More Like this

1. Totally Nonnegative Critical Varieties

   https://doi.org/10.1093/imrn/rnad084  

   Galashin, Pavel (April 2023, International Mathematics Research Notices)

   Abstract We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_f$$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of $$\operatorname{Crit}^{\geqslant 0}_f$$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety $$\operatorname{Crit}^{\geqslant 0}_{k,n}$$ is homeomorphic to the second hypersimplex $$\Delta _{2,n}$$. 

   more » « less
2. Higher Lorentzian Polynomials, Higher Hessians, and the Hodge–Riemann Relations for Graded Oriented Artinian Gorenstein Algebras in Codimension Two

   https://doi.org/10.1093/imrn/rnaf183  

   Macias-Marques, Pedro; McDaniel, Chris; Seceleanu, Alexandra (July 2025, International Mathematics Research Notices)

   Abstract We define $$i$$-Lorentzian polynomials in two variables, and characterize them as the real homogeneous Macaulay dual generators whose corresponding codimension two algebras satisfy the mixed Hodge–Riemann relations in degree $$i$$ on the positive orthant of linear forms. We further show that $$i$$-Lorentzian polynomials of degree $$d$$ are in one-to-one correspondence with totally nonnegative Toeplitz matrices of size $$(i+1)\times (d-i+1)$$. Using this latter characterization, we show that a certain subclass of real rooted polynomials called normally stable are $$i$$-Lorentzian for all $$i$$. Our results also lead to a new theorem on Toeplitz matrices: the closure of the set of totally positive Toeplitz matrices equals the set of totally nonnegative Toeplitz matrices. 

   more » « less
3. Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties Anthony M. Bloch & Steven N. Karp

   Anthony Bloch and Steven Karp (April 2023, Communications in Mathematical Physics)

   One can view a partial flag variety in ℂ𝑛 as an adjoint orbit 𝜆 inside the Lie algebra of 𝑛×𝑛 skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. The paper has three main parts: (1) We introduce the totally nonnegative part of 𝜆 , and describe it explicitly in several cases. We define a twist map on it, which generalizes (in type A) a map of Bloch, Flaschka, and Ratiu (Duke Math. J. 61(1): 41–65, 1990) on an isospectral manifold of Jacobi matrices. (2) We study gradient flows on 𝜆 which preserve positivity, working in three natural Riemannian metrics. In the Kähler metric, positivity is preserved in many cases of interest, extending results of Galashin, Karp, and Lam (Adv. Math. 397: Paper No. 108123, 1–23, 2022; Adv. Math. 351: 614–620, 2019). In the normal metric, positivity is essentially never preserved on a generic orbit. In the induced metric, whether positivity is preserved appears to depends on the spacing of the eigenvalues defining the orbit. (3) We present two applications. First, we discuss the topology of totally nonnegative flag varieties and amplituhedra. Galashin, Karp, and Lam (2022, 2019) showed that the former are homeomorphic to closed balls, and we interpret their argument in the orbit framework. We also show that a new family of amplituhedra, which we call twisted Vandermonde amplituhedra, are homeomorphic to closed balls. Second, we discuss the symmetric Toda flow on 𝜆 . We show that it preserves positivity, and that on the totally nonnegative part, it is a gradient flow in the Kähler metric up to applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu (1990). 

   more » « less
4. Positive Configuration Space

   https://doi.org/10.1007/s00220-021-04041-x  

   Arkani-Hamed, Nima; Lam, Thomas; Spradlin, Marcus (April 2021, Communications in Mathematical Physics)

   Abstract We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply thenonnegative configuration space. This space has a natural stratification bypositive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel. 

   more » « less
5. Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples

   https://doi.org/10.4230/LIPIcs.SoCG.2024.14  

   Basilio, Brannon; Lee, Chaeryn; Malionek, Joseph (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Finding a totally geodesic surface, an embedded surface where the geodesics in the surface are also geodesics in the surrounding manifold, has been a problem of interest in the study of 3-manifolds. This has especially been of interest in hyperbolic 3-manifolds and knot complements, complements of piecewise-linearly embedded circles in the 3-sphere. This is due to Menasco-Reid’s conjecture stating that hyperbolic knot complements do not contain such surfaces. Here, we present an algorithm that determines whether a given surface is totally geodesic and an algorithm that checks whether a given 3-manifold contains a totally geodesic surface. We applied our algorithm on over 150,000 3-manifolds and discovered nine 3-manifolds with totally geodesic surfaces. Additionally, we verified Menasco-Reid’s conjecture for knots up to 12 crossings. 

   more » « less