# On Ovoids, Cores, and Preserver Problems

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Marko Orel

University of Primorska; Institute of Mathematics, Physics, and Mechanics

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**Minisymposium:**
FINITE GEOMETRY

**Content:**
I will present two results from matrix theory (two preserver problems), which are closely related to (non)existence of ovoids in certain classical polar spaces. While in the first problem, the nonexistence of ovoids or spreads in polar spaces is just applied to deal with one of the key steps in the proof, the second problem is essentially equivalent to the (non)existence of ovoids. In both results several tools from graph theory that involves graph spectra, chromatic number of a graph, etc, are applied as well.