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I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and Homological Algebra), self study of Geometry (Manifolds, Differential Geometry, Riemannian Metrics, Curvature and Connections), elementary representation theory of finite dimensional semisimple Lie Algebras and complex analysis. Before I study TQFTs I will try to read Algebraic Topology, Characteristic Classes and K-Theory. On the physics side I am comfortable with scalar field theory and renormalization.
My first question is :

1.) Are there any other subjects that I should learn before embarking on a study of TQFTs from a Mathematics perspective ? In particular should I study Algebraic Geometry (e.g. Hartshorne or Huybretchs) before or concurrently with TQFTs ?

Also there are good reviews of this subject from the Physicists' perspective such as the famous review here and lots others listed here, however I wish to approach the subject from the Mathematics side. One way is to perhaps start with the original papers of Witten but as I mentioned I am more keen on learning the Mathematics side and preferably from a book or review article.

2.) Is there a good book or detailed review article useful for beginners interested in learning TQFTs from a mathematical perspective ?

TQFT is now a very vast subject and I would like a book or article dealing with an overview of the subject rather than a specialized sub area. I am aware of the book by Turaev but am not sure whether it is good for beginners or satisfies the above criteria.

I am more keen on learning the subject for its applications in manifolds rather than knot theory (for which a helpful answer is already available in this answer.)

It would be ideal if someone could suggest a roadmap to learning this subject. Thanks !

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    $\begingroup$ Have you watched Jacob Lurie's lectures on the classification of TQFTs. I thought he did a top-notch job of motivating their mathematical content, as well as, well, overviewing the classification. $\endgroup$ Commented Aug 13, 2014 at 20:21
  • $\begingroup$ @AlexYoucis No I have not. Can you please post a link ? Thanks ! $\endgroup$ Commented Aug 13, 2014 at 20:26
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    $\begingroup$ It would be useful to know some basic category theory (not lots, but maybe a few chapters from Categories for the Working Mathematician). $\endgroup$ Commented Aug 14, 2014 at 13:39
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    $\begingroup$ This is the first part: youtube.com/watch?v=Bo8GNfN-Xn4 $\endgroup$ Commented Aug 17, 2014 at 5:51
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    $\begingroup$ The paper by Atiyah in 1988 where he states and motivates the axioms for TQFTs is short & useful. $\endgroup$ Commented Sep 13, 2014 at 10:19

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