This paper characterizes a specific type of vertex coloring called "ordered Szlam colorings" in Euclidean spaces, providing necessary and sufficient conditions for a coloring to be classified as such.
This paper introduces a novel definition of noncrossing partitions for marked surfaces, generalizing Kreweras' classical construction, and explores the properties of the resulting noncrossing partition lattices, including their lattice structure, rank function, and lower intervals.
This research paper generalizes Buchweitz's Theorem, which relates singularity categories and stable categories of modules over Gorenstein rings, to the setting of N-complexes over exact categories.
This research paper proves the 1-dimensional Tangle Hypothesis, which provides a topological framework for constructing link invariants in any dimension, generalizing the Reshetikhin-Turaev invariants for framed links in 3-dimensional space.
This paper establishes the equivalence of various conditions to the weak parabolic Harnack inequality for general regular Dirichlet forms without a killing part, offering a significant advancement in the understanding of non-local operator behavior.
This paper introduces a generalized silting reduction technique for extriangulated categories and uses it to define picture categories for 0-Auslander exact dg categories, offering a new approach to understanding categorifications of cluster algebras.
This book introduces a novel framework for understanding higher-categorical diagrams, using the combinatorics of regular directed complexes and their morphisms to provide a powerful and flexible tool for reasoning about higher categories and their applications.
This note demonstrates a novel interpretation of the lattice of Dyck paths as a lattice of subcategories, specifically ω-torsion pairs, within the context of representation theory of finite-dimensional algebras.
This research paper investigates the minimum number of maximal independent sets (MIS) in trees and unicyclic graphs when both the order (number of vertices) and independence number (size of the largest independent set) are fixed.
This paper introduces the concept of "categorical valuative invariants" for polyhedra and matroids, which elevates traditional numerical invariants to exact sequences in additive categories, offering a deeper understanding of valuativity and enabling computations that respect matroid symmetries.