Given a finite collection of probability measures defined on subsets of a measurable space, how can we determine if they are compatible, in the sense that they can be realized as conditional distributions of a single probability measure on the full space? This formulation of the consistency problem for conditional probabilities is significant in Bayesian epistemology and probabilistic reasoning, as it describes the conditions under which a collection of agents can reach agreement by sharing information. We derive a necessary and sufficient condition under which joint compatibility is equivalent to pairwise compatibility. This condition is stated in terms of the cohomology of a simplicial complex constructed from the given probability measures, exposing a novel application of algebraic topology to Bayesian reasoning.

For a filtered simplicial complex $K$ embedded in $\mathbb{R}^{d+1}$, the merge tree of the complement of $K$ induces a forest structure on the persistent homology $H_d(K)$ via Alexander duality. We prove that the connected components of $\mathbb{R}^{d+1}\setminus K_r$ correspond to representative cycles for a basis of $H_d(K_r)$ which are volume-optimal. By keeping track of how these representatives evolve with the filtration of $K$, we can equip each interval $I$ in the barcode of $H_d(K)$ with a sequence of canonical representative cycles. We develop and implement an efficient algorithm to compute the progression of cycles in time $\mathcal{O}((\#K)^2)$. We apply functionals to these representatives, such as path length, enclosed volume, or total curvature. This way, we obtain a real-valued function for each interval, which captures geometric information about~$K$. Deriving from this construction, we introduce the \emph{generalized persistence landscapes}. Using the constant one-function as the functional, this construction gives back the standard persistence landscapes. Generalized landscapes can distinguish point clouds with similar persistent homology but distinct shape, which we demonstrate by concrete examples.

In a $k$-critical bifiltration, every simplex enters along a staircase with at most $k$ steps. Examples with $k>1$ include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a $k$-critical bifiltration into a $1$-critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations. We present two algorithms. The first one computes free resolutions corresponding to path graphs and assembles them to a chain complex by computing additional maps. The simple combinatorial structure of path graphs leads to good performance in practice, as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because long paths might yield dense boundary matrices in the output. Our second algorithm replaces the simplex-wise path graphs with ones that maintain short paths which leads to almost linear runtime and output size. We demonstrate that pre-computing a free resolution speeds up the task of computing a minimal presentation of the homology of a $k$-critical bifiltration in a fixed dimension. Furthermore, our findings show that a chain complex that is minimal in terms of generators can be asymptotically larger than the non-minimal output complex of our second algorithm in terms of description size.

We introduce a general theory of homological Milnor-Witt cycle modules over an excellent base scheme equipped with a dimension function, extending both Rost's cycle modules and Feld's theory over fields. To any such module we associate a Rost-Schmid type complex whose homology defines a Borel-Moore intersection theory with quadratic coefficients, satisfying homotopy invariance, localization, proper pushforwards, smooth pullbacks, and Gysin morphisms for essentially smoothable lci morphisms. Using duality data induced by pinning structures, we define cohomological Milnor-Witt modules and establish a duality equivalence between homological and cohomological theories. As applications, we extend Chow-Witt groups to schemes over general (possibly singular or arithmetic) bases, prove generalized Bloch formulas and representability results, and compute graded Chow-Witt groups over Dedekind schemes of finite type over the integers. In particular, we obtain finiteness results for Chow-Witt and related Milnor-Witt invariants in dimension at most one.

Topological Hochschild homology is a topological analogue of classical Hochschild homology of algebras and bimodules. Beliakova, Putyra, and Wehrli introduced quantum Hochschild homology (qHH) and used it to define a quantization of annular Khovanov homology as qHH of the tangle bimodules of Chen-Khovanov and Stroppel. After introducing quantum topological Hochschild homology (qTHH), we construct a new stable homotopy refinement of quantum annular Khovanov homology and show that it agrees with qTHH of the spectral Chen-Khovanov tangle bimodules of Lawson, Lipshitz, and Sarkar. We also show that this new stable homotopy refinement recovers the construction introduced in earlier work of Krushkal together with the first and third authors.

As the size of data increase, persistence diagrams often exhibit structured asymptotic behavior, converging weakly to a Radon measure. However, conventional vector summaries such as persistence landscapes are not well-behaved in this setting, particularly for diagrams with high point multiplicities. We introduce continuous persistence landscapes, a new vectorization defined on a special class of Borel measures, which we call q-tame measures. It includes both the persistence diagrams and their weak limits. Our construction generalizes persistence landscapes to a measure-theoretic setting, preserving the intrinsic structure of persistence measures. We show that this vector summary is bijective and L^1-stable under mild assumptions, and that the original measure can be uniquely reconstructed. This approach gives a more faithful description of the shape of data in the limit and provides a stable, invertible way to analyze topological features in large systems.

