Monday, Sept. 14Title: A decade of magnitude homologyAbstract: Magnitude homology was introduced a little over 10 years ago. Since then, the subject has grown by leaps and bounds. I'll offer a (personal and, with apologies, selective) overview of how the subject has evolved over that decade. I'll try to highlight some of the main themes, current developments, and (time permitting) future prospects.Title: Representation (co)homology of a small category Abstract: Let $\C_\bullet$ be a simplicial object in the category $\Cat$ of small categories. For a field $k$, taking the (split) Grothendieck groups of isomorphism classes of $k\C_n$-modules gives rise via restriction of the simplicial operators to a cosimplicial abelian group, and as such to a cochain complex. The cohomology, which we refer to as representation cohomology, is the object we discuss in this talk. In particular, to any small category $\C$, we associate a simplicial object in $\Cat$, where for each $n \geq 0$ the objects of the level $n$ category are the simplices of the nerve of $\C$. One may similarly define representation homology (which we shall not explore in detail for lack of time). We shall discuss some basic properties of the resulting representation cohomology and certain subobjects are then studied in detail. We will present some general theoretical computations in favourable cases as well as some computer aided computations. The project is joint work with Markus Klemetti, Henri Riihimäki and Daniel Sölch. Title: Eulerian magnitude homology: diagonality, injective words and regular path homologyAbstract: In this talk, we investigate the algebraic framework and combinatorial features of Eulerian magnitude homology. We begin by examining its diagonality properties, which lead to a characterization of complete graphs. We then introduce the regular magnitude-path spectral sequence (constructed as the spectral sequence of the (filtered) injective nerve of the reachability category) and discuss some implications of this construction, such as connections to regular path homology. This is joint work with Luigi Caputi.Title: Fundamental Groups of Permutation ComplexesAbstract: The complex of injective words Inj(n), going back to Farmer and Björner-Wachs, is the ordered simplicial complex whose facets are the permutations of n letters. Equivalently, it is the directed flag complex of the complete directed graph with n vertices. Chacholski, Levi and Meshulam have studied certain subcomplexes of Inj(n), called permutation complexes, generated by a collection of permutations. While it is known that every finite homotopy type can be realized by a permutation complex up to suspension, their topology is still far from understood. For example, it is not known whether the fundamental group of such a complex must necessarily be free. In the talk I will give a description of the fundamental group and discuss some consequences of this result, including the fact that permutation complexes generated by at most three permutations have free fundamental group.Title: Topology of Graph ColoringsAbstract: Chromatic homology theories categorify the chromatic polynomial and relate graph coloring to topological and combinatorial constructions such as configuration spaces, Khovanov link homology, broken-circuit models, and spanning trees. This talk will focus on the structure of these theories and the relations between them, as well as their generalizations to simplicial complexes and hypergraphs, where new torsion phenomena occur. Finally, we return to combinatorics and matroid theory by describing the extremal part of chromatic homology in terms of Crapo’s invariant.Title: Magnitude Homology of Hypergraphs via Constraint GraphsAbstract: In 2023, Bi, Li, and Wu initiated the study of magnitude homology for hypergraphs by extending the intercrossing distance to hyperedges. Building on their foundational work, this talk develops additional homological tools by studying hypergraph magnitude homology through an associated class of weighted graphs. This categorical perspective places graphs and hypergraphs in a unified framework and enables the construction of long exact Mayer–Vietoris sequences, inclusion–exclusion formulas for magnitude, and Künneth-type results. Applications to structured families of hypergraphs illustrate how techniques from graph magnitude homology extend naturally to the hypergraph setting.Title: From local to global in directed random graphs Abstract: Random graphs are a fundamental tool for studying complex interaction networks arising in biology, communication systems, or social science. Empirical biological networks, such as gene regulatory networks, exhibit distinctive structural features, including power-law degree distributions and the over-representation of specific local motifs, most notably feed-forward loops.Motivated by these observations, we investigate how local attachment rules influence the global topology of directed random graphs. We discuss a family of spatial preferential attachment models based on "the age-dependent random connection model" introduced by Gracar, Grauer, Lüchtrath, and Mörters in their 2019 paper. We show how suitable parameter choices allow them to reproduce key statistical and topological properties observed in empirical networks. In particular, we analyze the