Topology of Algebraic Hypersurfaces

University Of Utah, Salt Lake City UT

Investigators

Abstract

DMS-0104727 Grigory Mikhalkin This project will study complex and real algebraic hypersurfaces from a topological perspective. In particular it will study the interrelation of their topological and algebraic properties. This subject recently got a new powerful tool, the so-called theory of amoebas. Mikhalkin will work on further developing this theory and will make use of amoebas for investigating algebraic varieties. Amoebas have already proved to be an extremely useful tool in several areas of Mathematics, including Algebraic Geometry, Topology and Complex Analysis. Earlier Mikhalkin used this tool in the proof of topological uniqueness of maximal arrangements of plane real algebraic curves (in the context of Hilbert's 16th problem). A part of the project is to obtain a generalization of this result in higher dimensions. The other part of the project is based on the same technique but will investigate complex algebraic hypersurfaces by constructing and analyzing certain singular Lagrangian fibrations over PL-complexes. Such fibrations give a hope to extract topology hidden in the Hodge structure of algebraic varieties. Algebraic varieties and, in particular, real algebraic varieties are fundamental mathematical objects naturally occuring in modern-day Physics, Optics, Mechanics and Robotics. For instance, all possible positions of a mechanical linkage (e.g. an arm of a robot) form a real algebraic variety. The question of topological classification of plane real algebraic curves (1-dimensional varieties) posed by Hilbert over 100 years ago is still open. A recent advance by Mikhalkin in this problem relied on a new mathematical notion, that of amoeba, introduced by Gelfand, Kapranov and Zelevinski. Algebraic amoebas are regions in space which are associated to varieties. These concave regions with "tentacles" and "vacuoles" resemble in shape biological amoebas.