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This talk is about the asymptotic geometry of algebraic varieties defined by random systems of polynomial equations of a large degree N tending to infinity. Our theme is that there are statistical patterns in the zeros of polynomial systems which become more pronounced as the degree gets large (they are already visible by degree 10 or so). Here are two examples of such patters.

(i) Consider a random polynomial system of m equations in m variables, so that the joint zeros (intersection points of the m varieties) forms a discrete set. It becomes denser as N -> infinity, but if we magnify the picture by $\sqrt{N}$, the zeros become spaced apart by one unit on average. Do the zeros tend to repel like electrical particles? clump together like gravitating particles? or ignore each other like a neutral gas particles? The answer turns out to depend on $m$, reflecting the geometry of the discriminant variety.

(ii) Now constrain the Newton polytopes P of the polynomials. How does this affect the positions of their zeros? It turns out that for typical systems of m equations in m variables, the zeros concentrate in the inverse image of P under the moment map of projective space, and are sparse outside of it.

The heart of the methods involve semiclassical analysis applied in a setting of algebraic geometry. We hope to convey both the analytical and the algebro-geometric aspects of the topic to non-experts in each.