homology of the directed flag complexes associated to the random graphs. Further, we introduce a probability-informed filtration whose persistence homology informs about the expected Betti number of an instantiation of the random graph model.Tuesday, Sept. 15Title: Vistas on (infty,2)-categoriesAbstract: This talk is going to address various aspects of (infty,2)-categories (including at least a moral definition), briefly mentioning various model structure implementing them. I will particularly focus on a pasting theorem for (infty,2)-categories where graphs make a prominent (though maybe surprising) appearance. This is based on joint work with Martina Rovelli. Title: Thomason-Type Model Structures on Simplicial Complexes and GraphsAbstract: Matsushita proved the existence of a model structure on the category of loop graphs which is Quillen equivalent to spaces, and which reproduces x-homotopy theory. In my recent paper I factored his construction through the category of reflexive graphs and simplicial complexes, right transferring the model structure each time, showing that each is proper and each right transfer is a Quillen equivalence. In this talk I'll explain what this means conceptually and if there is time I will discuss the Z_2 graded situation.Title: Introduction to discrete homotopy theoryTitle: Homotopy n-types of cubical sets and graphsAbstract: In classical homotopy theory, graphs are treated as 1-dimensional CW complexes. But since neither continuous maps nor their continuous homotopies respect the discrete nature of graphs, a lot of combinatorial information is lost in this treatment. Discrete homotopy theory is specifically designed, by Barcelo and collaborators [BKLW01, BBdLL06], to study such discrete objects as graphs. The "discrete homotopy hypothesis" (DHH) is the assertion that the graph nerve functor N_\infty : Graph --> cSet defined in [CK24] induces an equivalence of homotopy theories between the discrete homotopy theory of graphs and the homotopy theory of cubical sets (a stand-in for the classical homotopy theory of spaces). One possible strategy towards proving the DHH is to first prove the "DHH for n-types". That is, to compare the homotopy n-types on both sides for finite n >= 0, which involves localizing at the class of n-equivalences (maps inducing isomorphisms on the first n+1, rather than all, homotopy groups). Once might then hope to prove the general case by some type of Postnikov tower convergence argument. In this talk, based on joint work with Chris Kapulkin [KM24], we will describe the abstract homotopical machinery that forms the basis for this proof strategy. More precisely, we will construct a model category structure on cSet where the weak equivalences are the cubical n-equivalences, and a companion fibration category structure on Graph whose weak equivalences are the discrete n-equivalences, and show that the graph nerve functor is an exact functor of fibration categories. The DHH for n-types then becomes a question about whether this functor is an equivalence of fibration categories. (NB: This question has been answered positively in a recent preprint [CK26] of Carranza and Kapulkin.)References:[BBdLL06] E. Babson, H. Barcelo, M. de Longueville, and R. Laubenbacher, Homotopy theory of graphs, J. Algebraic Combin. 24 (2006), no. 1, 31–44.[BKLW01] H. Barcelo, X. Kramer, R. Laubenbacher, and C. Weaver, Foundations of a connectivity theory for simplicial complexes, Adv. Appl. Math. 26 (2001), no. 1, 97–128.[CK24] D. Carranza and K. Kapulkin, Cubical setting for discrete homotopy theory, revisited, Compos. Math. 160 (2024), no. 12, 2856–2903.[CK26] D. Carranza and K. Kapulkin, Discrete homotopy hypothesis for n-types, 2026. preprint.[KM24] K. Kapulkin and U. Mavinkurve, Homotopy n-types of cubical sets and graphs, 2024. preprint.Title: The discrete homotopy hypothesis for n-typesAbstract: I will report on joint work with Chris Kapulkin [arXiv:2602.19293] making progress towards the discrete homotopy hypothesis by showing that the nerve functor from the category of graphs to the category of topological spaces induces an equivalence after localizing at n-equivalences. Our results imply that, given any topological space X and any non-negative integer n, there exists a graph G such that the discrete homotopy groups of G match the ordinary homotopy groups of X up to degree n. Moreover, our proof method allows us to compute many previously-unknown discrete homotopy groups, which we use to construct candidate models for the n-sphere in discrete homotopy theory.Title: Topological data analysis using discrete homologyAbstract: Persistent homology is a tool of Topological Data Analysis commonly used to detect the shape of data. When the data of interest comes from a metric space, for example a finite subset of R^n, the method is generally noise resistant. We show that this is not the case when the data fails the triangle inequality. To solve this issue, we propose a new method: persistence discrete homology. This method uses discrete cubical homology, which is a homology theory for simple undirected graphs. This allows us to take homology over a filtration of graphs rather than the filtration of simplicial complexes normally used. In this talk, we will introduce the classical method of persistent homology, discuss discrete cubical