An Introduction to Noncommutative Geometry

Authors

Professor Adam Nyman, Pomona College and Algebra Candidate

Document Type

Presentation Abstract

Presentation Date

3-8-2002

Abstract

Given a system of polynomial equations (in n unknowns) with real coefficients,

f₁ (x₁,...,x_n)= ... f_r(x₁,...,x_n)=0

can we find all real d x d solutions, i.e. can we find all n-tuples of real d x d matrices M₁,...,M_n such that

f₁(M₁,...,M_n)= ... =f_r(M₁,...,M_n)=0?

When d = 1 , solutions are elements of Rⁿ. The set of all solutions is a geometric object called a variety. Algebraic geometry is the study of the interplay between the geometry of the variety and the nature of the polynomials f₁,...,f_r

When d >1, it is often true that MN ≠ NM for d x d matrices M and N, so in this case, our equations are "noncommutative". Is there still a bridge between the worlds of algebra and geometry? We describe recent efforts to make sense of the notion "noncommutative variety". We shall see that, while some important noncommutative varieties don't have any points, they can be embedded in slightly larger spaces which have enough points so that they can be understood geometrically.

Additional Details

Friday, 8 March 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

Recommended Citation

Nyman, Professor Adam, "An Introduction to Noncommutative Geometry" (2002). Colloquia of the Department of Mathematical Sciences. 114.
https://scholarworks.umt.edu/mathcolloquia/114