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We give a complete description of the category of smooth complex representations of the multiplicative group of a central simple algebra over a locally compact nonarchimedean local field. More precisely, for each inertial class in the…
Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is…
We show that any smooth one-dimensional link in the real projective three-plane is the fixed-point locus of a smooth symplectic surface in the complex projective three-plane which is invariant under complex conjugation. The degree of the…
We construct (2+1)-dimensional lattice systems, which we call fusion surface models. These models have finite non-invertible symmetries described by general fusion 2-categories. Our method can be applied to build microscopic models with,…
By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of…
We obtain infinitely many (non-conjugate) representations of 3-manifold fundamental groups into a lattice in the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of…
We show that if n>5, PU(n-1,1) does not contain a cocompact arithmetic subgroup with the same Euler-Poincare characteristic (in the sense of C.T.C. Wall) as the complex projective space of dimension n-1, and show that if n=5, there are at…
We construct a compact PL 5-manifold $M$ (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, $\vee^{}_{1 1}S^2$, which is "spineless", meaning $M$ is not the regular neighborhood of any 2-complex PL embedded in…
A product of a K3 surface $S$ and a flat 3-dimensional torus $T^3$ is a manifold with holonomy $SU(2)$. Since $SU(2)$ is a subgroup of $G_2$, $S\times T^3$ carries a torsion-free $G_2$-structure. We assume that $S$ admits an action of…
The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct…
We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant…
We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain…
We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to…
We construct the moduli space of smooth hypersurfaces with level $N$ structure over $\mathbb{Z}[1/N]$. As an application we show that, for $N$ large enough, the stack of smooth hypersurfaces over $\mathbb{Z}[1/N]$ is uniformisable by a…
We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion…
Given a smooth curve defined over a field $k$ that admits a non-singular plane model over $\overline{k}$, a fixed separable closure of $k$, it does not necessarily have a non-singular plane model defined over the field $k$. We determine…
We classify curvature-adapted real hypersurfaces $M$ of non-flat quaternionic space forms $\mathbb HP^m$ and $\mathbb HH^m$ that are of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where…
In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the…
Discrete forms of the mean and directed curvature are constructed on piecewise flat manifolds, providing local curvature approximations for smooth manifolds embedded in both Euclidean and non-Euclidean spaces. The resulting expressions take…
We introduce a technique to construct gapped lattice models using defects in topological field theory. We illustrate with 2+1 dimensional models, for example Chern-Simons theories. These models are local, though the state space is not…