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                * Princeton Discrete Math Seminar *
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Date: Thursday 12th February, 3:00 in Fine Hall 224.
 
Speaker: Guangda Hu (Princeton)

Title: Sylvester-Gallai for arrangements of subspaces

In this work we study arrangements of k-dimensional subspaces V_1,...,V_n 
over the complex numbers. 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 any pair of subspaces intersect only at
zero  (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or
Kelly's theorem for complex numbers), which proves the k=1 case.  One of the
main ingredients in the proof is a strengthening of a Theorem of Barthe  (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 

Joint work with Zeev Dvir.

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