A fiber bundle expresses global phenomenon in terms of the output of local data.
Analogously, a geometric multi-scale modeling technique, aptly named fiber bundle dynamics, expresses macroscale data in terms of the output of microscale models.
For example: if the microscale model is molecular dynamics and the macroscale model is continuum hydrodynamics, then this formula is the Irving-Kirkwood formula that expresses stress in terms of the atomistic data from molecular dynamics. If the microscale model is replaced by Brownian dynamics, then this link is replaced by Kramer’s expression, etc.
“…in concurrent coupling methods, one does not compute the constitutive relation within the full range of these variables – only the values that actually occur in the simulation are needed, and these might be a very small subset of the entire range.”
— Ren, Seamless Multiscale Modeling of Complex Fluids via Fiber Bundle Dynamics
It’s interesting to think of (1) dependent types as fibrations and (2) a multi-scale model as a generalized transition system. Continue reading Some Thoughts on Dynamical Systems
Mathematics is a huge subject.
Category theory is one area of mathematics dedicated to exploring the commonality of structure between different branches of mathematics.
Categorical language allows us to ascend a layer of abstraction, and recognize the obvious underlying principles that guide seemingly unrelated concepts. Generality facilitates connections.
A category $C$ consists of:
The concept of composition follows naturally from the definition of path equivalence in graph theory: Two paths with the same source as destination are equal. Additionally, categories must satisfy the laws of associativity and identity.
Categories must obey 2 laws:
Two paths are equal if the source and destination of the paths are equal. With this in mind, we can represent the category laws of identity and associativity with diagrams.
Follow the arrows, recall that we write function composition backward! If we traverse $f$ then $g$, it is the convention to write $g \circ f$
Groups, together with group homomorphisms, form a category (we will discuss these next lecture for those who have not danced with abstract algebra).
Each of the natural numbers is a category:
Any property which can be expressed in terms of (category, objects, morphism, and composition):
An example of associating morphisms from $C$ to $D$ with the functor $F(C)$.
A few categories you have likely encountered before without recognizing it:
A categorical key to highlight some of the relationships between structure preserving maps:
Here are some more advanced examples for those who feel groovy and algebraic:
Already, criticism has be thrown at me for deciding to focus on pure mathematics. I am often told (with the best intentions) that I should go back to doing applied math/engineering. I’m guessing that other people who’ve decided to study theoretical subjects suffer from a similar lack of respect.
Hopefully, this post will be helpful for you, my friends. Perhaps, to send to your families, engineers, etc. to save you time when you are sick of answering the same questions — laced with misunderstanding. Perhaps, to find consolation and recognize that you are not alone.
This post (admittedly written with the muses of frustration and maths evangelism) can be generalized to theoretical endeavors of any flavor.
Why do I want to study pure math? The same reason that the scientist is providing above.
There is an itch in minds of the analytic and aesthetically inclined that mathematics scratches like no other.
I encourage you to read A Mathematician’s Apology and What is it Like to Understand Advanced Mathematics?.
It doesn’t matter if you’re good at mathematics or not if you enjoy doing it and have people willing to help you learn.
You’ve picked something that you enjoy doing, and you’re rolling with it.
Thereafter, if you do it for long enough, you’ll become good at it.
Because of its abstractness, mathematics is universal in a sense that other fields of human thought are not. It finds useful applications in business, industry, music, historical scholarship, politics, sports, medicine, agriculture, engineering, and the social and natural sciences. The relationship between mathematics and the other fields of basic and applied science is especially strong.
Here’s an exploration of Which Mathematical Ideas Have Done The Most To Change History?.
0. Maths develops the logical mindset, assiduous and aesthetic values needed to implement regular concise conversation and clear explanation. It teaches us to not accept hand-wavey proofs, but to derive them ourselves from first principles.
1. The ability to analyze a problem as a structure, work it out step by step and solve it generally translates between fields, allowing one trained in maths to easily move to other puzzle-based disciplines. For example, a maths person can transfer their skillset to computer programming by seeing code and programming languages as a collection of structures and rules governing their interactions.
2. An understanding of maths allows us to appreciate our environment by observing patterns and connections between objects that we wouldn’t otherwise see. After studying a field such as Model Theory, you begin to see deep connections (across fields of study) in the abstractions of seemingly unrelated concepts. Limiting yourself to the current ideas of one discipline makes original research unnecessarily difficult. I find that most original research is really just connecting past ideas into a new idea that is more than the sum of its parts.
3. Some visually oriented mathematicians overlay representations of abstract concepts on their environments as they go about daily life. This visual overlay of abstract stimuli increases the ability to appreciate things in their own right and builds visual intuition. However, overlaying mathematics on life differs significantly from making a mathematical model of your surroundings: the latter is far more concrete. Overlaying connections on stochastic stimuli is an enrichment to perception, whereas modeling stimuli is separate from it.
Visual intuition is important for research on the leading edge of science and mathematics (especially in geometry). Formalisms come after a topic is well understood.
4. The ability to describe mental pictures in a generalized symbolic fashion is incredibly powerful. The symbolic language of mathematics allows us to represent concepts generally without extraneous information, which permits each viewer to perceive the equations according to which mode of translation best suits their thought processes.
At this point, I’ll cease indulging my evangelism and go back my studies.