**YouTube recording:** https://www.youtube.com/watch?v=2ga1XlPid9c

**Time: **Thursday, Mar 16, 3pm ET (beware of U.S./Canada time change on Mar 12!)**Zoom: **https://uwaterloo.zoom.us/j/94977150195

pwd=LzRHN3NVcWtGbWlOVGNmakJQenF1Zz09**Passcode:** kroeker1**Speaker:** Matthew Kroeker, University of Waterloo**Title: **Unavoidable Flats in Matroids Representable over a Prime Field**Abstract: **The Sylvester-Gallai Theorem says that every rank-3 real-representable matroid contains a two-point line. A high-dimensional generalization of this result, due to Hansen, implies that every simple real-representable matroid of sufficiently high rank contains a rank-k independent flat. One only needs to look as far as the binary projective plane to see that such a result cannot hold for the class of matroids representable over a finite field. In light of this, we ask whether it is possible to determine a small list of “unavoidable” rank-k flats guaranteed to exist in a simple GF(q)-representable matroid of sufficiently high rank. We will answer this question for the case of prime fields: in particular, we show that, for any prime p and positive integer k, any simple GF(p)-representable matroid of sufficiently high rank contains a rank-k flat which is either independent, or is a projective or affine geometry. This is joint work with Jim Geelen.