Research

Much of my recent work is a collaboration with R. Field and R. Shankar that explores invariants of smooth 7-manifolds, such as the Eells–Kuiper and t-invariants. To compute these invariants I have taken a foray into orbifolds, particularly Kawasaki's orbifold extension of the Atiyah–Singer index theorem. Part of our work establishes the Inverse Soul Conjecture for homotopy 7-spheres.

Beyond this, 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.

Below you can find my published papers, slides from various talks, information about my work with Atiyah–Floer conjecture, my undergraduate and graduate theses, and also some notes about a variety of topics.

Papers

A. Bao, A. Chakraborty, D.L. Duncan, J. Larson, K. McBride. Representation varieties of RAAGs. (2025). (Submitted) PDF (Preprint)
A. Bao, A. Chakraborty, D.L. Duncan, J. Larson, K. McBride. Representation varieties and genus-three Torelli maps. (2025). (Submitted) PDF (Preprint)
D.L. Duncan, I. Hambleton. Existence of mASD connections on 4-manifolds with cylindrical ends. Communications in Analysis and Geometry, Volume 32, No 8, 2024. PDF
D.L. Duncan. Triangles & princesses & bears, oh my! A journey from a puzzle to the Schrödinger equation. The Mathematical Intelligencer (2022). DOI: 10.1007/s00283-022-10219-5 PDF
D.L. Duncan, W.J. Engelbrecht. Characterizing immutable sandpiles: A first look. Discrete Math., Vol. 345, Issue 1, 2022. PDF (Preprint)
J.E. Ducey, D.L. Duncan, W.J. Engelbrecht, J.V. Madan, E.Piato, C.S. Shatford, A.Vichitbandha. Critical group structure from the parameters of a strongly regular graph. J. of Comb. Theory, Series A, Vol. 180, 105424, 2021. PDF (Preprint)
D.L. Duncan. The Yang–Mills flow for cylindrical end 4-manifolds. Indiana Univ. Math. J., Vol. 69, Issue 3, pp. 1007–1071, 2020. PDF (Preprint)
D.L. Duncan. The Chern–Simons invariants for doubles of compression bodies. Pac. J. Math. 280, 17–39, 2016. PDF (Preprint)

Slides from Various Talks

Gluing mASD connections on cylindrical end 4-manifolds. Gauge Theory Virtual Seminar, 2021. Virtual.
Bundle splittings on boundary-punctured disks. Brandeis Topology Seminar, 2020. Virtual.
Heat flows for cylindrical-end manifolds. 2016 CMS Winter Meeting. Niagara, ON.
The quilted Atiyah–Floer conjecture and the Yang–Mills heat flow. 2015 SIAM Conference on Analysis of Partial Differential Equations. Scottsdale, AZ.
Gauge theoretic invariants of surface products. 2015 CMS Winter Meeting. Montreal, QC.
The quilted Atiyah–Floer conjecture and the Yang–Mills heat flow. 2015 AMS Spring Meeting. Washington DC.
From instantons to quilts with seam degenerations. 2014 CMS Winter Meeting. Hamilton, ON.

Atiyah–Floer Project

Here are some papers pertaining to my Atiyah–Floer project.

Higher-rank instanton cohomology and the quilted Atiyah–Floer conjecture. This paper describes the conjecture and outlines an approach.
Compactness results for neck stretching limits of instantons. Just prior to publishing, I found a crucial technical error in the proof of Proposition 4.9 that I have not been able to fix, so I withdrew the paper from the arXiv and the publication pipeline. The details are discussed in the red comments in v5 in the link.
An index relation for the quilted Atiyah–Floer conjecture. The proof of Proposition 3.5 relies on Proposition 4.9 of the Compactness paper above. As such, the results of this paper should be considered as conjectural until that Compactness paper is fixed. For more details, see the red comments in v3 in the link. I withdrew this paper from the arXiv and the publication pipeline as a consequence of this error.

Theses

Compactness Results for the Quilted Atiyah–Floer Conjecture. PhD Thesis, 2013. Supervised by Chris Woodward at Rutgers University.
Constructions Regarding Integration in the Plane and the Rotation of Segments. Senior Undergraduate Thesis, 2006. Supervised by Jim Morrow at the University of Washington.

Various Notes

Here is a collection of notes I have put together on a variety of topics. Some of them are short computations, while others are detailed proofs or constructions. Most of them began as notes to myself, but perhaps someone else will find something useful here. In cases where the title is not sufficiently self-explanatory, I have given a brief description of the contents.

The character variety of the thrice-punctured sphere
Chern classes and Hopf fibrations: Complex and quaternionic...let's try to get the signs right!
The Chern–Weil formula in dimension 4 and the degree for general gluings
Flat connections and holonomy. This details a proof that the holonomy provides an identification between the moduli space of flat connections and the character variety.
The Hodge star on the 3-sphere
An introduction to the Loewner equation and SLE. This was completed when I was an undergraduate, under the supervision of Steffen Rohde and Joan Lind. The .tex file seems to have been lost in some digital purgatory, and so I am unfortunately unable to make any further corrections or edits.
Natural connections on Lie groups
Newton's iteration for quadratic functions
A nontrivial sphere bundle in U(2) coming from the trace
A note on product manifolds and shrinking à la Kaluza–Klein
On the components of the gauge group for PU(r)-bundles
The osculating helix. The osculating circle is used to give a geometric interpretation of the curvature of a curve in 3-space. The osculating helix is a refinement that also encodes the torsion. This note explores some of the details. It was originally prepared for an undergraduate course on differential geometry.
A proof that TS2 has c1 = 2 (with some pictures!)
Rotations, SO(3), and RP3
The tangent space to the spaces of smooth maps and sections
Toroiding a prism: When does this preserve volume? Suppose you take a prism and bend it into a toroid. If the base of the prism is a circle, then this operation preserves volume. Why?
Which real vector bundles over S3 admit complex structures?

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