**abstract:**
The Sylvester-Gallai theorem states that a configuration of points in
R^{n} in which every pair is in some collinear triple, has to lie on a
single line. I will discuss some recent results that go beyond this
theorem, dealing with more robust scenarios. The first type of results
deal with configurations in which **many** pairs are in collinear
triples. For example: If a constant fraction of the pairs are in a
collinear triple then there is a constant fraction of the points that
lie in a subspace of constant dimension. Another type of results deal
with configurations in which there are many **almost**, or
epsilon-collinear triples (that is, points that are epsilon-close to
being on a line). We will see that, under some uniformity conditions
on the distances between points, one can find a subspace of low
dimension that approximates all (or most) of the points. All of these
results, which deal with points in complex space, are obtained by
understanding the rank (or, more generally, the number of small
singular values) of sparse complex matrices with specific patterns of
zeros*non-zeros.
*

*Based on joint works with Albert Ai, Boaz Barak, Subhangi Saraf, Avi Wigderson and Amir Yehudayoff.
*

Mon 3 Sep, 10:30 - 11:30, Aula Dini

Tue 4 Sep, 14:00 - 15:00, Aula Dini

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