# Noncommutative algebraic topology

Picture the spaces you know from everyday life: a grid of points or continuous lines. These are commutative spaces. These spaces include spheres, circles, Euclidean space, Hilbert space, Hausdorff space, etc. For any two continuous functions f and g on the space, the product f(x)g(x) is equal to the product g(x)f(x) for all points x in the space. Commutative algebra provides a way to describe the algebraic structure of commutative spaces. Now, think of spaces that don’t follow these familiar rules. These are noncommutative spaces. They are described by noncommutative algebra. In these spaces, the way we multiply elements together doesn’t always follow the usual order. This quirk leads to intriguing and often intricate algebraic structures. To make sense of the maths in these unconventional spaces, mathematicians use several tools, one of them is noncommutative algebraic topology. Topology deals with the properties of objects that are stretched, twisted, and bent. For example, a square and a circle are topologically equivalent because they can be continuously deformed into each other.

## What is Noncommutative Algebraic Topology

Noncommutative algebraic topology is a branch of mathematics that unites noncommutative algebra, which examines algebraic structures in which the order of element multiplication is not always commutative, with algebraic topology, which investigates topological spaces and their invariants.

## Examples of Noncommutative Spaces

• Noncommutative Matrix Algebras: Matrices, which are arrays of numbers, behave in a way where the order of multiplication matters. This noncommutative property is crucial in quantum physics and control theory.
• Noncommutative Quantum Mechanics: In quantum physics, properties like position and momentum don’t follow the usual rules of multiplication. This noncommutative behaviour is fundamental in understanding the behaviour of tiny particles.
• Noncommutative Function Spaces: Imagine spaces where mathematical functions don’t multiply like regular numbers. These spaces help us model complex systems in mathematics and science.
• Quantum Groups: These are like mathematical groups, but their multiplication doesn’t follow the usual rules. Quantum groups are used in physics and algebra for various applications.
• Noncommutative Geometry in String Theory: In the world of string theory, even space itself can act noncommutatively at very small scales. This concept helps us explore the fundamental nature of the universe.
• Noncommutative Torus: Picture a unique shape like a doughnut, where the way you multiply things together doesn’t follow standard rules. This is a way of exploring geometry with unconventional math.
• Noncommutative Fractals: Think of intricate and irregular patterns, like snowflakes. Noncommutative geometry helps us understand these complex shapes when traditional geometry falls short.
• Quantum Plane: This is a two-dimensional noncommutative space where the usual rules of multiplication don’t apply. It’s used in quantum mechanics and mathematical physics to study noncommutative phenomena at the quantum level.

## Applications of Noncommutative Algebraic Topology

So, why should you care about noncommutative algebraic topology in your everyday life? Well, here’s the exciting part: these maths isn’t just for the textbooks. It has real-world applications, too! For instance, it plays a role in improving communication networks and even helps solve complex problems in physics.

### Quantum Field Theory

Noncommutative algebraic topology helps us understand how tiny particles interact at the quantum level.

### Quantum Gravity

It also aids in figuring out how gravity works at the smallest scales.

### String Theory

By connecting maths and string theory, it delves into the fundamental nature of the universe.

### Quantum Mechanics

It’s handy for figuring out how things behave at the tiniest levels in quantum mechanics. It helps us understand how really tiny things, like atoms and particles, behave.

### Statistical Mechanics

It helps us understand complex physical systems and how they work.

### Interdisciplinary Maths

It connects with other maths areas like differential geometry and representation theory to give us a broader view of mathematical physics.

### Information Theory

In information theory, it helps us understand how data moves around, making communication systems better.

### Quantum computing

It helps us with quantum computing and keeping our data safe with quantum cryptography.

### Cryptography

It helps us makes sure our codes and secrets stay secure, so no one can snoop on our messages.

### Control Theory

In control theory, it helps us make sure that things work smoothly and don’t go haywire. It’s also a big help in the world of quantum technology.

In conclusion, noncommutative algebraic topology helps us understand these unique spaces and teams up with quantum mechanics to solve scientific mysteries. Plus, it has real-world applications that impact our daily lives. As this field continues to evolve, who knows what exciting discoveries lie ahead?

## References

https://gow.epsrc.ukri.org/NGBOViewGrant.aspx?GrantRef=EP/I026703/1

https://arxiv.org/pdf/math/0305165.pdf

https://www.maths.ed.ac.uk/~v1ranick/books/nlat.pdf