Noncrossing Partitions of Marked Surfaces: A Generalization of Kreweras' Noncrossing Partitions
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.
The Stabilized Bounded N-Derived Category of an Exact Category: Relating Singularity Categories and Stable Categories of N-Complexes
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.
The Tangle Hypothesis: A Topological Approach to Link Invariants in Higher Dimensions
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.
Equivalent Conditions for the Weak Parabolic Harnack Inequality for Non-Local Dirichlet Forms
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.
Silting Reduction and Picture Categories in 0-Auslander Extriangulated Categories: A Framework for Studying Categorifications of Cluster Algebras
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.
A Categorical and Combinatorial Approach to Higher-Dimensional Diagrams and Their Applications
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.
Interpreting the Lattice of Dyck Paths as a Lattice of Subcategories Using ω-Torsion Pairs
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.
The Minimum Number of Maximal Independent Sets in Graphs with Given Order and Independence Number
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.
Categorifying Valuative Invariants of Polyhedra and Matroids via Categorical Convolution
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.
On the Two-Color Rado Number for a Specific Linear Equation
This research paper investigates the two-color Rado numbers for a specific class of linear equations, providing exact values in some cases and upper and lower bounds in others, thereby generalizing previous findings.
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