Quotient Spaces, Tori and Finance

The quotient space of a topological space is formed by partitioning the space into equivalence classes based on some specified equivalence relation. Basically, you are bucketing your space by classes, and the elements of each class are equivalent (in a precise way though).

??One common way to visualize this is by considering the torus.

?? The torus can be thought of as a quotient space because it is formed by taking a rectangular region in the plane and identifying opposite sides in a specific way. This identification process leads to a geometric object that has a different topology than the original rectangular region. In the case of the torus, the opposite sides of the rectangle are glued together in a way that creates a surface with a single hole - the characteristic shape of a torus.

??Mathematically speaking, the torus is the result of taking a flat square in the plane and identifying points on the top and bottom edges as well as points on the left and right edges. This process of identification creates a surface that wraps around itself and forms a closed, compact shape with a hole in the middle - the torus.

??By defining this identification process, we are essentially creating equivalence classes of points on the rectangle based on the identified edges. The resulting space, which is the torus in this case, is then a quotient space because it is formed by partitioning the original rectangle into these equivalence classes based on the identified points. This partitioning and gluing process characterize the torus as a quotient space with distinct topological properties.

?? Think about it: when you are classifying "equivalent" elements (as we do e.g. in machine learning and artificial intelligence), you might be quotienting your original space, resulting in a quotient space with very interesting properties! ??

Perhaps without having noticed it, quotient spaces really are abundant in your environment! In particular, in quantitative finance, the concept of quotient spaces and periodic functions (T periodic functions are elements of the torus IR/TZ, do you see why?) can have various applications:

1. Time Series Analysis: Financial data often exhibit periodic patterns, such as daily, weekly, or seasonal fluctuations. By utilizing the concept of quotient spaces, analysts can effectively model and analyze these periodic trends in time series data to make informed decisions.
2. Fourier Analysis: Periodic functions play a crucial role in Fourier analysis, which is widely used in quantitative finance for signal processing and modeling financial data. Understanding quotient spaces can provide insights into the mathematical foundations of Fourier analysis and its applications in finance.
3. Options Pricing: Quotient spaces and periodic functions can be leveraged to develop more sophisticated models for options pricing. By incorporating periodicity and symmetries into option pricing models, analysts can better capture the complexities of financial markets and improve the accuracy of pricing derivatives.
4. Cryptocurrency and Blockchain: elliptic curves operate on finite fields, be isomorphic to the famous quotient space Z/pZ with p be a prime number. In addition, the halving process can be described through a pattern identified in Z/9Z (see Some Fundamentals of Mathematics of Blockchain: Amazon.co.uk: Riposo, Julien: 9783031313226: Books).

By exploring the connections between quotient spaces, periodic functions, and quantitative finance, researchers and analysts can enhance their understanding of market dynamics and develop more robust financial models for decision-making.

?Bonus: video games?

Have you ever played Sonic The Hedgehog 3? The world structure of Special Stages are... tori!

Can you guess the world structure in Super Mario Bros. Classic?

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