What is the most singular possible singularity? What can we say about its geometric and algebraic properties? This seemingly naive question has a sensible answer in characteristic p. The ``$F$-pure threshold", which is an analog of the log canonical threshold, can be used to ``measure" how bad a singularity is. The $F$-pure threshold is a numerical invariant of a point on (say) a hypersurface---a positive rational number that is $1$ at any smooth point (or more generally, any F-pure point) but less than one in general, with ``more singular" points having smaller $F$-pure thresholds. We explain a recently proved lower bound on the F-pure threshold in terms of the multiplicity of the singularity. We also show that there is a nice class of hypersurfaces--which we call ``Extremal hypersurfaces"---for which this bound is achieved. These have very nice (extreme!) geometric properties. For example, the affine cone over a non Frobenius split cubic surface of characteristic two is one example of an ``extremal singularity". Geometrically, these are the only cubic surfaces with the property that {\bf every} triple of coplanar lines on the surface meets in a single point (rather than a ``triangle" as expected)--a very extreme property indeed.

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