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Igor Stamenkovic
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Oxford University - basic quantum mechanics course, 27 lecture videos

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A classic of physics -- the first systematic presentation of Einstein's theory of relativity.

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A foundations of mathematics for the 21st century

It's here!   For decades, mathematicians been dreaming of an approach to math where different proofs that x = y can be seen as different paths in a space.   It's finally been made precise, thanks to Vladimir Voevodsky and a gang of mathematicians who holed up for a year at the Institute for Advanced Studies, at Princeton. 

I won't try to explain it, since that's what the book does.  I'll just mention a few of the radical new features:

•  It includes set theory as a special case, but it's founded on more general things called 'types'.  Types include sets, but also propositions.  Proving a proposition amounts to constructing an element of a certain type.  So, proofs are no longer 'outside' the mathematics being discussed, they're inside it just like everything else.

• The logic is 'constructive', meaning that to prove something exists amounts to giving a procedure for constructing it.  As a result, the whole system can be and is being computerized with the help of programs like COQ and AGDA.

• Types can be seen as 'spaces', and their elements as 'points'.  A proof that two elements of a type are equal can be seen as constructing a path between two points.  Sets are just a special case: the '0-types', which have no interesting higher-dimensional aspect.  There are also types that look like spheres and tori!  Technically speaking, the branch of topology called homotopy theory is now a part of logic!  That's why the subject is called homotopy type theory.

• Types can also be seen as infinity-groupoids.  Very roughly, these are structures with elements, isomorphisms between elements, isomorphisms between isomorphisms, and so on ad infinitum.  So, a certain chunk of the important new branch of math called 'higher category theory' is now part of logic, too.

• The most special contribution of Voevodsky is the univalence axiom.  Very very roughly, this expands the concept of 'equality' so that it's just as general as the hitherto more flexible concept of 'isomorphism' - or, if you know some more fancy math, 'equivalence'.   Mathematicians working on homotopy theory and higher category theory have known for decades that equality is too rigid a concept to be right - for certain applications.  The univalence axiom updates our concept of equality so that it's good again!    

Since this is all about foundations, and it's all quite new, please don't ask me yet what its practical applications are.  Ask me in a hundred years.  For now, I can tell you that this is the 'upgrade' that the foundations of math has needed ever since the work of Grothendieck.  It's truly 21-century math.

It's also a book for the 21st century, because it's escaped the grip of expensive publishers!  While it's 600 pages long, a hardback copy costs less than $27.  Paperback costs less than $18, and an electronic copy is free!  

#spnetwork #homotopytheory #ncategories #logic #foundations  

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At last - a textbook on category theory for scientists!  And it's free!

• David Spivak, Category Theory for Scientists,

"This course is an attempt to extol the virtues of a new branch of mathematics, called category theory, which was invented for powerful communication of ideas between different fields and subfields within mathematics. By powerful communication of ideas I actually mean something precise. Different branches of mathematics can be formalized into categories. These categories can then be connected together by functors. And the sense in which these functors provide powerful communication of ideas is that facts and theorems proven in one category can be transferred through a connecting functor to yield proofs of an analogous theorem in another category. A functor is like a conductor of mathematical truth."

"I believe that the language and toolset of category theory can be useful throughout science. We build scientific understanding by developing models, and category theory is the study of basic conceptual building blocks and how they cleanly fit together to make such models. Certain structures and conceptual frameworks show up again and again in our understanding of reality. No one would dispute that vector spaces are ubiquitous. But so are hierarchies, symmetries, actions of agents on objects, data models, global behavior emerging as the aggregate of local behavior, self-similarity, and the effect of methodological context."

"Some ideas are so common that our use of them goes virtually undetected, such as set-theoretic intersections. For example, when we speak of a material that is both lightweight and ductile, we are intersecting two sets. But what is the use of even mentioning this set-theoretic fact? The answer is that when we formalize our ideas, our understanding is almost always clarified. Our ability to communicate with others is enhanced, and the possibility for developing new insights expands. And if we are ever to get to the point that we can input our ideas into computers, we will need to be able to formalize these ideas first."

"It is my hope that this course will offer scientists a new vocabulary in which to think and communicate, and a new pipeline to the vast array of theorems that exist and are considered immensely powerful within mathematics. These theorems have not made their way out into the world of science, but they are directly applicable there. Hierarchies are partial orders, symmetries are group elements, data models are categories, agent actions are monoid actions, local-to-global principles are sheaves, self-similarity is modeled by operads, context can be modeled by monads."

David Spivak asks readers from different subjects for help in finding new ways to apply category theory to those subjects.  And that's the right attitude to take when reading this book.  I've found category immensely valuable in my work.  But it took effort to learn category theory and see how it can apply to different subjects.  People are just starting to figure out these things, so don't expect instant solutions to the problems in your own favorite field.  But Spivak does the best job I've seen so far at explaining category theory as a general-purpose tool for thinking clearly.

Thanks to +Charlie Clingen for pointing this out!

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Richard Feynman explains how a train stays on the tracks.

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Single variable calculus textbook published in 1917. There is also the solutions manual:

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Professor Coleman's wit and teaching style is legendary and, despite all that may have changed in the 35 years since these lectures were recorded, many students today are excited at the prospect of being able to view them and experience Sidney's particular genius second-hand.
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