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We are looking to run a working seminar about the Yang-Mills story. We hope that our seminars is of interest to analysts (working with curvatures and Ricci flows on Riemannian manifolds), the algebraists (working on Lie algebras and Lie groups), and, of course, math physicists. If we manage to understand something about the Clay problem in this direction, that would be great, but it is not strictly necessary.

The seminar is to run for a year. It involves talks by graduate students, postdocs, and faculty (almost all from the maths department). I'm looking for helpful references.

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    $\begingroup$ Are you referring to the classical Yang-Mills theory or to the quantum Yang-Mills theory (or both)? $\endgroup$ – user25309 Apr 28 '16 at 13:17
  • $\begingroup$ @user25309 Just out of curiousity Is the clay institue problem about both classical and quantum YM? $\endgroup$ – Saal Hardali Apr 28 '16 at 14:54
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    $\begingroup$ @SaalHardali: The Clay problem is about quantum YM. See claymath.org/sites/default/files/yangmills.pdf (especially section 2). In fact, the Clay problem is about formulating and studying quantum Yang Mills theory and showing that it does indeed behave differently from the classical YM in certain regimes (as one expects in physics). $\endgroup$ – Willie Wong Apr 28 '16 at 15:25
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    $\begingroup$ If you are interested in quantum (and not classical) Yang-Mills and are starting from zero background, I suggest you forget about classical YM and in fact YM altogether and learn instead about the 2d Gross-Neveu model at large N and the 2d sigma model where the situation is better understood and the same behavior is expected as in YM4: asymptotic freedom in the UV and mass gap in the IR. See the lectures by David Gross in the IAS course for mathematicians and Coleman's book "Aspects of Symmetry" for a good place to start. $\endgroup$ – Abdelmalek Abdesselam Apr 29 '16 at 12:56
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If your goal is to get some understanding of the Clay Problem, you can't really go wrong with first reading the official problem statement and then reading the papers referred to in the document.


On the other hand, if your goal is not the quantum problem but more the classical problem, for the geometers and algebraists a good starting point is of course Donaldson's Geometry of four-manifolds which contains a lot of classical results in the direction (and you can use the reference list to find the original papers should you wish). Donaldson's more recent survey can also be a point of departure.

Since you mentioned that there are participants interested in curvature and Ricci flows, you can also consider discussing the results related to the Yang-Mills heat flow. The standard references are

If you also want to discuss the hyperbolic initial value problem, then a good (classical) place to start would be

There has been a lot of development/improvement/extensions since then (see the References link in mathscinet) but they tend to get very technical very fast (into the details of PDE theory for wave equations) and may be less interesting for your stated audience.

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You could start with Terence Tao's Local well-posedness of the Yang-Mills equation in the Temporal Gauge below the energy norm, available online via the arXiv.

A second paper by Tao and Gang Tian on the subject is A singularity removal theorem for Yang-Mills fields in higher dimensions, also available via the arXiv.

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    $\begingroup$ (1) These papers are about classical Yang Mills, and so somewhere far from the Clay problems (2) Terry's LWP paper are probably of limited interest to people not strictly working within PDEs. $\endgroup$ – Willie Wong Apr 28 '16 at 15:27
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Useful references can be found at https://ncatlab.org/nlab/show/Yang-Mills+theory See also http://arxiv.org/abs/0911.1013 (Mass in Quantum Yang-Mills Theory, by L.D. Faddeev).

From the physics perspective, classical Yang-Mills theory is covered in the book V. Rubakov "Classical Theory of Gauge Fields", and the quantum theory in the books L.D. Faddeev, A.A. Slavnov "Gauge Fields: An Introduction To Quantum Theory" and I.J.R. Aitchison "An Informal Introduction to Gauge Field Theories".

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