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Academy of Mathematics and Systems Science, CAS Colloquia & Seminars：Toric Algebras and Matroids
Toric代数 拟阵 复曲面代数
2023/11/13
In the first lecture I will give an introduction to toric algebras. A toric algebra can be seen as a monomial subalgebra, or as a quotient of a polynomial ring by a binomial prime ideal. We will discu...
We show that a certain class of varieties with origin in Physics,
generates (additively) the Denef-Loeser ring of Motives. In particular, this
disproves a conjecture of Kontsevich on the number of p...
One of the foundations of oriented matroid theory is the topological representation theorem of Folkman and Lawrence [8]. It says that an oriented (simple) matroid can be realized uniquely as an arrang...
For any linear quotient of a sphere, X = Sn−1/Γ, where Γ is an elementary abelian p–group, there is a corresponding Fp representable matroid MX which only depends on the isometry class of X. Whe...
Complete enumeration of small realizable oriented matroids
Complete enumeration small realizable oriented matroids Combinatorics
2012/4/18
Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensiona...
Sperner property and finite-dimensional Gorenstein algebras associated to matroids
Sperner property finite-dimensional Gorenstein algebras matroids Commutative Algebra
2011/9/21
Abstract: We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. We ...
A catalogue of small regular matroids and their Tutte polynomials
catalogue of small regular matroids Tutte polynomials Combinatorics
2011/8/29
Abstract: A catalogue of all non-isomorphic simple connected regular matroids ${\cal M}$ of cardinality $n \leq 15$ is provided on the net. These matroids are given as binary matrix matroids and are s...
Mutual information, matroids and extremal dependencies
Mutual information matroids extremal dependencies
2011/2/24
In this paper, it is shown that the rank function of a matroid can be represented by a “mutual
information function” if and only if the matroid is binary. The mutual information function
considered ...
On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids
hypergraphs associated geometric graphs matroids
2010/11/24
The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whos...
It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infini...
Sparse paving matroids, basis-exchange properties, and cyclic flats
Sparse paving matroids basis-exchange properties cyclic flats
2010/11/9
We provide evidence for five long-standing, basis-exchange conjectures for matroids by proving them for the enormous class of sparse paving matroids. We also explore the role that these matroids may p...
Characterizations of transversal and fundamental transversal matroids
Matroid transversal matroid fundamental transversal matroid
2010/12/8
A result of Mason, as refined by Ingleton, characterizes transversal matroids as the matroids that satisfy a set of inequalities that relate the ranks of intersections and unions of nonempty sets of c...
Triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ and Tropical Oriented Matroids
Triangulations $\Delta_{n-1} \times \Delta_{d-1}$ Tropical Oriented Matroids
2010/12/13
Develin and Sturmfels showed that regular triangulations of n1 d1 can be thought
as tropical polytopes. Tropical oriented matroids were de ned by Ardila and Develin, and were
...
Lattice-theoretic Characterization of Secret Sharing Representable Connected Matroids
cryptographic protocols secret sharing
2010/7/14
Necessary and sufficient conditions for a connected matroid to be secret sharing (ss-)representable are obtained. We show that the flat lattices of ss-representable matroids are closely related with w...
On codes, matroids and secure multi-party computation from linear secret sharing schemes
Multi-party computation multiplicative linear secret sharing schemes efficient error correction.
2009/3/27
Error correcting codes and matroids have been widely used in the study of ordinary secret
sharing schemes. In this paper, we study the connections between codes, matroids, and a special
class of sec...