The focus of my research is on certain partial differential equations (PDEs) that arise in 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. I am particularly interested in how these PDEs behave under certain geometric limits of the underlying manifolds.

More specifically, in gauge theory I am interested in the instanton and Yang-Mills equations, and in symplectic geometry I am interested in the holomorphic curve and harmonic map equations. The solution sets of these equations 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 equations and their associated invariants. Perhaps the most famous of these relationships are the Atiyah-Floer conjectures, which have motivated much of my work.

More specifically, in gauge theory I am interested in the instanton and Yang-Mills equations, and in symplectic geometry I am interested in the holomorphic curve and harmonic map equations. The solution sets of these equations 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 equations and their associated invariants. Perhaps the most famous of these relationships are the Atiyah-Floer conjectures, which have motivated much of my work.

**Publications and preprints**

- D. Duncan, ''The Chern-Simons invariants for the double of a compression body''.
*Pac. J. Math.***280**, 17-89, 2016.__PDF__ - D. Duncan, ''Compactness results for neck-stretching limits of instantons''.
*J. Symp. Geo.*(Accepted), 2016.__PDF__ - D. Duncan, ''The Yang-Mills flow for cylindrical end 4-manifolds''. Preprint.
__PDF__ - D. Duncan, ''An index relation for the quilted Atiyah-Floer conjecture''. Preprint.
__PDF__

**Slides from various talks**

- Heat flows for cylindrical end manifolds (CMS 2016; Niagara)
__PDF__ - The quilted Atiyah-Floer conjecture and the Yang-Mills heat flow (SIAM 2015; Scottsdale)
__PDF__ - Gauge theoretic invariants of surface products (CMS 2015; Montreal)
__PDF__ - From instantons to quilts with seam degenerations (CMS 2014; Hamilton)
__PDF__

**Recorded talks**

**Various notes**

- Higher-rank instanton cohomology and the quilted Atiyah-Floer conjecture.
__PDF__ - Regularity of split-type Hamiltonians for quilts.
__PDF__ - On the components of the gauge group for PU(r)-bundles.
__PDF__ - Newton's iteration for quadratic functions.
__PDF__ - Flat connections and holonomy.
__PDF__ - An introduction to the Loewner equation and SLE. (Completed when I was an undergraduate, under the supervision of Steffen Rohde and Joan Lind.)
__PDF__

**Theses**

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