Publications and Notes

Papers

D.L. Duncan, I. Hambleton. Existence of mASD connections on 4-manifolds with cylindrical ends. Communications in Analysis and Geometry (Accepted March 2023) PDF (Preprint)
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.

Recorded Talks

Below are links to video recordings of some of my talks. My apologies if some of the videos are no longer available; they are not hosted by me.

The Chern–Simons invariants for general compact Lie groups. Duke Geometry and Topology Seminar, 2018. Durham, NC.
Boundary-singular holomorphic curves from instantons. Fields Institute Thematic Program on Geometric Analysis, 2017. Toronto, ON.
A new proof of a theorem of Dostoglou and Salamon. 2016 BIRS Conference on Interactions of Gauge Theory with Contact and Symplectic Topology in Dimensions 3 and 4. Banff, AB.

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.

Which real vector bundles over S4 admit complex structures?.
Natural connections on Lie groups.
The Hodge star on the 3-sphere.
A note on product manifolds and shrinking à la Kaluza–Klein.
A proof that TS² has c₁ = 2 (with some pictures!).
A nontrivial sphere bundle in U(2) coming from the trace.
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.
On the components of the gauge group for PU(r)-bundles.
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.
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.
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?