Related papers: Random Latin squares and 2-dimensional expanders
We say that a $d$-regular graph is a $\gamma$-expander if for every not too large set of vertices $S$, there are at least $\gamma d |S|$ edges leaving $S$, and we say that a graph $G$ is $\gamma$-far from bipartite if at least $\gamma e(G)$…
We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions…
For any integers $n\geq 2$ and $1\leq r\leq n-1$ satisfying $3r\geq 2n-1$, we show that the expected volume fraction of a random degree $d$ complex submanifold of $\C\mathbb{P}^n$ of codimension $r$ where the bisectional holomorphic…
For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the…
High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of…
The joint spectral radius of a bounded set of $d \times d$ real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn…
Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S obtained by starting with the full (d-1)-dimensional skeleton of S and then adding each d-simplex independently with probability p=c/n. We compute an explicit…
The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the…
Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n \geq 1$, there exists a constant $\epsilon(n)>0$ such that for any $0 \leq k <n$ the $k$-th coboundary expansion…
We show that a simple scoring-based tie-breaking can help improve lower bounds for the expansion (aka isoperimetric number) of random regular graphs with small even degrees. Specifically, for degrees 4, 6 and 8, we show that, with high…
We show that every orthogonal polyhedron of genus at most 2 can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal…
A partial transversal $T$ of a Latin square $L$ is a set of entries of $L$ in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any…
Any non-split complex supermanifold is a deformation of a split supermanifold. These deformations are classified by group orbits in a non-abelian cohomology. For the case of a split supermanifold with no global nilpotent even vector fields,…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for…
This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We show the condition of local spectral expansion has several nice implications. For…
A $d$-dimensional annulus graph with radii $R_1$ and $R_2$ (here $R_2\ge R_1\ge 0$) is a graph embeddable in $\mathbb R^d$ so that two vertices $u$ and $v$ form an edge if and only if their images in the embedding are at distance in the…
We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the…
In this paper, we further explore the local-to-global approach for expansion of simplicial complexes that we call local spectral expansion. Specifically, we prove that local expansion in the links imply the global expansion phenomena of…
The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality. In the first part of the paper we concentrate…