Related papers: Van Kampen-Flores theorem for cell complexes

We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k=2) and…

It is shown that Euler's theorem for graphs can be generalized for 2-complexes. Two notions that generalize cycle and Eulerian tour are introduced (``circlet'' and ``Eulerian cover''), and we show that for a strongly-connected, pure…

Hamiltonian Floer theory plays an important role for finding periodic solutions of Hamilton's equation, which can be seen as a generalization of Newton's equation. Generalizing Newton's equation to Laplace's equation with non-linearity, we…

Williamson's theorem states that if $A$ is a $2n \times 2n$ real symmetric positive definite matrix then there exists a $2n \times 2n$ real symplectic matrix $M$ such that $M^T A M=D \oplus D$, where $D$ is an $n \times n$ diagonal matrix…

We introduce sequentially $S_r$ modules over a commutative graded ring and sequentially $S_r$ simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying…

Recent progress in understanding modulus stabilization in string theory relies on the existence of a non-renormalization theorem for the 4D compactifications of Type IIB supergravity which preserve N=1 supersymmetry. We provide a simple…

In his 2011 paper, Teleman proved that a cohomological field theory on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable complex curves is uniquely determined by its restriction to the smooth part $\mathcal{M}_{g,n}$, provided that…

An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant…

We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that…

We show that every topological n-manifold M admits a locally flat closed embedding $\iota\colon M \hookrightarrow \mathbb{R}^{2n+1}$ and is a retract of some neighbourhood $U \subseteq \mathbb{R}^{2n+1}$

The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes…

We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 \pi…

Let $X$ be an open Riemann surface. We prove an Oka property on the approximation and interpolation of continuous maps $X \to (\mathbb{C}^*)^2$ by proper holomorphic embeddings, provided that we permit a smooth deformation of the complex…

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map…

We study a Floer-theoretic approach to harmonic maps from the two-torus into non-flat K\"ahler manifolds. Building on the complex-regularized polysymplectic (CRPS) formalism of [BF24], which provides a Hamiltonian description of harmonic…

Gauged linear sigma-models at critical coupling on Riemann surfaces yield self-dual field theories, their classical vacua being described by the vortex equations. For local models with structure group ${\rm U}(r)$, we give a description of…

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the…

We describe embeddings of $n$-dimensional Lorentzian manifolds, including Friedmann-Lema\^itre-Robertson-Walker spaces, in $\mathbb{R}^{n+2}$ such that the metrics of the submanifolds are inherited by a restriction from that of…

We prove that if an (n-1)-dimensional torus acts symplectically on a 2n-dimensional manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel theorem. The case…

A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embedding of the underlying graph in a closed surface with the added property that for every…