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David L. Duncan

Publications and source records attributed to David L. Duncan.

11 recordsLinked to original sources
arXiv2025

Representation varieties of RAAGs

We investigate the $G$-representation varieties of right-angled Artin groups (RAAGs) for various Lie groups $G$. We show these varieties are connected for a large class of such $G$, including $\mathrm{SU}(n), \mathrm{Sp}(n)$ and $\mathrm{U}(n)$, while they are generally not connected for other large classes, such as $\mathrm{SO}(n)$ and $\mathrm{Spin}(n)$ for $n \geq 3$. When $G = \mathrm{SO}(3)$ we determine the number of connected components of the representation variety associated to any RAAG that is also a 3-manifold group.

math.GT↗
arXiv2025

Representation varieties and genus-three Torelli maps

We consider the family of Torelli homeomorphisms on a genus-three surface given by powers of a fixed bounding pair map. For each such homeomorphism $ϕ$ we determine the number of connected components of the fixed point set of the induced map on the representation variety of the surface, as well as the number of connected components of the representation variety of the mapping torus of $ϕ$.

math.GT↗
arXiv2023

Existence of mASD connections on 4-manifolds with cylindrical ends

Taubes' gluing theorems establish the existence of ASD connections on closed, oriented 4-manifolds. We extend these gluing results to the mASD connections of Morgan-Mrowka-Ruberman on oriented 4-manifolds with cylindrical ends. As a corollary, we obtain an ASD-existence result in the presence of degenerate asymptotic flat connections.

math.DG↗
arXiv2020

Characterizing immutable sandpiles: A first look

By working with coefficients in $\mathbb{Z}$ or $\mathbb{R}$, one can define two different notions of stability for a sandpile on a graph. We call a sandpile immutable when these notions agree. Our main results give linear-algebraic characterizations for large classes of immutable sandpiles.

math.CO↗
arXiv2019

Critical group structure from the parameters of a strongly regular graph

We give simple arithmetic conditions that force the Sylow $p$-subgroup of the critical group of a strongly regular graph to take a specific form. These conditions depend only on the parameters $(v, k, λ, μ)$ of the strongly regular graph under consideration. We give many examples, including how the theory can be used to compute the critical group of Conway's $99$-graph and to give an elementary argument that no $srg(28,9,0,4)$ exists.

math.CO↗
arXiv2019

An index relation for the quilted Atiyah-Floer conjecture

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. Each of these is equipped with a chain level grading. We show that the gradings agree.

math.SG↗
arXiv2016

The Chern-Simons invariants for the double of a compression body

Given a 3-manifold that can be written as the double of a compression body, we compute the Chern-Simons critical values for arbitrary compact connected structure groups. We also show that the moduli space of flat connections is connected when there are no reducibles.

math.GT↗
arXiv2016

The Yang-Mills flow for cylindrical end 4-manifolds

We establish various existence and uniqueness results for the Yang-Mills flow on cylindrical end 4-manifolds. We also show long-time existence and infinite-time convergence under certain hypotheses on the underlying data.

math.DG↗
arXiv2015

Higher-rank instanton cohomology and the quilted Atiyah-Floer conjecture

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

math.SG↗
arXiv2013

On the components of the gauge group for PU(r)-bundles

We discuss a general procedure for using characteristic classes to study the components of the gauge group for a principal G-bundle. To illustrate this, we work out the case where G is the projective unitary group.

math.DG↗