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Where can I find some good introductory reading to the topic of Topological Modular Form or some related topic? Thanks.

SamC
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  • Have you looked at wikipedia? That seems like a good place to start. – Tobias Kildetoft Aug 28 '15 at 12:42
  • I had, but there are a lots of references, and I don't know which one to start first. I want to start from reading prerequisites first before go into the core of the topic. – SamC Aug 28 '15 at 12:49
  • Then you should probably add to this question what your background is. How familiar are you with cohomology? Modular forms? – Tobias Kildetoft Aug 28 '15 at 12:55
  • No, I don't have any knowledge about cohomology, hootopy, elliptic curve nor modular form. This is my first year of grad school, and the subjects that I did so far are differential geometry and measure theory. At the moment taking Riemann surfaces and representation theory. My actual project is starting next year, I just want to use the time from now to get the prerequisites done and get an idea what TMF is. – SamC Aug 28 '15 at 13:05
  • Then I must say I am curious how you came to be interested in the topic. – Tobias Kildetoft Aug 29 '15 at 10:59

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I think the best reference for a complete introduction (with a touch into advanced topics) is the book from Diamond and Shurman : A first course in modular forms. It treats the subject in a really nice algebraic and geometric point of view. I'm more of an analysis person but algebra brings into play a lot of beautiful theorems about the structure of the space of modular forms that is in some sense unaccessible from a purely analytic point of view. Take a look at it ! If you want a shortened version of the book, take a look at Marc Masdeu course at Warwick : http://homepages.warwick.ac.uk/~masmat/files/teaching/modforms.pdf

I read his notes and they are quite really complete for a first reading.

If you want something easy to start with, there is : Modular forms and dirichlet series in number theory from Apostol which is fully analytic and describes all you have to know for modular forms on $SL_2(\mathbb{Z})$. I think that if you want to go slowly, read this one first then Masdeu's notes to have some kind of background. Masdeu's notes generalize to modular forms on congruence subgroups and establishes the link of modular forms with elliptic curve via the modularity theorem at the end.

Enjoy !

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