Related papers: The positive orthogonal Grassmannian
''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent…
We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann--Pl\"ucker functions. The equivalence is motivated by Cayley's identities expressing principal and almost-principal minors of a…
The hypergeometric amplitude is a one-parameter deformation of the Veneziano amplitude for four-point tachyon scattering in bosonic string theory that is consistent with $S$-matrix bootstrap constraints. In this article we construct a…
A plabic graph is a planar bicolored graph embedded in a disk, which satisfies some combinatorial conditions. Postnikov's boundary measurement map takes the space of positive edge weights of a plabic graph $G$ to a positroid cell in some…
The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Pl\"ucker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a…
We study planar gluon scattering amplitudes and Wilson loops in non-commutative gauge theory. Our main results are: 1. We find the map between observables in non-commutative gauge theory and their holographic dual. In that map, the region…
Let $G$ be the interior domain of a piecewise analytic Jordan curve without cusps. Let $\{p_n\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal over $G$ with respect to the area measure, with each $p_n$ having leading…
This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $\tau$-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite…
The set of Gaussian Type Orbitals g(n1,n2,n3) of order (n+1)(n+2)/2, of common n=n1+n2+n3<=7, common center and exponential, is customized to define a set of 2n+1 linear combinations t(n,m) (-n<=m<=n) such that each t(n,m) depends on the…
Orthogonal spreads in orthogonal spaces of type $V^+(2n+2,2)$ produce large numbers of rank $n$ dual hyperovals in orthogonal spaces of type $V^+(2n,2)$. The construction resembles the method for obtaining symplectic spreads in $V(2n,q)$…
We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n,…
Given integers $n \geq k \geq d$, let $X_{n,k,d}$ be the moduli space of $n$-tuples of lines $(\ell_1, \dots, \ell_n)$ in $\mathbb{C}^k$ such that $\ell_1 + \cdots + \ell_n$ has dimension $d$. We give a quotient presentation of the…
Matroid varieties are the closures in the Grassmannian of sets of points defined by specifying which Pl\"ucker coordinates vanish and which don't --- the set of nonvanishing Pl\"ucker coordinates forms a well-studied object called a…
In this paper we consider the polynomial sequence $(P_{n}^{\alpha,q}(x))$ that is orthogonal on $[-1,1]$ with respect to the weight function $x^{2q+1}(1-x^{2})^{\alpha}(1-x), \alpha>-1, q\in \mathbb N$; we obtain the coefficients of the…
In this paper we consider a well-known construction due to Gromov and Lawson, Schoen and Yau, Gajer, and Walsh which allows for the extension of a metric of positive scalar curvature over the trace of a surgery in codimension at least $3$…
The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of R^n whose nonzero Pluecker coordinates all have the same sign. Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that V is…
A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n…
In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.
In this article, we give a combinatorial approach to the exponents of the Moore spaces. Our result states that the projection of the $p^{r+1}$-th power map of the loop space of the $(2n+1)$-dimensional mod $p^r$ Moore space to its atomic…
We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval,…