homology and how it can be used for persistence, and compare the two methods. We show that persistent discrete homology is better suited to analyze data not coming from metric spaces. This talk is based on joint work with Chris Kapulkin, and the corresponding paper can be found here: arxiv.org/html/2506.15020.Title: The discrete homotopy hypothesis for directed graphsAbstract: We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to as A-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an ∞-category, which we denote by DGra∞. Our main result shows that DGra∞ is equivalent to the ∞-category of spaces. This is joint work with Briony Eldridge, Sergei O. Ivanov, Shing-Tung Yau, and Mengmeng Zhang. Wednesday, Sept. 16Title: Homotopy and Homology of DigraphsAbstract: In the cycle of papers in 2012-2015, Grigor’yan, Lin, Muranov and Yau developed effective methods for constructing (co)homolgy theories on discrete sets. Usually in this approach objects of various categories of graphs are considered in framework of classical algebraic topology. In this talk we discuss homotopy and homology theories in digraph theory and relations between them. We also describe the transfer of these theories to other categories of graph theory.Title: Avalanche HomologyAbstract: We introduce a new homology theory for graphs and digraphs called avalanche homology, which is based on the Abelian sandpile model. This homology theory captures not just the structure of the graph but the dynamics of the graph. We prove some combinatorial results on avalanche homology for simple digraphs, and present some links and comparisons to other theories such as nerve complexes of circular arcs and burning homology. Finally we introduce a persistent variant of avalanche homology which captures the evolution of the dynamics.Title: Higher-order rich clubs and configuration models on general directed hypergraphsAbstract: Detecting structure in complex networks, especially those arising from physical systems, is a central problem across the sciences. One approach is via rich club analysis, which identifies important vertices using a centrality metric and measures whether those vertices are more tightly interconnected than expected by chance. While informative, this approach captures only pairwise interactions, missing out on higher-order ones known to shape the structure and function of many complex systems. We propose a hyper-rich club pipeline that asks whether central vertices are more tightly interconnected than expected by chance through hyperedges encoding higher-order interactions, which also enables the inclusion of important, often omitted, directional information. Its concrete construction depends on explicit choices the domain scientist fixes according to their research goals. Particular choices recover the existing rich club notions for graphs and undirected hypergraphs, and yield the first such notion for different versions of directed hypergraphs. We demonstrate that the pipeline recovers meaningful structure in data by studying networks of very different origins: connectomes, temporal networks of infectious spread, networks of poems, and the XGI hypergraph database, in each case detecting structure the standard graph rich club misses. This is joint work with J.P. Smith, C. Hacker, J. Lazovskis, F. Unger, K.M. Smith Thursday, Sept. 17Title: The $\times$-fundamental groupoid and its compatible graph homotopy theories. Abstract: We begin with the definition of the $\times$-fundamental groupoid in a very concrete way, with examples and pictures. Then we take a step back to consider $\times$-homotopy theory and discuss some of its major features. We will see that this offers an alternate way to define the fundamental groupoid, as $\times$-homotopy classes of walks, and that our original definition gives a $\times$-homotopy invariant. The original $\times$-homotopy theory has been modified in several ways, extending the class of weak equivalences (partially in an attempt to obtain a model structure, which does not exist in the original theory). We will look at a couple of these, one created with path objects and one with a simplicial complex called the Box complex, and see how they build off of the original theory. Despite these two extensions being independent and taking totally different directions, they both retain a close connection to where we started: the $\times$-homotopy fundamental groupoid is a homotopy invariant for both of these spin-off theories. Title: Cubical sets and directed graphsAbstract: Directed graphs admit a variety of homological and homotopical invariants, which capture different features of their directed structure. Understanding how these invariants are related is a natural and useful problem. In previous work, we proved a one-dimensional Hurewicz theorem connecting the r-fundamental groups of a directed graph, in the sense of Di–Ivanov–Mukoseev–Zhang, with the (1,0)-entry of Asao's magnitude–path spectral sequence.In this talk, I will explain another Hurewicz-type theorem for directed graphs. This theorem relates the higher homotopy groups introduced by Li–Wu–Yau–Zhang to the singular cubical homology groups studied by Grigor'yan–Jimenez–Muranov. The proof comes from a more general Hurewicz theorem for discrete homotopy groups of cubical sets. I will also compare our