Topology: The Secret Ingredient In The Latest Theory of Everything | Hacker News Reader

Topology: The Secret Ingredient In The Latest Theory of Everything

technologyreview.com

34 pointsvectorbunny17 comments

Of course, no theory is worth more than bag of beans unless it makes testable predictions about the universe.

Wen says that his theory has significant implications for the states of matter that existed soon after the Big Bang but doesn't develop the idea into specific predictions.

It's always so disappointing to look for the rubber-meeting-the-road part of a theory and see that the tires are still spinning a few feet above the ground.

In what sense is this article using the word "theory"? In the scientific sense of "as true as things come" (e.g. theory of gravity, theory of relativity), or in the pop scientific sense of "plausible conjecture"?

Seems like the latter. Little wonder we hear people saying evolution is "just a theory" when science reporting can't even use decent terminology.

I am usually very weary of this kind of "science for the general public" articles. We all know how the "computer science for the general public" articles sound like – I usually doubt whether I would get an accurate picture after reading them, if I didn't have a technical background in CS. Since I don't know anything about advanced physics myself, I will not indulge in any "ah, I see" kind of reaction after reading this article.

E.g.: I have no idea how accurate my interpretation of the phrase "quantum ether" is; it's like an empty container, and as a non-specialist I could put lots of non-sense in it.

I agree. I'm a mathematician working in quantum topology, and this article seems to equal parts vague and incoherent.

As for this "Wow! Topology impacts physics! Useful for the first time!" is total bullshit. The interaction between topology and physics goes back a century, or centuries. The work of Poincare, the father of modern topology, for example, was motivated by studying PDEs and the behavior of dynamical systems. More spectacular is the work of the last 30 years, starting with the work of Donaldson in the 80s, and later work by Witten, between gauge theory (the mathematical formalism of the standard model) and the topology of 4-dimensional spaces. It is beautiful story, much like the origins of calculus or the application of ideas from differential geometry in the foundations of general relativity.

Random question, is category theory helpful in working out equations with quantum topology?

Short answer: Yes.

Long answer: I'm not quite sure what you mean by "working out equations". Category theory is useful. Categories were introduced by Eilenberg and MacLane to formalize certain structures were studying in the context of algebraic topology (homology, cohomology and homotopy are all functors from topological to algebraic categories, and there are lots of examples of natural transformations on these functors). Algebraic topology is a foundation for quantum/low-dimensional topology, so categories naturally arise in using the tools from algebraic topology.

In quantum topology, many of the natural invariants one studies are in some sense TQFTs: Topological Quantum Field Theories. The modern formulation of TQFTs is due to Atiyah. In this formulation, a TQFT is a functor from a certain kind of topological category called a cobordism category (in analogy with physics, one can think of the objects as spacetimes and the morphisms as spacetime evolution) to certain algebraic categories (which one can think of as quantum state spaces and evolution operators). These TQFTs are often not just functors, but n-functors: the categories in question are not just regular categories, but higher categories or infinity categories.

Going off-topic, what would a good "computer science for the general public" article look like to you? I imagine it would not be about something new, but something well-established and how it relates to the real world. Eg, the principles of process control as applied to logistics.

I say the whole dichotomy between the day-to-day vocabulary and technical/scientific vocabularies should be dropped altogether. This dichotomy is responsible for the cliquey attitude of scientists on one end; the treatment of sciences as esoteric by the laymen on the other end; the labelling of science enthusiasts as "hipsters"; etc. All of which is ridiculous.

Today, the life conditions of the layman are given by modern sciences, but he insists in using an antiquated vocabulary in speaking about his experiences and the world. Hence some people strongly feel in terms of "computer wizard" instead of "computer literate"; "static matter" instead of "processes"; etc. We really should stop divorcing sciences from our day-to-day life, vocabularies and feelings; it's unnatural what is going on. We look like cavemen with mobile phones, and more than half a century after the formulation of cybernetics, our daily epistemologies are still largely Aristotelian!

So a well written scientific article for the general public would avoid translating the scientific discoveries into the antiquated terms. "Quantum ether"? What a heroic way to crap on the enormous shift from Newton's "fundamental emptiness" to Einstein's "fundamental plenum". Similarly for CS articles, careless translations should be avoided; instead, we should borrow in our common speaking as much as possible from scientific vocabularies, and make those terms fundamental: they serve and represent us much, much better, because they package the most up-to-date results of our enquires into the structure of the world.

Sciences give us factors of sanity, let's not waste them.

Edit: cosmetics.

Ah, but you'd have to spend a few hundred words explaining "plenum". I have no clue what that is, and I'd need at the very least a simple analogy. Science writing can lose clarity in the pursuit of concision and precision.

In the past, topology was little more than an amusing diversion for mathematicians doodling about the difference between donuts and dumplings.

...what?

Obviously it is super-important in all sorts of areas of mathematics, but I can't think of any branches of physics that explicitly invoke any interesting topology. (Except probably string theory.)

(And perhaps some areas of solid state do?)

There are lots of examples. For instance, you can interpret Maxwell equations as certain statements about de Rham cohomology classes of certain 2-forms on 4-manifolds. Fiber bundles are used to describe local symmetries in gauge theories, and since fiber bundles are connected with homotopy groups, they also find their use. Even K-theory is used, in string theory. The list could easily go on.

EDIT: fixed mistake.

The list could go on ... if you are a mathematician. If you are a physicist (not working on string theory), you rarely use topological concepts - if at all. While working towards my Ph.D. on some topic in finite temperature effects in gauge theories, I did not know anything about fiber bundles and still know nothing about de Rham cohomology classes. My non-theorist physicists friends knew even less about those mathematical topics.

Sure. That's why math is great: after you have spent enough time familiarizing with concepts, you can easily understand so many things in terms of concepts you already feel very comfortable with.

You see electromagnetism, realize that electromagnetic field is just 2-form on 4-manifold, and see that Maxwell equations just state that both this form and its Hodge dual are closed, so that they represent de Rham cohomology classes. Of course, you can also represent this result in a classical way, but creating a bridge translating physical concepts into well studied mathematical frameworks has the advantage enabling you to also pass the bridge in the other direction: sometimes you can find physical interpretation concepts that arised in the abstract setting.

So, we know that forms representing electromagnetic fields are closed, but what does it mean in physics when they are exact? How to interpret Mayer-Vietoris sequence in electromagnetism terms? What the induced maps do with forms representing electromagnetic fields, and how to interpret homotopy? What about Poincare duality?

It's such a great feeling to realize that your favourite toy, after playing with it for years, is actually able to do a whole lot more stuff than you were previously aware of.

Sure, but in no point in the pursuit of a PhD in theoretical physics did I actually use any of that machinery. Nor did it ever come up in any papers I've read, nor any talk I've seen.

The point is this -- if you want to extend our model of the universe, knowing the more abstract machinery is useful; by generalising your understanding (and a hell of a lot of math is basically generalising to more abstract systems) you gain insight into how alternative models might be constructed. If you want to understand our current model, even at a quite deep level, you can get by without knowing any topology at all.

I hear another very interesting application of topology was in human psychology, by the famous Kurt Lewin. He wrote a book, "Principles of Topological Psychology"; I haven't read it yet, but it sounds very interesting.

Wait, the "latest" theory of everything? Topology's been in use in String Theory since, well, since it was developed, along with M-Theory, et al. Not to mention some parts of the standard model, but it's all by no means a new or "recent" discovery.

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