Program

Real algebraic geometry -- the geometry of varieties defined by systems of real polynomial equations -- is a classical subject presently encompassing many distinct lines of inquiry. This program will cover modern developments in real algebraic geometry and its applications emphasizing topological aspects of this subject and its relations to other fields of mathematics. These relations arise as real algebraic varieties appear naturally in various mathematical contexts and, in particular, in applied mathematics, and there continue to be important interactions with these subjects. Besides the traditional directions of topological classification of real algebraic varieties, we mean to focus on enumerative problems and relations to convex geometry via the theory of amoebas and tropical geometry. This will include many recent and notable advances in real algebraic geometry, as well as some of its most important open problems. Of particular emphasis will be the following topics. Real algebraic curves. (The pictures below are of two constructions of real plane curves exploiting convexity.) Real flexible (J-holomorphic) curves. Higher-dimensional real varieties. Amoebas. Tropical algebraic geometry. Real Solutions to systems of equations. (A survey on geometric aspects of this subject.) Algorithms for real algebra via semi-definite programming. Real algebraic geometry in applications.