Gauge theory and algebraic geometry

Abstract

Gauge theory is the study of differential equations involving connections on vector bundles over manifolds using methods from differential geometry, algebraic geometry and geometric analysis. This is area of research emerged in the late 1970s, especially after works by Michael Atiyah, Nigel Hitchin and simon Donaldson, among others, as a mathematical model for classical field theory in mathematical physics, and had inumerous applications to more tradicional areas of mathematics, like differential topology, algebraic geometry, complex geometry, Riemannian geometry, geometric analysis and representation theory.At the core of the theory is a 1-1 correspondence between solutions of certain equations (e.g. the Yang--Mills anti-self-duality equation) and holomoprhic vector bundles with certain properties (e.g. stability). The research group in Gauge Theory and Algebraic Geometry of the Department of Mathematics at IMECC-UNICAMP focuses on the interaction between these two areas, also working on specific problems in each of them. (AU)