We consider the action of the projective group $PGL(3,\mathbb{R})$ on the $n$-fold first-order jet space of point configurations on the plane. Using the method of moving frames, we construct an explicit complete generating set for the field of absolute first-order joint projective differential invariants $\mathcal{I}_{n,0}$ for any $n \ge 3$. This approach provides a unified construction for all $n$, immediately ensuring functional independence of the fundamental invariants and yielding formulas suitable for both symbolic and numerical implementation. Next, we study the field of relative first-order invariants $\mathcal{I}_n$ with Jacobian multiplier. It is shown that the invariantization of the Jacobian under the projective action yields a primitive element of the field extension $\mathcal{I}_n / \mathcal{I}_{n,0}$. Finally, we introduce a multiplicative cochain complex $C^\bullet$ associated with the action of $PGL(3,\mathbb{R})$ on the jet space, and show that the invariantization operator induced by the moving frame generates an explicit contracting homotopy. This provides a constructive proof of the vanishing of higher cohomology and an interpretation of the "defect" of invariantization as an exact cocycle in $C^\bullet$.

The concept of effective resistance, originally introduced in electrical circuit theory, has been extended to the setting of graphs by interpreting each edge as a resistor. In this context, the effective resistance between two vertices quantifies the total opposition to current flow when a unit current is injected at one vertex and extracted at the other. Beyond its physical interpretation, the effective resistance encodes rich structural and geometric information about the underlying graph: it defines a metric on the vertex set, relates to the topology of the graph through Foster's theorem, and determines the probability of an edge appearing in a random spanning tree. Generalizations of effective resistance to simplicial complexes have been proposed in several forms, often formulated as matrix products of standard operators associated with the complex. In this paper, we present a twofold generalization of the effective resistance. First, we introduce a novel, basis-independent bilinear form, derived from an algebraic reinterpretation of circuit theory, that extends the classical effective resistance from graphs. Second, we extend this bilinear form to simplices, chains, and cochains within simplicial complexes. This framework subsumes and unifies all existing matrix-based formulations of effective resistance. Moreover, we establish higher-order analogues of several fundamental properties known in the graph case: (i) we prove that effective resistance induces a pseudometric on the space of chains and a metric on the space of cycles, and (ii) we provide a generalization of Foster's Theorem to simplicial complexes.

Let $M$ be the disk or a compact, connected surface without boundary different from the sphere $S^2$ and the real projective plane $\mathbb{R}P^2$, and let $N$ be a compact, connected surface (possibly with boundary). It is known that the pure braid groups $P_n(M)$ of $M$ are bi-orderable, and, for $n\geq 3$, that the full braid groups $B_n(M)$ of $M$ are not bi-orderable. The main purpose of this article is to show that for all $n \geq 3$, any subgroup $H$ of $B_n(N)$ that satisfies $P_n(N) \subsetneq H \subset B_n(N)$ is not bi-orderable.

These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor formalisms; the 2-category of cohomological correspondences, and resulting simplifications in the proofs of Poincaré--Verdier duality results; the relation between 6-functor formalisms and ``geometric rings''; many examples of 6-functor formalisms, both old and new.

In this paper, we propose a method for constructing a colored $(d+1)$-operad $\mathbf{seq}_d$ in $\mathrm{Sets}$, in the sense of Batanin [Ba1,2], whose category of colors (=the category of unary operations) is the category $\Theta_d$, dual to the Joyal category of $d$-disks [J], [Be2,3]. For $d=1$ it is the Tamarkin $\Delta$-colored 2-operad $\mathbf{seq}$, playing an important role in his paper [T3] and in the solution this http URL. to the Deligne conjecture for Hochschild cochains. We expect that for higher $d$ these operads provide a key to solution to the the higher Deligne conjecture, in the (weak) $d$-categorical context. For general $d$ the construction is based on two combinatorial conjectures, which we prove to be true for $d=2,3$. We introduce a concept of a generalised Joyal disk, so that the category of generalised Joyal $d$-disks admits an analogue of the funny product of ordinary categories. (For $d=1$, a generalised Joyal disk is a category with a ``minimal'' and a ``maximal'' object). It makes us possible to define a higher analog $\mathcal{L}^d$ of the lattice path operad [BB] with $\Theta_d$ as the category of unary operations. The $\Theta_d$-colored $(d+1)$-operad $\mathbf{seq}_d$ is found ``inside'' the desymmetrisation of the symmetric operad $\mathcal{L}^d$. We construct ``blocks'' (subfunctors of $\mathcal{L}^d$) labelled by objects of the cartesian $d$-power of the Berger complete graph operad [Be1], and prove the contractibility of a single block in the topological and the dg condensations. In this way, we essentially upgrade the known proof given by McClure-Smith [MS3] for the case $d=1$, so that the refined argument is generalised to the case of $\Theta_d$. Then we prove that $\mathbf{seq}_d$ is contractible in topological and dg condensations (for $d=2,3$, and for general $d$ modulo the two combinatorial conjectures).

