Mathematicians build complex structures with low connectivity in high dimensions
Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree
Discrete Mathematics
Summary
The paper deals with creating special geometric objects called Cayley complexes, which are like networks built using algebraic rules. The authors found new ways to build these objects in higher dimensions that have relatively few connections but still maintain strong mathematical properties related to how tightly connected parts of the structure are. This could help in understanding complex systems and improving future mathematical constructions. They use ideas from algebra and geometry, like points on curves, to guide their designs.
Cayley complexhigh-dimensional expandereigenvaluesvertex-linkspectral normalgebraic curvefinite fielddegreeaffine function
Authors
Songtao Mao
Abstract
We construct an explicit infinite family of simple two-dimensional Cayley complexes over $\mathbb{F}_2^n$ whose degree is polynomial in $n$ and whose nontrivial vertex-link eigenvalues lie in $[-λ,λ]$ for every fixed $λ>0$. For every fixed $d\ge2$, we also obtain an explicit infinite family of weighted $d$-dimensional Cayley complexes over $\mathbb{F}_2^n$ with codimension-two local spectral norm at most $1/d$ and Cayley degree $Θ_d(n)$. Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.