Cyclic Ordering of Split Matroids: Proving a Special Case of Gabow's and Kajitani-Ueno-Miyano's Conjectures
Split matroids whose ground set can be partitioned into pairwise disjoint bases admit a cyclic ordering where each basis forms a contiguous interval.
Matroid Colorings of Komiya Covers with Applications to Discrete Geometry and Fair Division
This paper presents a novel theorem that generalizes the KKM theorem using matroid colorings, leading to new results in discrete geometry and fair division problems.
Solvable and Nilpotent Matroids: Exploring Their Realizability, Irreducible Decomposition, and Defining Equations
This paper introduces solvable and nilpotent matroids, exploring their properties like realizability and the irreducibility of their realization spaces, and delves into the challenge of defining their associated varieties using tools like Grassmann-Cayley algebra.
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.
Characterizing Real-Representable Matroids With Large Average Hyperplane-Size
Real-representable matroids with large average hyperplane-size exhibit a specific structure: either every hyperplane contains one of a small set of lines, or a large portion of the matroid is contained within a degenerate subset.
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