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### Most useful isomorphisms you know?

Posted: Thu Mar 21, 2019 11:43 pm UTC
So, working on the Ph.D., I find myself proving various isomorphisms pretty regularly...which got me wondering as to which ones are actually really useful/powerful to have. Thinking over the ones I know, the biggest that stands out is in Algebraic Topology, the isomorphism between singular and simplicial Homology. For those who don't know, general results are very easy to prove in singular topology, but it's next to impossible to compute with for any practical purposes, while the reverse holds true for simplicial homology.

I'm sure there's lots of others out there though?

### Re: Most useful isomorphisms you know?

Posted: Fri Mar 22, 2019 9:21 am UTC
This may be too trivial for your taste, but having isomorphisms between any (discrete, finite) piece of information and a binary representation is pretty useful if you like having general-purpose computers.

### Re: Most useful isomorphisms you know?

Posted: Thu Mar 28, 2019 5:51 pm UTC
Since you mentioned algebraic topology I've always loved the isomorphism between an arbitrary group and the fundamental group of its presentation complex.

### Re: Most useful isomorphisms you know?

Posted: Sun Mar 31, 2019 1:00 am UTC
Reminds me of a quote by someone - "Mathematics is the art of calling different things by the same name"
Algebraic topology by itself is a nice 'isomorphism' - though not really; in the sense that we can look at spaces as algebraic structures and algebraic structures as spaces (really an adjunction I guess -?)

### Re: Most useful isomorphisms you know?

Posted: Wed Apr 10, 2019 10:05 pm UTC
I like how random variables tend to be related to the same abstract probability space. Do you want to find a random point in the Mandelbrot set? Awesome! All you need is to apply the right deterministic function to a uniform random variable on the interval [0,1], and you got it! You can even do this in reverse if you are terrible at darts and have a Mandelbrot set-shaped dartboard!

### Re: Most useful isomorphisms you know?

Posted: Wed Apr 10, 2019 11:46 pm UTC
cyanyoshi wrote:I like how random variables tend to be related to the same abstract probability space. Do you want to find a random point in the Mandelbrot set? Awesome! All you need is to apply the right deterministic function to a uniform random variable on the interval [0,1], and you got it! You can even do this in reverse if you are terrible at darts and have a Mandelbrot set-shaped dartboard!

You might have to be pretty good at darts to simulate a uniform random distribution.

### Re: Most useful isomorphisms you know?

Posted: Wed May 15, 2019 9:08 am UTC
The complex numbers and the algebraic closure of the p-adic rationals are isomorphic as fields. Very useful if you like l-adic cohomology.