Related papers: Ramanujan Complexes and bounded degree topological…

In this paper we show that graphs of "neighbourly" cubical complexes -- cubical complexes in which every pair of vertices spans a (unique) cube -- have good expansion properties, using a technique based on multicommodity flows. By showing…

Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra).…

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

Reidl, S\'anchez Villaamil, and Stravopoulos (2019) characterized graph classes of bounded expansion as follows: A class $\mathcal{C}$ closed under subgraphs has bounded expansion if and only if there exists a function $f:\mathbb{N} \to…

We consider $d$-dimensional simplicial complexes which can be PL embedded in the $2d$-dimensional euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is…

We prove a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to…

We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly…

A notion of an $i$-banner simplicial complex is introduced. For various values of $i$, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary…

We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group…

Bounded cohomology of groups was first studied by Gromov in 1982. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic…

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the…

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale…

We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets…

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…

Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…

Let $\Gamma$ be a group of type $F_n$ and let $X$ be the $n$ skeleton of the universal cover of a $K(\Gamma,1)$ simplicial complex with finite $n$ skeleton. We show that if $\Gamma$ is strongly $n$-Kazhdan, then for any family of finite…

In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is…

In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m…

For every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic $\mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$ vertices that is $\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in…

Merge-width, recently introduced by Dreier and Toru\'nczyk, is a common generalisation of bounded expansion classes and twin-width for which the first-order model checking problem remains tractable. We prove that a number of basic…