Rutgers Discrete Mathematics Seminar

Title: Sylvester-Gallai for Arrangements of Subspaces

Speaker: Guangda Hu, Princeton University

Date: Monday, February 9, 2015 11:00 am

Location: CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ


In this work we study arrangements of k-dimensional subspaces V_1,...,V_n subset C^ell. Our main result shows that, if every pair V_a,V_b of subspaces is contained in a dependent triple (a triple V_a,V_b,V_c contained in a 2k-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that V_a cap V_b = {0} for every pair (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the k=1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. [BDWY-pnas]. One of the main ingredients in the proof is a strengthening of a Theorem of Barthe [Bar98] (from the k=1 to k>1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem). Joint with Zeev Dvir.