# An Introduction to Noncommutative Geometry

## Document Type

Presentation Abstract

## Presentation Date

3-8-2002

## Abstract

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

*f*_{1} (*x*_{1},...,*x*_{n})= ... *f*_{r}(*x*_{1},...,*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 *M*_{1},...*,M*_{n} such that

*f*_{1}(*M*_{1},...,*M*_{n})= ... =*f*_{r}(*M*_{1},...,*M*_{n})=0?

When *d* = 1 , solutions are elements of **R**^{n}. 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 *f*_{1},...,*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.

## 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

## Additional Details

Friday, 8 March 2002

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)