Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

Zeev Dvir
Institute for Advanced Study
October 19, 2010

A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that for $m\geq n$, the rank of any $(q,k,t)$-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least $n - (qtn/2k)^2$ .

Using this result we derive the following applications:

Invariants of Graphs, Their Associated Clique Complexes and Right-Angled Coxeter Groups

Michael Davis
The Ohio State University; Member, School of Mathematics
October 19, 2010

Associated to any simplicial graph there is a right-angled Coxeter group. Invariants of the Coxeter group such as its growth series or its weighted L^2 Betti numbers can be computed from the graph's clique complex (i.e., its flag complex).

Metaphors in Systolic Geometry

Larry Guth
University of Toronto; Institute for Advanced Study
October 18, 2010

The systolic inequality says that if we take any metric on an n-dimensional torus with volume 1, then we can find a non-contractible curve in the torus with length at most C(n). A remarkable feature of the inequality is how general it is: it holds for all metrics.

A Unified Framework for Testing Linear-Invariant Properties

Arnab Bhattacharyya
Massachusetts Institute of Technology
October 18, 2010

In a sequence of recent papers, Sudan and coauthors have investigated the relation between testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Linear-invariance is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties.