Gregory Mikhalkin

Title: "Geometry of holomorphic amoebas and real algebraic geometry"

The amoeba of a holomorphic curve in the plane with (complex) coordinates x and y is its image under the map (x,y) -> (log|x|,log|y|). This name (given to such objects by Gelfand, Kapranov and Zelevinski) is due to the shape of such objects, they have "tentacles" and "vacuoles" responsible for the algebraic properties of the curve. In a more general setup, the amoeba map of a hypersurface in a toric variety is the restriction of the moment map to this hypersurface. For a curve in a surface it is a map of a complex curve to a real plane and its folds carry a lot of information. If the holomorphic curve is defined over real numbers then the real locus is contained in the folds. It turns out that amoebas of real curves are generically pinched (there are whole circles mapped to a point under the amoeba map and the pinching persists under real perturbations).

Amoebas provide a new tool for the study of topology of real algebraic curves of given degree (the first part of Hilbert's 16th problem). In my talk I will use amoebas to get new restrictions on the position of ovals of a real algebraic curve of given degree. If time permits I will also discuss the applications for real surfaces, 3-folds and higher-dimensional varieties.