The h-vector of a matroid is an important invariant that has been the subject of intense study in recent years. A still open conjecture of Stanley posits that the h-vector of a matroid is a pure O-sequence, meaning that it can be obtained by counting faces of a pure multicomplex. Merino established Stanley's conjecture for the case of cographic matroids via chip-firing on graphs and the notion of a G-parking function. Inspired by these constructions, we introduce the notion of a "cycle system" for a matroid M - a family of circuits of M with overlap properties that mimic cut-sets in a graph. A choice of cycle system on M defines a collection of integer sequences that we call "coparking functions", which we show are in bijection with the set of bases of M. This leads to a proof of Stanley's conjecture for the case of matroids that admit cycle systems, which, for instance, include graphic matroids of cones as well as $K_{3,3}$-minor free graphs. Joint work with Scott Corry, Solis McClain, David Perkinson, and Lixing Yi.

I will present a method for computing resonance varieties, Alexander invariants, and Chen ranks of spaces that are not formal but admit finite-type cdga models. The method is based on Koszul linearization, which replaces the classical algebraic constructions underlying Alexander-type invariants with functorial algebraic objects built directly from the cdga, thereby shifting the role of the cohomology ring to the full model.

A key feature of this approach is the existence of functorial spectral sequences that interpolate between invariants computed from cohomology and those arising from the cdga model, with higher differentials encoding iterated Massey products. This framework yields extensions of several results from the formal setting, including the fact that cohomology controls the first-order behavior at the origin of the resonance varieties, as well as explicit formulas for infinitesimal Alexander invariants and Chen ranks in terms of the model. Discrepancies between cohomological and model-theoretic invariants thus provide computable obstructions to formality.

The constructions are functorial with respect to cdga morphisms and provide effective tools for computation. Applications include nilpotent Lie algebras and elliptic configuration spaces, as well as consequences for detecting non-formality.

Inspired by the elementary collapses/expansions and simple homotopy theory of CW-complexes, we consider various notions of discrete homotopy defined by making prescribed local changes on a graph (and digraph). This includes the I-homotopy type of a graph introduced by Chen, Yau, and Yeh; the s-homotopy of graphs introduced by Boulet, Fieux, and Jouve; as well as the $\times$-homotopy of graphs introduced by the author. We seek to place these constructions in a uniform setting, and also relate them to cylinder objects and internal hom structures in the category of graphs. We will see how the notion of a homomorphism complex (which has applications to obstruction theories for graph homomorphisms) plays a role in understanding the various theories. Parts of this are joint work with Takahiro Matsushita and Anurag Singh.