Graph Neural Networks (GNNs) have shown remarkable success across various scientific fields, yet their adoption in critical decision-making is often hindered by a lack of interpretability. Recently, intrinsically interpretable GNNs have been studied to provide insights into model predictions by identifying rationale substructures in graphs. However, existing methods face challenges when the underlying rationale subgraphs are complex and varied. In this work, we propose TopInG: Topologically Interpretable Graph Learning, a novel topological framework that leverages persistent homology to identify persistent rationale subgraphs. TopInG employs a rationale filtration learning approach to model an autoregressive generation process of rationale subgraphs, and introduces a self-adjusted topological constraint, termed topological discrepancy, to enforce a persistent topological distinction between rationale subgraphs and irrelevant counterparts. We provide theoretical guarantees that our loss function is uniquely optimized by the ground truth under specific conditions. Extensive experiments demonstrate TopInG's effectiveness in tackling key challenges, such as handling variform rationale subgraphs, balancing predictive performance with interpretability, and mitigating spurious correlations. Results show that our approach improves upon state-of-the-art methods on both predictive accuracy and interpretation quality.

For an étale groupoid, we define a pairing between the Crainic-Moerdijk groupoid homology and the simplex of invariant Borel probability measures on the base space. The main novelty here is that the groupoid need not have totally disconnected base space, and thus the pairing can give more refined information than the measures of clopen subsets of the base space. Our principal motivation is $C^*$-algebra theory. The Elliott invariant of a $C^*$-algebra is defined in terms of $K$-theory and traces; it is fundamental in the long-running program to classify simple $C^*$-algebras (satisfying additional necessary conditions). We use our pairing to define a groupoid Elliott invariant, and show that for many interesting groupoids it agrees with the $C^*$-algebraic Elliott invariant of the groupoid $C^*$-algebra: this includes irrational rotation algebras and the $C^*$-algebras arising from orbit breaking constructions studied by the first listed author, Putnam, and Strung. These results can be thought of as establishing a refinement of Matui's HK conjecture for the relevant groupoids.

We study Bott and Cattaneo's $\Theta$-invariant of 3-manifolds applied to $\mathbb{Z}\pi$-homology equivalences from 3-manifolds to a fixed spherical 3-manifold. The $\Theta$-invariants are defined by integrals over configuration spaces of two points with local systems and by choosing some invariant tensors. We compute upper bounds of the dimensions of the space spanned by the Bott--Cattaneo $\Theta$-invariants and of that spanned by Garoufalidis and Levine's finite type invariants of type 2. The computation is based on representation theory of finite groups.

We consider a twisting function from a 1-reduced simplicial set $X$ to a simplicial group $G$. We prove in detail that the associated Szczarba operators induce a simplicial map from the triangulation of the cubical cobar construction of $X$ to $G$. This confirms a result due to Minichiello-Rivera-Zeinalian and gives, as pointed out by these authors, a conceptual proof of the fact that the dga map $\Omega\,C(X) \to C(G)$ induced by Szczarba's twisting cochain is comultiplicative.

Reflexive dg categories were introduced by Kuznetsov and Shinder to abstract the duality between bounded and perfect derived categories. In particular this duality relates their Hochschild cohomologies, autoequivalence groups, and semiorthogonal decompositions. We establish reflexivity in a variety of settings including affine schemes, simple-minded collections, chain and cochain dg algebras of topological spaces, Ginzburg dg algebras, and Fukaya categories of cotangent bundles and surfaces as well as the closely related class of graded gentle algebras. Our proofs are based on the interplay of reflexivity with gluings, derived completions, and Koszul duality. In particular we show that for certain (co)connective dg algebras, reflexivity is equivalent to derived completeness.

Discrete exterior calculus offers a coordinate--free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present a wedge product on 2--dimensional pseudomanifolds, whose faces are any polygons. We prove that this polygonal wedge product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. We thus extend previously studied discretizations of wedge products from simplicial or quadrilateral meshes to surface meshes whose faces are arbitrary simple polygons. We also prove that our discrete wedge product corresponds to a cup product of cochains on 2--pseudomanifolds. By rigorously justifying our construction we add another piece to ever evolving discrete versions of exterior calculus.

We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field $k$. We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients.

Path homology plays a central role in digraph topology and GLMY theory more general. Unfortunately, the computation of the path homology of a digraph $G$ is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In this paper we introduce an inductive method of constructing elements of the path homology chain modules $\Omega_n(G;R)$ from elements in the proceeding two dimensions. This proceeds via the formation of what we call upper and lower \emph{extensions}, that are parametrised by certain labeled multihypergraphs which we introduce and call \emph{face multihypergraphs}. When the coefficient ring $R$ is a finite field the inductive elements we construct generate $\Omega_*(G;R)$. With integral or rational coefficients, the inductive elements generate at least $\Omega_i(G;R)$ for $i=0,1,2,3$. Since in low dimensions the inductive elements extended over labeled multigraphs coincide with naturally occurring generating sets up to sign, they are excellent candidates to reduce to a basis. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph $G$. We employ inductive elements to construct a sequence of digraphs whose path Euler characteristic can differ arbitrarily depending on the choice of field coefficients. In particular, answering an open question posed by Fu and Ivanov.