Schedule and Abstracts

Abstract: The Ehrhart series of a lattice polytope P is a combinatorial gadget that counts the number of lattice points of P and of its dilations. The Hilbert series of a simplicial complex Σ counts how many monomials supported on faces of Σ exist in each possible degree. The aim of this talk is to introduce equivariant versions of such constructions, where we are not just interested in counting but we also want to record how the action of a finite group affects such collections of lattice points or monomials. Inspired by previous results by Betke–McMullen, Stembridge, Stapledon and Adams–Reiner, we will investigate which extra combinatorial features of the group action give rise to “nice” rational expressions of the equivariant Hilbert and Ehrhart series, and how the two are sometimes related.

References

[1]  A. Adams and V. Reiner, A colorful Hochster formula and universal parameters for face rings, J. Commut. Algebra, 15 (2023), 151–176.

[2]  U. Betke and P. McMullen, Lattice points in lattice polytopes, Monatsh. Math., 99 (1985), 253–265.

[3]  A. Stapledon, Equivariant Ehrhart theory, Adv. Math., 226 (2011), 3622–3654.

[4]  A. Stapledon, Equivariant Ehrhart theory, commutative algebra and invariant triangulations of polytopes, arXiv preprint arXiv:2311.17273 (2023).

[5]  J. R. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math., 106 (1994), 244–301.

Abstract. Magnitude was first introduced by Leinster in 2008 as a notion analogous to the Euler characteristic of a category. Magnitude homology was defined in 2014 by Hepworth and Willerton as a categorification of magnitude in the context of simple undirected graphs, and although the construction of the boundary map suggests that magnitude homology groups are strongly influenced by the graph substructures, it is not straightforward to detect such subgraphs.

In this talk, I will describe the work done by Chad Giusti and myself toward elucidating the connection between magnitude homology of simple graphs equipped with the hop metric and their combinatorial structure. The approach we take is to observe that a large portion of the magnitude chain complex is redundant, in the sense that the chains reflect combinatorial structure already recorded by chains of lower bigrading. To leverage this observation, we define the subcomplex of eulerian magnitude chains EMCk,l(G), supported on trails with no repeated landmarks. Focusing on the k = l line where the list of landmarks completely determines a trail, we obtain strong relationships between the (k, k)-eulerian magnitude homology groups and the structure of a graph. We accomplish this in Theorem 1 by decomposing such cycles into a generating set described in terms of their structure graphs, which encode how terms in the differentials of the constituent chains cancel. We are thus able to characterize subgraphs of a graph that support non-trivial cycles in EMHk,k(G) in terms of the corresponding structure graphs, providing a framework for computing these groups for graphs of interest.

In the interest of exploring what features of a graph the (eulerian) magnitude homology groups capture, we turn our attention to classes of random graphs: Erdős-Rényi (ER) random graphs and random geometric graphs on the standard torus. In each context, we derive a vanishing threshold for the limiting expected rank of the (k, k)-eulerian magnitude homology in terms of the density parameter. Further, adapting tools from Kahle and Meckes we develop a characterization of the limiting expected Betti numbers of the (k,k)-eulerian magnitude homology groups in terms of density. In this talk, I will focus on the results in the context of ER graphs.

I will finally discuss the homotopy type of the eulerian magnitude chain complex by highlighting its connection with the complex of injective words.

References

[1] C. Giusti, G. Menara, Eulerian magnitude homology: subgraph structure and random graphs, arXiv:2403.09248 (2024).

[2] T. Leinster,The Euler characteristic of a category, Documenta Mathematica (13) (2008) 21– 49.

[3] R. Hepworth, S. Willerton, Categorifying the magnitude of a graph. Homotopy, Homology and Applications 16(2) (2014) 1–30.

[4] M. Kahle, E. Meckes, Limit theorems for Betti numbers of random simplicial complexes. Homology, Homotopy and Applications 15(1) (2013) 343–374.

Abstract:

Abstract: Threshold-linear networks (TLNs) are common models in theoretical neuroscience that are useful for modeling neural activity and computation in the brain. They are simple, recurrently-connected networks with a rich repertoire of nonlinear dynamics including multista- bility, limit cycles, quasiperiodic attractors, and chaos. In this talk, I will give a brief introduction to TLNs, and then show how ideas from sheaf theory and oriented matroids provide valuable insights into the connection between network architecture and dynamics.

Abstract:

Abstract:

Abstract. Oriented matroids encode the combinatorial structure of arrangements of pseudo- spheres, generalizing arrangements of hyperplanes in real vectorspaces. Salvetti complexes of oriented matroids represent a strictly wider class of homotopy types with respect to complements of hyperplane arrangements in complex vectorspaces (e.g., one obtains fundamental groups that cannot appear in the case of hyperplane arrangements).

To every real hyperplane arrangement and, more generally, to every oriented matroid, we associate a “tope-pair poset”. This poset is homotopy equivalent to the Salvetti complex and carries some extra structure that makes it a useful tool for combinatorial topology.

For instance, the tope-pair poset supports a free action of Z4 that discretizes the diagonal C∗-action on complexified arrangement’s complements, and it affords a proof of the fact that the integer cohomology of the Salvetti complex of any oriented matroid is given by the Orlik-Solomon algebra of the underlying matroid.

In the talk I will define the tope-pair poset, survey some of its features, and outline its connections to Artin groups and combinatorial fibrations. The results presented in the talk are joint work with Michael Falk.

Abstract:

Abstract:

Abstract. Given an abstract graph Γ = (V, E), define the chromatic configuration space to be 

ConfΓ(X)={(x1,···,xn)∈X|V| |xi ̸=xj if{i,j}∈E(Γ)}.

When the graph is complete, we get the classical configuration space of pairwise distinct points. We will show that ConfΓ(X) splits, after only one suspension, into a bouquet of spheres, the number of spheres is computed precisely and is given in terms of some chromatic numbers of the graph. Our tools are poset topology and a geometric description of the homology classes. Similar ideas apply to a related kind of configuration spaces.

This work is joint with Moez Bouzouita (University of Tunis).