
Topology: The Secret Ingredient In The Latest Theory of Everything - vectorbunny
http://www.technologyreview.com/view/429528/topology-the-secret-ingredient-in-the-latest/?ref=rss
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crusso
_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.

~~~
jamesbritt
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.

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xk_id
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.

~~~
cottonseed
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.

~~~
runT1ME
Random question, is category theory helpful in working out equations with
quantum topology?

~~~
cottonseed
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.

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

...what?

~~~
shardling
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?)

~~~
xyzzyz
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.

~~~
aroberge
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.

~~~
xyzzyz
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.

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xk_id
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.

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LolWolf
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.

