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We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is…
In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we…
Given an $n$-gon, the poset of all collections of pairwise non-crossing diagonals is isomorphic to the face poset of some convex polytope called \textit{associahedron}. We replace in this setting the $n$-gon (viewed as a disc with $n$…
Friezes patterns are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their…
In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-injective objects to be a torsion class. By doing so, we propose a notion of $n$-hereditary categories. We also define and…
The Chern-Fulton class is a generalization of Chern class to the realm of arbitrary embeddable schemes. While Chern-Fulton classes are sensitive to non-reduced scheme structure, they are not sensitive to possible singularities of the…
Let $\rho: G \to \operatorname{GL}(V)$ be a rational representation of a reductive linear algebraic group $G$ defined over $\mathbb C$ on a finite dimensional complex vector space $V$. We show that, for any generic smooth (resp. $C^M$)…
In this paper we study triangular matrix categories using the theory of recollements of abelian categories. Given a triangular matrix category we construct two canonical recollements. We show that if certain funtors of these recollements…
Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander--Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with…
Given a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and an $\mathbb{E}_{\infty}$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{Fun}(M,…
Torsion theories play an important role in abelian categories and they have been widely studied in the last sixty years. In recent years, with the introduction of pretorsion theories, the definition has been extended to general…
Geiss, Leclerc and Schr\"oer introduced a class of 1-Iwanaga-Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. They also proved that…
Frieze patterns have attracted significant attention recently, motivated by their relationship with cluster algebras. A longstanding open problem has been to provide a combinatorial model for frieze patterns over the ring of integers modulo…
The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last…
To any nodal curve $C$ is associated the degree class group, a combinatorial invariant which plays an important role in the compactification of the generalised Jacobian of $C$ and in the construction of the N\'eron model of the Picard…
The combinatorial approach to knot theory treats knots as diagrams modulo Reidemeister moves. Many constructions of knot invariants (e.g., index polynomials, quandle colorings, etc.) use elements of diagrams such as arcs and crossings by…
Notions of guardedness serve to delineate the admissibility of cycles, e.g. in recursion, corecursion, iteration, or tracing. We introduce an abstract notion of guardedness structure on a symmetric monoidal category, along with a…
We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's…
Frieze patterns have been introduced by Coxeter in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values…
This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…