Eran Nevo
This thesis focuses on algebraic shifting and its applications to f-vector theory of simplicial complexes and more general graded posets. In particular, several approaches and partial results concerning the g-conjecture for simplicial spheres are presented here.
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Partial Algebraic Shifting
本文研究外代数中非一般矩阵下的一致超图与单纯复形部分代数移位，回答了 Kalai 提出的若干问题，并给出保持 Betti 数的充分条件且证明其精确性。
An Empirically Fast Las Vegas Algorithm for Algebraic Shifting
本文提出了一种高效的拉索瓦算法，用于在正特征性下计算超图和单纯复形的外代数位移。
Volume rigidity and algebraic shifting
本文研究单纯复形的体积刚性，通过外移位识别其刚性，并证明高维体积刚性不具超图稀疏刻画。
Matroids arising from algebraic shifting
本文研究了通过代数移位得到的matroid结构，揭示了移位超图与matroid基集的关系，并给出了相关matroid的分类和例子。
In the present paper, exterior algebraic shifting and symmetric algebraic shifting of bipartite graphs and chordal graphs are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that, for $a>3$ and $b>3$, the exterior algebraic shifted graph of the complete bipartite graph $K_{a,b}$ of size $a,b$ is different from the symmetric algebraic shifted graph of $K_{a,b}$. Second, we will show that the exterior algebraic shifted graph of any chordal graph $G$ is coincident with the symmetric algebraic shifted graph of $G$.
The typical algebraic shifting of a surface
本文研究随机图加细与随机曲面 Delaunay 三角剖分的外代数移位，证明其典型极限由 Betti 数或亏格决定，并给出边收缩普适性工具。
A family $\mathcal{A}$ of sets is {\it $t$-intersecting} if the cardinality of the intersection of every pair of sets in $\mathcal{A}$ is at least $t$, and is an {\it $r$-family} if every set in $\mathcal{A}$ has cardinality $r$. A well-known theorem of Erdős, Ko, and Rado bounds the cardinality of a $t$-intersecting $r$-family of subsets of an $n$-element set, or equivalently of $(r-1)$-dimensional faces of a simplex with $n$ vertices. As a generalization of the Erdős-Ko-Rado theorem, Borg presented a conjecture concerning the size of a $t$-intersecting $r$-family of faces of an arbitrary sim…
We try to understand the behavior of exterior algebraic shifting with respect to basic constructions on simplicial complexes, like union and join. In particular we give a complete combinatorial description of the shifting of a disjoint union, and more generally of a union along a simplex, in terms of the shifting of its components. As a corollary, we prove the following, conjectured by Kalai: $Δ(K \cup L) = Δ(Δ(K) \cup Δ(L))$, where $K,L$ are complexes, $\cup$ means disjoint union, and $Δ$ is the exterior shifting operator. We give an example showing that replacing the operation 'union' with t…
Recently, Nevo introduced the notion of strongly edge decomposable spheres. In this paper, we characterize the algebraic shifted complex of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley--Reisner ideal of Gorenstein* complexes having the strong Lefschetz property in characteristic 0.
In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let $σ$ and $τ$ be simplicial complexes and $σ* τ$ their join. Let $J_σ$ be the exterior face ideal of $σ$ and $Δ(σ)$ the exterior algebraic shifted complex of $σ$. Assume that $σ* τ$ is a simplicial complex on $[n]=\{1,2,...,n\}$. For any $d$-subset $S \subset [n]$, let $m_{\preceq_{rev} S}(σ)$ denote the number of $d$-subsets $R \in σ$ which is equal to or smaller than $S$ w.r.t. the reverse lexicographic order. We will prove that $m_{\preceq_{rev} S}(Δ({σ* τ}))\geq m_{\preceq_{…
Gluck (1975) has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that linklessly embeddable graphs are generically 4-stress free. Both of these results are corollaries of the following theorem: every K_{r+2}-minor free graph is generically r-stress free for 0<r<5. (This assertion is false for r>5.) We give an equivalent formulation of this theorem in the language of symmetric algebraic shifting and show that its analogue for exterior algebraic shifting also holds. Some further extensions are detailed.
It is shown that conditions stronger in a certain sense than color-shifting cannot be placed on the class of colored complexes without changing the characterization of the flag f-vectors.
A recent framework for generalizing the Erdos-Ko-Rado Theorem, due to Holroyd, Spencer, and Talbot, defines the Erdos-Ko-Rado property for a graph in terms of the graph's independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erdos-Ko-Rado property to an arbitrary simplicial complex. An advantage of working in simplicial complexes is the availability of algebraic shifting, a powerful shifting (compression) technique, which we use to verify a conjecture of Holroyd and Talbot in the case of sequentially Cohen-Maca…
We show that the $h$-vector of a $2$-dimensional PS ear-decomposable simplicial complex is a pure $\mathcal{O}$-sequence. This provides a strengthening of Stanley's conjecture for matroid $h$-vectors in rank $3$. Our approach modifies the approach of combinatorial shifting for arbitrary simplicial complexes to the setting of $2$-dimensional PS ear-decomposable complexes, which allows us to greedily construct a corresponding pure multicomplex.
Given an infinite field $\mathbb{k}$ and a simplicial complex $Δ$, a common theme in studying the $f$- and $h$-vectors of $Δ$ has been the consideration of the Hilbert series of the Stanley--Reisner ring $\mathbb{k}[Δ]$ modulo a generic linear system of parameters $Θ$. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of $h_{d-1}^\mathfrak{a}(Δ)$, the dimension over $\mathbb{k}$ in degree $d-1$ of $\mathbb{k}[Δ]/(Θ)$, for any complex $Δ$ of dimension $d-1$. I…
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