Title: The integer homology threshold in random simplicial complexes

Abstract:
A classic result of Erd\\H{o}s and R\\'{e}nyi is that the connectivity threshold in their random graph model is at $(\\log n)/n$. That is if one samples a graph $G$ from the model $G(n, p)$ with $p$ less than $(\\log n)/n$ then almost certainly $G$ is disconnected, while $p > (\\log n)/n$ implies that $G$ almost certainly is connected. As graph connectivity is a homological property, it generalizes nicely to higher-dimensional simplicial complexes as homological connectivity, the vanishing of all homology groups in positive codimension. As such, the threshold for homological connectivity has been studied in the Linial--Meshulam model of random simplicial complexes. A paper of Meshulam and Wallach from 2006 develops the cocycle counting technique to establish the threshold for homological connectivity over a fixed finite field for random simplicial complexes. However the question of homological connectivity over the integers remained open. In this talk I will give an overview of the Linial--Meshulam model and then discuss joint work with Elliot Paquette to adapt the cocycle counting technique to establish the threshold for homological connectivity over the integers.

Computing Convex Hulls of Trajectories

Abstract: Motivated by the problem of designing chemical reactors we study convex hulls of trajectories generated by polynomial dynamical systems. Real algebraic curves constitute a special case. We numerically determine whether the convex hull of such a curve is forward closed. That is, if the dynamics started from any point within the hull generate trajectories remaining inside the convex hull. To this end, we compute polyhedral approximations of the convex hull. We give results on the limiting behavior of those approximations. From the polyhedral approximations we derive information about stratified families of faces on the boundary of the convex hull. These stratifications help to partition the boundary into regions of inward and outward pointing dynamics. Experiments have been conducted for dimensions up to 4.

Title: Binomial edge ideals of bipartite graphs

Abstract: Binomial edge ideals are ideals generated by binomials corresponding to the edges of a graph. They generalize the ideals of 2-minors of a generic matrix with two rows and arise naturally in Algebraic Statistics. We give a combinatorial classification of Cohen-Macaulay binomial edge ideals of bipartite graphs providing an explicit construction in graph-theoretical terms. In the proof we use the dual graph of an ideal, showing in our setting the converse of Hartshorne’s Connectedness theorem.

As a consequence, we prove for these ideals a Hirsch-type conjecture of Benedetti-Varbaro.

This is a joint work with Davide Bolognini and Francesco Strazzanti.

Title: Cluster partitions and fitness landscapes of the Drosophila fly microbiome

Abstract: Beerenwinkel et al.(2007) suggested studying fitness landscapes via regular
subdivisions of convex polytopes. Building on their approach we propose cluster
partitions and cluster filtrations of fitness landscapes as a new mathematical
tool. In this way, we provide a concise combinatorial way of processing metric
information from epistatic interactions. Using existing Drosophila microbiome
data, we demonstrate similarities with and differences to the previous approach.
As one outcome we locate interesting epistatic information where the previous
approach is less conclusive. Joint works with Holger Eble, Lisa Lamberti and
Will Ludington.

Title: The moduli space of Harnack curves

Abstract: tba