Related papers: Mukai lifting of self-dual points in $\mathbb{P}^6…

In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general…

In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian…

We introduce a notion of ``$n$-dual'' to a simplicial vector space for $n\ge 0$. Coming with it, there is a canonical pairing, which we show to be non-degenerate up to homotopy for homotopy $n$-types. As a result this notion of duality is…

David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman…

We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of geometric invariant theory. We apply this to two…

Let U be a real form of a complex semisimple Lie group, and tau, sigma, a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and…

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial…

Let $V$ be an $n$-dimensional left vector space over a division ring $R$ and $n\ge 3$. Denote by ${\mathcal G}_{k}$ the Grassmann space of $k$-dimensional subspaces of $V$ and put ${\mathfrak G}_{k}$ for the set of all pairs $(S,U)\in…

Mukai showed that the GIT quotient $\operatorname{Gr}(7,16) /\!/ \operatorname{Spin}(10)$ is a birational model of the moduli space of Deligne-Mumford stable genus 7 curves $\overline{M}_7$. The key observation is that a general smooth…

In recent years it has been shown that many, and possibly all, integrable systems can be obtained by dimensional reduction of self-dual Yang-Mills. I show how the integrable systems obtained this way naturally inherit bihamiltonian…

Let $X$ be a vertex subset of a graph $G$. Then $u, v\in V(G)$ are $X$-positionable if $V(P)\cap X \subseteq \{u,v\}$ holds for any shortest $u,v$-path $P$. If each two vertices from $X$ are $X$-positionable, then $X$ is a general position…

We solve an open problem in spectral geometry: the construction of finite-dimensional, discrete geometries coordinatized by non-simple, exceptional Jordan algebras. The approach taken is readily generalisable to broad classes of…

We hypothesize that the vacuum is spontaneously shifted from the self-dual point in the moduli space, at which all dimensions are compact, and that the true vacuum results from the decompactification of the four space-time dimensions and…

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate…

We apply the discontinuous Galerkin finite element method with a degree $p$ polynomial basis to the linear advection equation and derive a PDE which the numerical solution solves exactly. We use a Fourier approach to derive polynomial…

This paper is a sequel of arXiv:2109.06394. In this paper, we consider a kind of inverse problem of multipliers. The problem is to count number of isospectral correspondences, correspondences which has the same combination of multipliers.…

A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question…

Following a systematic analysis of existing results, we investigate when complete interlacing between the zeros of distinct polynomial sequences, $\{\mathcal{P}_n\}$ and $\{\mathcal{G}_n\}$ can be achieved by using a naturally arising extra…

Motivated by a question from V. Arnold about self-dual curves in projective spaces, we study {\cal M}_{m,n,k}: the moduli space of m-self-dual n-gons in {\mathbb P}^k. This paper lays out an explicit construction of self-dual polygons, and…

(Anti)self-dual solutions of the scale invariant SU(2) gauged Grassmanian model are sought. A stronger (anti)selfduality condition for this system is defined, referred to as strong self-duality, and spherically symmetric solutions of this…