Random Algebraic Geometry and Random Amoebas

Abstract: Random algebraic geometry studies variable properties of typical algebraic varieties as opposed to invariant properties or extremal properties. For instance, a complex algebraic projective plane curve is always topologically connected, which is an invariant property; a real algebraic projective plane curve of degree $d$ has, by a classical theorem of Harnack, at most $\displaystyle{g+1=(d-1)(d-2)/2+1}$ connected components where $g$ denotes genus, which is an extremal property; whereas a random real algebraic projective degree $d$ plane curve in a suitable precise sense (to be explained in the talk) has an expected number of connected components of order $d$. In this talk, I will first present the setup and some of the main known results of the field of random algebraic geometry. I will then proceed to discuss some of our results on the expected properties of amoebas of random complex algebraic varieties, based on a joint work with Turgay Bayraktar, and another joint work with Jean-Yves Welschinger.

algebraic geometry

Audience: researchers in the discipline

ODTU-Bilkent Algebraic Geometry Seminars