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David L. Duncan Associate Professor Department of Mathematics & Statistics Roop Hall Room 323 MSC 1911 James Madison University Harrisonburg, Virginia 22807 E-Mail: duncandl_at_jmu.edu Office Hours (Fall 2023): M 11:20-12:20 (in Roop G025) T 11-12 (in Roop G025) I am also available by appointment Curriculum Vita (Last Updated August, 2023) |
The focus of much of my research is on certain partial differential equations (PDEs) coming out of gauge theory and symplectic geometry. These PDEs arose in the latter quarter of the 20th Century, and are some of the most powerful tools we have for understanding the topology of 3- and 4-manifolds. Some of my early work was focused on gaining a better understanding of how these PDEs behave under certain geometric limits of the underlying manifolds.
More specifically, in gauge theory I am interested in analyzing various features of the Yang–Mills and Chern–Simons functionals. In symplectic geometry the analogous objects are the energy and action functionals. The critical points of these functionals are precisely the solutions of the aforementioned PDEs, and these can often be packaged algebraically to produce manifold invariants (e.g., Floer homology, the Donaldson invariants, and the Gromov–Witten invariants). There are deep relationships between these PDEs and their associated invariants. Perhaps the most famous of these relationships are articulated in the various Atiyah–Floer conjectures (special cases of which are now theorems), which motivated much of my PhD and postdoctoral work.
I have also explored problems involving graph theory (sandpiles and the critical group), representation varieties, the mASD equation of Morgan–Mrowka–Ruberman, and geodesic equations in various geometrically-natural spaces, among other things.
For more details on my research, see the following link and the references therein.
For more detailed information about previous student projects I have supervised, see my CV.
Here is some general information about undergraduate research in mathematics and statistics.
Official Degree Requirements:
Math & Stats Courses:
Other:
I find that often these activities reinforce each other: I am a better researcher when I teach, since teaching encourages me to consider questions I may not otherwise have asked. I am a better teacher when I am active as a researcher, since the research mentality encourages me to look at old material in new ways.
I view the mentoring of undergraduate research as lying at the intersection of teaching and research. I have seen that undergraduates who thoroughly engage in interesting research problems in mathematics understand the material at a much deeper level than one generally achieves in a standard classroom environment. Moreover, such students develop an autonomy for learning, gain an appreciation for knowledge-generation, and refine critical thinking skills to a degree rarely seen at the undergraduate level. As such, I hold the mentoring of undergraduate research in high regard.
I believe a good research problem for early researchers in mathematics should be (i) tractable and (ii) interesting (to someone other than the researcher). These are subjective and often competing properties that are difficult to simultaneously achieve, particularly at the undergraduate level. I think this goes a long way to explain why there are relatively few opportunities for undergraduates to engage in research in mathematics, as compared to many other disciplines. I personally believe that we in the mathematics community can do far better at generating meaningful undergraduate research experiences for a much wider range of students, and I think much of this starts by identifying as many good research problems for early researchers as we can.
The above poses a challenge: Find as many tractable and interesting undergraduate research problems as I can.
I am always on the lookout for such problems, and I have discovered a little trick that has helped me identify a number of them. Perhaps it will help others too, so I want to share it. The trick is that I look for pieces of my own research projects that an undergraduate can understand with a few weeks of background training. Such a problem tends to automatically be interesting (by construction it is related to broader or deeper questions), so the only constraint to be satisfied is that it needs to be tractable. If an undergraduate can understand the statement of the problem, then its likelihood of being tractable is often much higher (of course, as we know from Fermat, this is no guarantee; it is a challenge, after all).
To wrap this up, let me take a step outside of my department and briefly discuss the school I call home: James Madison University (JMU). One of the main reasons I chose to work at JMU is precisely its focus on the undergraduate experience, while simultaneously having resources to support advanced research. Indeed, JMU appreciates and respects excellence in teaching and excellence in research at the same level, and it also provides many opportunities and resources for undergraduates to engage in research.
