Noncommutative geometry

Noncommutative geometry is a cool area of mathematics that looks at spaces in a new way. In regular geometry, we describe spaces using rules for how things work together. But in noncommutative geometry, these rules can change. Doing one step and then another might give a different answer than if we did them in the opposite order. This helps mathematicians study spaces that are too tricky for regular geometry.

One important part of noncommutative geometry is the study of operator algebras. These are groups of actions on special spaces called Hilbert spaces. They help us learn about the shape of noncommutative spaces.

A well-known example is the noncommutative torus, which became important in the 1980s. It helped create new ideas like noncommutative versions of vector bundles, connections, and curvature—things we usually use to study everyday shapes. This field shows how algebra and geometry can work together in amazing ways.

Motivation

Noncommutative geometry extends a key idea in mathematics: the link between spaces and the functions on them. Normally, functions on a space can be added and multiplied in a simple way—this creates a commutative algebra. For example, functions that describe a smooth shape can help us reconstruct that shape.

The goal of noncommutative geometry is to generalize this idea to situations where functions do not follow this simple rule—their multiplication is noncommutative. This means changing the order of multiplication can give a different result. By studying these more complex structures, mathematicians aim to create new kinds of "spaces" and better understand the relationship between algebra and geometry.

The field connects to physics, where ideas from noncommutative geometry help study symmetry in physical systems. It also has roots in older mathematical theories about how groups act on spaces.

Noncommutative C*-algebras, von Neumann algebras

The duals of non-commutative C*-algebras are often called non-commutative spaces. This idea comes from the Gelfand representation, which links regular C*-algebras to locally compact Hausdorff spaces. For any C*-algebra S, we can create a topological space Ŝ; more information can be found in the spectrum of a C*-algebra.

There is also a connection between measurable spaces and von Neumann algebras. When these algebras are not commutative, they are referred to as non-commutative measure spaces. This shows how noncommutative geometry extends ideas from regular geometry into more abstract settings.

Noncommutative differentiable manifolds

A smooth Riemannian manifold is a special kind of space studied in geometry. To understand its shape, mathematicians use something called a spectral triple.

This idea helps mathematicians study noncommutative Riemannian manifolds. These are special spaces that use a triple with an algebra, a Hilbert space, and an operator. Researchers are exploring many examples of these interesting spaces.

Noncommutative affine and projective schemes

In noncommutative geometry, mathematicians study spaces using special kinds of algebraic structures. Regular geometry uses commutative rings, where the order of multiplication does not matter. Noncommutative geometry uses associative rings, where the order can matter.

These structures help build "noncommutative affine schemes." They are like building blocks that can be combined to form more complex objects.

This idea also works in projective geometry. Mathematicians like Michael Artin and J. J. Zhang have developed ways to handle "noncommutative projective schemes." These share many properties with regular projective schemes. These ideas help explore new kinds of geometric spaces using algebra.

Invariants for noncommutative spaces

Some important questions in mathematics ask how we can take ideas from topology — which studies shapes and spaces — and use them for more abstract algebraic structures. A key person, Alain Connes, introduced a new way to study these structures using something called cyclic homology. This connects to another area called algebraic K-theory.

Researchers have also extended ideas about characteristic classes — numbers that help describe geometric properties — to new types of mathematical objects called spectral triples. This work uses advanced tools and helps mathematicians find important numbers that describe these structures. It builds on classic results like the Chern character.

Examples of noncommutative spaces

In quantum mechanics, scientists change the space that describes where particles are and how they move. In this new space, the usual rules do not work. This helps us understand how very small particles act.

Another example is the noncommutative torus. It is a twist on the normal circle shape and helps test ideas in this part of math. We also see examples in how leaves grow on stems, number patterns, and other areas that make these special spaces.

Connection

In the sense of Connes

A Connes connection is a way to study geometry using special math rules. These rules are different from the ones we use every day. It was created by Alain Connes and later expanded by Joachim Cuntz and Daniel Quillen.

This idea helps us understand complicated spaces. It does this by making a map that follows special guidelines. This map is like the one we use to study shapes in regular geometry. It lets mathematicians find new and interesting patterns in numbers and shapes.