result with the Hurewicz theorem for cubical Kan complexes due to Carranza–Kapulkin–Tonks.This is based on joint work with Daisuke Kishimoto.Title: Quasi-flag manifolds and moment graphsAbstract: The equivariant rational cohomology of flag manifolds is closely related to the algebra of quasi-invariants and quasi-covariants, which are generalizations of invariant polynomials for finite reflection groups. In this talk, we will explain how to obtain a mod-$p$ cohomological decomposition (except for $p=2$) of a flag manifold along its moment graph, and how to generalize it to construct quasi-flag manifolds. These results are obtained by constructing rational models of the new spaces in terms of coaffine stacks — derived stacks introduced by Toën and Lurie as an algebro-geometric framework for rational homotopy theory — which are derived version of the ordinary varieties of quasi-invariants, obtained from the same construction in the $\infty$-category of coaffine stacks rather than in affine schemes.Title: Spectral sequences via presheavesAbstract: In this talk we consider the relative category of spectral sequences with E_r-equivalences, that is, maps which are quasi-isomorphisms on the r page. As a category it fails to be complete and cocomplete. This motivates the introduction of the category of linear whitness books, a presheaf category that contains the category of spectral sequences and enjoys better (homotopical) properties. We also use the presheaf approach to define two décalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying the prior use of the term décalage in connection with spectral sequences. This is joint work with Sarah Whitehouse.Title: Homotopy theories via the magnitude-path spectral sequenceAbstract: The terms 'discrete homotopy theory' and 'A-homotopy theory' refer to a particular homotopy theory for graphs introduced by Babson, Barcelo, de Longueville and Laubenbacher in 2004, and developed over the years since then by Kapulkin, Muranov and others. That theory admits a quantitative variant, due to Asao, in which one tracks the lengths of homotopies between maps, leading to an infinite hierarchy of homotopy theories for graphs. In this talk, I will report on an ongoing collaboration with Muriel Livernet and Sarah Whitehouse in which we seek presentations for various associated homotopy categories, using their techniques to construct model structures on categories of spectral sequences and multicomplexes. Title: Homotopy theory of hypergraphsAbstract: In their paper “A Homotopy Category for Graphs”, 2020, Chih and Scull develop a homotopy theory for the category of undirected graphs (perhaps with loops, without multiple edges) and strict graph morphisms. They introduce the concept of a “spider pair” to establish a concrete equivalent characterization of their homotopy relation and use it to define fold morphisms and stiff graphs. Basing upon their work, we will analyze in what sense this can be similarly done for the category of hypergraphs. Furthermore, we will examine the relationship between Chih and Scull's homotopy theory of graphs and the generalized homotopy theory of hypergraphs.Friday, Sept. 18Title: Topology in Pseudotopological Spaces: What, Why, and How Abstract: Pseudotopological spaces are a generalization of topological spaces which contain reflexive graphs, topological spaces, and metric spaces endowed with a privileged scale. First introduced by Choquet in 1948, they and several of their subcategories have recently been shown to admit a cornucopia of interesting algebraic invariants. We present an overview of results obtained in homotopy and homology for pseudotopological spaces, motivated by and with applications to TDA and homotopy theories for graphs. Title: Breaking Model Structures with GraphsAbstract: In this talk, we identify three failure mechanisms in a (discrete) homotopy theory that obstruct its realisation as a Quillen model structure. We first discuss each mechanism as a “moral obstruction,” then illustrate it with a formal example, using x-homotopy on graphs as a proof of concept. We subsequently show how these arguments extend to other categories of graphs and to other notions of graph homotopy. In particular, this framework recovers several known non-existence results for model structures, including the one for x-homotopy on simple graphs (Goyal–Santhanam), A-homotopy on reflexive graphs (Carranza–Kapulkin–Kim), and simplicial homotopy on simplicial sets (folklore). Title: Magnitude homology of geodetic graphsAbstract: Equipped with the shortest-path metric, a graph can be regarded as a metric space, and hence as an enriched category in the sense of Lawvere. This viewpoint allows us to apply category-theoretic techniques to the study of graphs. The magnitude of a graph, introduced by Leinster as an analogue of the Euler characteristic of finite categories, is one such invariant. It is a formal power series whose coefficients are categorified by Hepworth–Willerton’s magnitude homology.Computing magnitude homology is an intriguing but challenging problem. In this talk, we give a complete calculation of magnitude homology for geodetic graphs, i.e. graphs with unique shortest paths, using the derived-functor formulation developed by S. O. Ivanov and the speaker. This is based on joint work with S. Wakatsuki.