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I advance that I'm not a mathematician but I'm an undergraduate student of mathematics. In my courses at university I have studied a bit of Differential Geometry, in particoular differential geometry of curves and surfaces (in $\mathbb{R}^n$). I know that there are many research topics about manifolds, Lie groups atc. My question is the following:

In our days what are the most important research topics about curves and surfaces (in $\mathbb{R}^n$)?

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    $\begingroup$ Probably a good question to ask one of the mathematicians at your university. $\endgroup$
    – Gerry Myerson
    Commented Jul 24, 2016 at 22:53
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    $\begingroup$ Not every student is so lucky that the professors working in the field they're interested in are willing/have time to give a project and mentor it. $\endgroup$
    – 54321user
    Commented Jul 24, 2016 at 23:41
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    $\begingroup$ I would say that the most actively studied topics are about how surfaces behave under so-called geometric heat flows and about characterizing critical points (especially the global minima) of certain energy functionals involving the curvature of a surface. The recent work of Marques and Neves on the Willmore conjecture is a good example of this. $\endgroup$
    – Deane Yang
    Commented Jul 25, 2016 at 0:53
  • $\begingroup$ Also, in convex geometry, which is the study of convex bodies in Euclidean space, which in turn is equivalent to the study of their boundaries (convex hypersurfaces). Here, there are still unanswered important questions in integral geometry, which is not differential geometry but is very closely linked to it. $\endgroup$
    – Deane Yang
    Commented Jul 25, 2016 at 0:55
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    $\begingroup$ The Caratheodory conjecture still receives attention, in spite of claims of resolution. en.wikipedia.org/wiki/Carath%C3%A9odory_conjecture $\endgroup$
    – Ian Agol
    Commented Jul 25, 2016 at 2:51

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