Oleg Osipovich
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My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory!   It's free:

https://arxiv.org/abs/1612.09375

It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.

Huh?  What's a 'universal property'?

In category theory, we try to describe things by saying what they do, not what they're made of.  The reason is that you can often make things out of different ingredients that still do the same thing!  And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do

A universal property amounts to a precise description of what an object does.

Universal properties show up in three closely connected ways in category theory, and Tom's book explains these in detail:

through representable functors (which are how you actually hand someone a universal property),

through limits (which are ways of building a new object out of a bunch of old ones),

through adjoint functors (which give ways to 'freely' build an object in one category starting from an object in another).

If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom's book!  It's not easy to learn this stuff - but it's good for your brain.  It literally rewires your neurons.

Here's what he wrote, over on the category theory mailing list:

.............................................................................

Dear all,

My introductory textbook "Basic Category Theory" was published by Cambridge University Press in 2014.  By arrangement with them, it's now also free online:

https://arxiv.org/abs/1612.09375

It's also freely editable, under a Creative Commons licence.  For instance, if you want to teach a class from it but some of the examples aren't suitable, you can delete them or add your own.  Or if you don't like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros.  Just go the arXiv, download, and edit to your heart's content.

There are lots of good introductions to category theory out there.  The particular features of this one are:

• It's short.
• It doesn't assume much.
• It sticks to the basics.﻿
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This is the most astonishing martial-arts video I have ever seen.

Stuff you had no idea was even possible ... and he makes it look easy.﻿
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A free online course on chaos theory

Chaos theory is the study of physical systems whose motion depends very delicately on how they start out.  There's a lot of deep geometry here, and  has started a free online course on the subject!

There's a lot of hype about chaos theory, but Predrag is a good physicist, and he's written a good free textbook on the subject, so this is the real deal.

To register, just go to his webpage here.  The course started a week ago but you can still join in.  It lasts 8 weeks.  It'll use his book, links to explanatory videos, and weekly homework assignments, which include some computer programming. For the assignments you can use any computational tools you want, but he'll provide you with stuff written in Python.  There are no tests.

He encourages you to register even if you won't do the homework: you can talk to other students on the course forum.

Some administrators from his university tried to shut this course down at the last minute, probably because it's free.  I'm glad he fought them off and prevailed.

This course is called Nonlinear dynamics 1: Geometry of chaos, and here are the topics:

Topology of flows - how to enumerate orbits, Smale horseshoes
Quantitative dynamics - periodic orbits, local stability
The role of symmetries in dynamics

There will also be a second, more advance 8-week course called Nonlinear dynamics 2: Chaos rules, with these topics:

Transfer operators - statistical distributions in dynamics
Spectroscopy of chaotic systems
Dynamical zeta functions
Dynamical theory of turbulence

The prerequisites for this first course are a basic background in linear algebra, calculus, ordinary differential equations, probability theory, classical mechanics, and statistical mechanics.  You'll need to able to work with equations involving vectors and matrices, differentiate simple functions, and understand what a probability distribution is.   You will learn to write programs in Python. ﻿
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Hack your way to the Galaxy... #DontPanic Online Coding Contest - Saturday, October 25th http://ow.ly/D3dFu﻿
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Nifty papers I wrote that nobody knows about (Part 4: Complex Langevin equation)
This is the last installment of the "Nifty Papers" series. Here are the links to Part1 , Part2 , and Part 3 . For those outside the computational physics community, the following words don't mean anything:  " The Sign Problem " For those others that have en...﻿
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