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Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a…

We prove necessary and sufficient conditions for the existence of non-trivial Steenrod actions on the mod-$2$ cohomology of 4-dimensional toric orbifolds. As applications, the stable homotopy type and the gauge groups of a $4$-dimensional…

We show that in codimension at least 3, spaces of locally flat topological embeddings of manifolds are correctly modelled by derived spaces of maps between their configuration categories (under mild smoothability conditions). That general…

Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the "interesting" surface…

Given a 4-manifold X and an imbedding of T^{2} x B^2 into X, we describe an algorithm X --> X_{p,q} for drawing the handlebody of the 4-manifold obtained from X by (p,q)-logarithmic transforms along the parallel tori. By using this…

The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number,…

Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus…

For an ample line bundle $\mathcal{L}$ on a complete toric surface $X$, we consider the subset $V_{\mathcal{L}} \subset \vert \mathcal{L} \vert$ of irreducible, nodal, rational curves contained in the smooth locus of $X$. We study the…

Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the torus, with the exceptional graph class called "Petersen-like", is 3-edge-colorable. This means every (non-trivial) toroidal snark can be obtained from…

Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the…

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSL(2,R). Goldman conjectured that when the surface is closed and of genus bigger than…

There are two main approaches to building locally flat embedded surfaces in 4-manifolds: direct methods which geometrically manipulate a given map of a surface, and more indirect methods using surgery theory. Both rely on Freedman-Quinn's…

The splitting number of a plane irreducible curve for a Galois cover is effective to distinguish the embedded topologies of plane curves. In this paper, we define a connected number of any plane curve for a Galois cover whose branch divisor…

We give an overview of various recent results concerning the topology of symplectic 4-manifolds and singular plane curves, using branched covers and isotopy problems as a unifying theme. While this paper does not contain any new results, we…

We present constructions of simply connected symplectic 4-manifolds which have (up to sign) one basic class and which fill up the geographical region between the half-Noether and Noether lines.

We establish a correspondence between trisections of smooth, compact, oriented $4$--manifolds with connected boundary and diagrams describing these trisected $4$--manifolds. Such a diagram comes in the form of a compact, oriented surface…

In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of…

We prove an equidistribution result for iterated preimages of curves by a large class of rational maps $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP}^2$ that cannot be birationally conjugated to algebraically stable maps. The maps, which…

An embedding of a graph on a translation surface is said to be \emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation…