Georgia Institute of Technology College of Sciences

We say a partially ordered set $P$ is a tree poset if its Hasse diagram, the graph drawn by joining $x$ with $y$ if there is no $z$ such that $x > z > y$, is a tree. In this talk, we will be discussing a tool for embedding tree posets $P$ into subsets of the Boolean lattice, and some applications of it to counting copies of $P$ in subsets of the Boolean lattice, counting $P$-free subsets of the Boolean lattice, and largest $P$-free subsets of the Boolean lattice. This talk is based on joint work with Tao Jiang, Sam Spiro, and Liana Yepremyan. 

In this talk, we will discuss the construction of exotic 4-manifolds using Lefschetz fibrations over S^2, which are obtained by finite order cyclic group actions on Σg. We will first apply various cyclic group actions on Σg for g>0, and then extend it diagonally to the product manifolds ΣgxΣg. These will give singular manifolds with cyclic quotient singularities. Then, by resolving the singularities, we will obtain families of Lefschetz fibrations over S^2. Following the resolution process, we will determine the configurations of the singular fibers and the monodromy of the total space. In some cases, deformations of the Lefschetz fibrations give rise to nice applications using the rational blow-down operation, which provides exotic examples. This is a joint work with A. Akhmedov and M. Bhupal.

The degeneracy of a graph is a measure of sparseness that appears in many contexts throughout graph theory. In extremal graph theory, it is known that graphs of bounded degeneracy have Ramsey number which is linear in their number of vertices (Lee, 2017). Also, the degeneracy gives good bounds on the Turán exponent of bipartite graphs (Alon--Krivelevich--Sudokav, 2003). Extending these results to hypergraphs presents a challenge, as it is known that the naïve generalization of these results -- using the standard notion of hypergraph degeneracy -- are not true (Kostochka--Rödl 2006). We define a new measure of sparseness for hypergraphs called skeletal degeneracy and show that it gives information on both the Ramsey- and Turán-type properties of hypergraphs.

Based on joint work with Jacob Fox, Maya Sankar, Michael Simkin, and Yunkun Zhou

I'll review a few known results about quantum graphs that maximize or minimize eigenvalues or combinations of eigenvalues, and will then concentrate on ratios of eigevalues under topological constraints on the graph.  In particular a new discovery with James Kennedy and Gabriel Ramos is that the largest ratio of the first two eigenvalues of the Laplacian on a finite tree graph with Dirichlet conditions at the ends is achieved by equilateral stars. Some related, Weyl-sharp estimates of arbitrary eigenvalue ratios can be obtained using similar ideas.  If time permits, I will also describe some optimal results about differences and other combinations of eigenvalues.