartite graphs and determine $\mathrm{wsat}(n, K_{t,t})$ whenever $n > 3t-4$. For fixed $1

Determining when two objects have “the same shape” is difficult; this difficulty depends on the dimension we are working in. While many of the same techniques work to study things in dimensions 5 and higher, we can better understand dimensions 1, 2, and 3 using other methods. We can think of 4-dimensional space as the “bridge” between low-dimensional behavior and high-dimensional behavior. One way to understand the possibilities in each dimension is to examine objects called cobordisms: if an (n+1)-dimensional space has an ``edge,”  then that edge is itself an n-dimensional space. We say that two n-dimensional spaces are cobordant if together they form the edge of an (n+1)-dimensional space. Using the idea of spaces related by cobordism, we can form a group. In this way, we can attempt to understand higher dimensions using clues from lower dimensions and organize this information using algebra. In this talk, I will discuss different types of cobordism groups and how to study them using tools from a broad range of mathematical areas.