Thoughts on math and science

Here I just include some things I learned that I did not read in books that I think are valuable lessons. Some of these might be obvious, but I feel like they took me a long time to digest. I learned most of these from my advisors, professors, and colleagues.

Structure and property

A given mathematical object can be endowed with additional structure. This structure is additional data. For instance, a vector space can be endowed with an inner product. A property is a feature of a mathematical object. For instance, every vector space has a basis. A choice of a basis is structure, but the existence is a property.

Definition vs. construction

A definition of a mathematical object need not assume its existence. A construction provides a proof for the existence of some mathematical object. Sometimes, a property may characterize a mathematical object or a class of mathematical objects. In this case, a definition of a mathematical object can be defined in terms of its characterizing properties. For example, a definition of a tensor product of vector spaces is given by its universal property. A construction of a tensor product of vector spaces is given in terms of generators and relations. As another example, entropy is typically written down as a formula. However, a proper definition of entropy is some function satisfying certain (physically reasonable) postulates. The formula is a particular construction of this definition and in certain cases it is unique (up to a choice of units).

Invariants are functors

What are invariants? Given a mathematical structure, the collection of these mathematical structures and all of their relations (morphisms and so on) form some sort of a category. Invariants are functors from this category to any other category. Homotopy theory provides another interesting perspective on invariants. In homotopy theory, one is interested in invariantsĀ under homotopy. It turns out there is a universal invariant in the sense that every other invariant factors through this one. More precisely, let $latex C$ be a model category or a category with weak equivalences. Let $latex \mathrm{ho}C$ be the associated homotopy category obtained by localizing $latex C$ with respect to the weak equivalences. Then the localization $latex L : C \to \mathrm{ho}C$ satisfies the property that for any other functor $latex F: C \to D$ that sends weak equivalences to isomorphisms, there exists a unique functor $latex \mathrm{ho} C \to D$ making the obvious diagram commute. I first learned this from Dan Freed and Mike Hopkins in their article “Chern-Weil Forms and Abstract Homotopy Theory.”

"You can never verify a theory, you can only verify its predictions." – Venkataraman Balakrishnan