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I am a recent PhD student trying to settle into a research topic. Even though I have a current project I am working on, I am not particularly enjoying it and would like to switch. Before braving the lion's den and telling my supervisor that I would like to change projects, I figure it would be prudent to do some research on current research.

To be more precise, I have developed a small fascination with the moduli space of flat connections on Riemann surfaces, but I do not know the current state of research in that field. I hope that my questions are not so vague as to bring down the wrath of the overflow gods, but they are as follows:

  1. How active is current research in the moduli space of flat connections? Some scholarly searching reveals there are still papers being published in the field, but has the research talent narrowed to specific experts?
  2. Is the field still in a state where it would be reasonable to do a PhD and actually get novel results in a reasonable amount of time? (I ask only because I have colleagues whose work lies in the Langland's program, and they seem to have spent the majority of their graduate careers just trying to get the point where they can ask a reasonable question.)

My apologies if the question seems infantile. It's just that there is a mountain ahead of me and I cannot seem to see past the first ridge.

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  • $\begingroup$ The Geometric Langlands Program, which is a reasonably active field, has something to do with sheaves on this moduli space. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 7, 2013 at 22:51
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    $\begingroup$ Maybe one could say that Geometric Langlands is advancing so quickly that it is important to have access to a "big expert" if one wants to work on that. $\endgroup$
    – Eleanor Von Hohlandsbourg
    Commented Jun 8, 2013 at 2:34
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    $\begingroup$ I think the author's reference to the Langlands Program was just for purposes of giving an example of problems not immediately accessible to graduate students at the outset. I suspect the author is not seeking an answer involving the Langlands Program. $\endgroup$
    – Peter Crooks
    Commented Jun 10, 2013 at 15:20
  • $\begingroup$ Study the symplectic volume of moduli space of flat connections and its asymptotic formula historically come back to Witten.For example Witten (and later Bismut by different approach)showed that the symplectic volume of moduli of flat connections on Riemann surface $S$ of genus $g$ can be written by Yang-Mills functional $$Vol=\lim_{t\to 0}\int_\mathcal A\exp\left(-\frac{\int_S|F_A|^2}{4t}\right)d\mathcal A$$ See .E. Witten. On quantum gauge theories in two dimensions. Comm. Math. Phys., 141:153–209, 1991 $\endgroup$
    – user21574
    Commented Jun 5, 2017 at 17:10

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