arXiv subjects

Dylan Galt

Publications and source records attributed to Dylan Galt.

4 recordsLinked to original sources

arXiv2024

An Invariant for Transverse Coassociative 4-Folds

We define a $\mathbb{Z}_2$-valued invariant for transversely-intersecting coassociative $4$-folds equipped with spin structures. Our main result shows this invariant provides an obstruction to separating two such coassociatives through a family of transverse coassociative deformations. We further prove that there is a canonical generalized connected sum of two such transverse coassociatives whose diffeomorphism type is determined by our invariant. When one coassociative is a graph over the other, we relate our invariant to the parity function in near-symplectic geometry. Finally, we discuss conjectural consequences for non-compactness phenomena and compute our invariant for the $Sp(1)$-invariant coassociatives discovered by Harvey and Lawson.

math.DG↗

arXiv2024

Dimension Reduction of Generalized ASD Instantons

We study generalized anti-self-dual instantons defined over Riemannian manifolds equipped with a parallel codimension-$4$ differential form. In particular, for product Riemannian manifolds possessing such a form, we study dimension reduction phenomena, finding a topological criterion for bundles which, when satisfied, allows for a complete characterization of dimension reduction for the corresponding moduli space of generalized ASD instantons. By establishing an integrability result for families of connections, we then deduce explicit descriptions for these moduli spaces, including those of Hermitian Yang--Mills connections, $G_2$-, and $\Spin(7)$-instantons. When one factor in the product is a $4$-manifold, we establish well-behaved compactifications for these moduli spaces.

math.DG↗

arXiv2023

Cohomology at Infinity and the Well-Tempered Complex

We prove the existence of a sequence of commutative diagrams generalizing existing results on the cohomology of the Borel-Serre boundary and well-rounded retract to the context of the well-tempered complex. Our main theorem provides a method for computing in finite terms the action of Hecke operators on the equivariant cohomology of an arithmetic subgroup $\Gamma$ of the special linear group $SL_n$.

math.NT↗

arXiv2021

Computing Hecke operators for arithmetic subgroups of $\text{Sp}_4$

We outline an algorithm for computing Hecke operators on equivariant cohomology $H^\ast_{\Gamma_{\text{Sp}}}(X_{\text{Sp}};\rho)$ for the symplectic group $\text{Sp}_4(\mathbb{R})$. To do this, we define a new acyclic cell complex for $\text{Sp}_4(\mathbb{R})$ and make use of the well-tempered complex.

math.NT↗