Homotopy is the study of geometric regions through examining paths that can be drawn within the region.
Given two paths with common endpoints, if one path can be continually deformed into the other without changing the endpoints and remain in the defined region, the two paths are homotopic.
Homotopy has been applied to computational methods for algebraic equations. Specifically, Homotopy continuation is used in systems where a Homotopy is formed between two polynomial systems and the isolated solutions of one are continued onto the other. In practice continuation is done using numerical predictor-corrector methods, such as an ODE predictor method like Runge-Kutta and correction with Newton-Raphson. Homotopy continuation theoretically guarantees the computation of all the solutions to a given polynomial system, while in practice all solutions may not be guaranteed.
I hope this post influences you to learn more about Homotopy. Thank you for reading!
A fiber bundle is a space that is locally a product space but may have different global topological structure.
In simple terms, fiber bundles provide a convenient way to take products of topological spaces. This can be used to build complex spaces from simpler spaces.
The Möbius strip is a nontrivial bundle over the circle.
Another nontrivial bundle is the Klein bottle, which can be viewed as a “twisted” circle bundle over another circle.
Fiber bundles are used in physics to represent Gauge theories and constrained vector fields. More specifically, fiber bundles are important global structures in physical fields and are thus relevant to Gauge theories in electricity and magnetism, quantum electrodynamics, quantum chromodynamics and consequently Yang-mills theory.
I hope this post inspires you to learn more about these fascinating topological structures. Thank you for reading!
Knot theory is a field in topology that involves the mathematical study of knots. A mathematical knot is a topological embedding of a circle, which is similar to the conventional notion of knots. The biggest difference between a mathematical knot and conventional knot is that mathematical knots are closed.
Knot theory has applications in encrypted systems, chemical graph theory, molecular biology, statistical mechanics and quantum physics.
In the context of cryptography, cryptographic protocols can be designed based on the difficulty of decomposing complex mathematical knots. …