Research Fellows Directory
Professor John Stafford
University of Manchester
Nature is inherently noncommutative: applying operations (like rotations and reflections or even putting on one's shoes and socks) in different orders leads to different outcomes. A classic example from quantum physics is Heisenberg's Uncertainty Principle which after rescaling is codified by the equation qp - pq = 1. Noncommutative algebra seeks to understand these (and more sophisticated) algebraic concepts. For example, the Weyl Algebra or Algebra of Quantum Mechanics is the set of all ``polynomials" in the variables p and q, where we use the rule qp = 1 + pq to multiply polynomials.
My research interests are primarily in noncommutative algebra, most especially in noncommutative algebraic geometry: an area that has flourished recently with increasing applications to other areas of mathematics and physics. Algebraic geometry is one of the deepest areas of mathematics; it analyzes solutions of polynomial equations and their associated geometric constructs. For example, solutions of x^2 - y^3 = 0 in the plane define a curve called a cusp. Noncommutative algebraic geometry is concerned with the interaction between (projective) algebraic geometry and noncommutative algebra and is very successful in using algebraic geometry to understand noncommutative structures. A major problem that I and my students are working on is to classify and understand noncommutative projective surfaces and we have made recent substantial progress.