Ben Lund, Maximal 3-wise intersecting families

A family $\mathcal F$ of subsets of {1,2,…,n} is called maximal k-wise intersecting if every collection of at most k members from $\mathcal F$ has a common element, and moreover, no set can be added to $\mathcal F$ while preserving this property. In 1974, Erdős and Kleitman asked for the smallest possible size of a maximal k-wise intersecting family, for k≥3. We resolve this problem for k=3 and n even and sufficiently large.

This is joint work with Kevin Hendrey, Casey Tompkins, and Tuan Tran.

Ben Lund, Limit shape of lattice Zonotopes

A convex lattice polytope is the convex hull of a set of integral points. Vershik conjectured the existence of a limit shape for random convex lattice polygons, and three proofs of this conjecture were given in the 1990s by Bárány, by Vershik, and by Sinai. To state this old result more precisely, there is a convex curve $L \subset [0,1]^2$ such that the following holds. Let $P$ be a convex lattice polygon chosen uniformly at random from the set of convex lattice polygons with vertices in $[0,N]^2$. Then, for $N$ sufficiently large, $(1/N)P$ will be arbitrarily close (in Hausdorff distance) to $L$ with high probability. It is an open question whether there exists a limit shape for three dimensional polyhedra.

I will discuss this problem and some relatives, as well as joint work with Bárány and Bureaux on the existence of a limit shape for lattice zonotopes in all dimensions.

Ben Lund, Perfect matchings and derangements on graphs

We show that each perfect matching in a bipartite graph G intersects at least half of the perfect matchings in G. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of derangements and permutations on graphs. We give several related results and open questions. This is joint work with Matija Bucic, Pat Devlin, Mo Hendon, and Dru Horne.

Ben Lund, Point-plane incidence bounds

In the early 1980s, Beck proved that, if P is a set of n points in the real plane, and no more than g points of P lie on any single line, then there are $\Omega(n(n-g))$ lines that each contain at least 2 points of P. In 2016, I found a generalization of this theorem, giving a similar lower bound on the number of planes spanned by a set of points in real space. I will discuss this result, along with a number of applications and related open problems.

Welcome Ben Lund and Tuan Tran, new research fellows in the IBS Discrete Mathematics Group

The IBS discrete mathematics group welcomes Dr. Ben Lund and Dr. Tuan Tran, new research fellows at the IBS discrete mathematics group from August 1, 2020.

Ben Lund received his Ph.D. from the Department of Mathematics at Rutgers University in 2017 under the supervision of Prof. Shubhangi Saraf. Before joining the IBS, he was a postdoc at Princeton University and a postdoc at the University of Georgia.

Tuan Tran received his Ph.D. from the Department of Mathematics at the Freie Universität Berlin in 2015 under the supervision of Prof. Tibor Szabó. Before joining the IBS, he was a lecturer at Hanoi University of Science and Technology, a postdoc at ETH Zürich, and a postdoc at Czech Academy of Sciences. He won the IBS Young Scientist Fellowship.