Convex bodies: the brunn–minkowski theory

How Many People Can the Earth Support? By Joel E. Cohen. Norton: 1995. Pp. 532. $30, £22.50. UK publication date, 27 March.

Book of numbers

The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. When k+m=n, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when k=1, the amplituhedron is a cyclic polytope. While the case m=4 is most relevant to physics, the amplituhedron is an interesting mathematical object for any m. In this paper we study it in the case m=1. We start by taking an orthogonal point of view and define a related "B-amplituhedron" B(n,k,m), which we show is isomorphic to A(n,k,m). We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron A(n,k,1) using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that A(n,k,1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that A(n,k,1) is homeomorphic to a ball.

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The $m=1$ Amplituhedron and Cyclic Hyperplane Arrangements

The well-known moment map maps the Grassmannian $Gr_{k+1,n}$ and the positive Grassmannian $Gr^+_{k+1,n}$ onto the hypersimplex $\Delta_{k+1,n}$, which is a polytope of codimension $1$ inside $\mathbb{R}^n$. Over the last decades there has been a great deal of work on matroid subdivisions (and positroid subdivisions) of the hypersimplex; these are closely connected to the tropical Grassmannian and positive tropical Grassmannian. Meanwhile any $n \times (k+2)$ matrix $Z$ with maximal minors positive induces a map $\tilde{Z}$ from $Gr^+_{k,n}$ to the Grassmannian $Gr_{k,k+2}$, whose image has full dimension $2k$ and is called the $m=2$ amplituhedron $A_{n,k,2}$. As the positive Grassmannian has a decomposition into positroid cells, one may ask when the images of a collection of cells of $Gr^+_{k+1,n}$ give a dissection of the hypersimplex $\Delta_{k+1,n}$. By dissection, we mean that the images of these cells are disjoint and cover a dense subset of the hypersimplex, but we do not put any constraints on how their boundaries match up. Similarly, one may ask when the images of a collection of positroid cells of $Gr^+_{k,n}$ give a dissection of the amplituhedron $\mathcal{A}_{n,k,2}$. In this paper we observe a remarkable connection between these two questions: in particular, one may obtain a dissection of the amplituhedron from a dissection of the hypersimplex (and vice-versa) by applying a simple operation to cells that we call the T-duality map. Moreover, if we think of points of the positive tropical Grassmannian $\mbox{Trop}^+Gr_{k+1,n}$ as height functions on the hypersimplex, the corresponding positroidal subdivisions of the hypersimplex induce particularly nice dissections of the $m=2$ amplituhedron $\mathcal{A}_{n,k,2}$. Along the way, we provide a new characterization of positroid polytopes and prove new results about positroidal subdivisions of the hypersimplex.

The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron

Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems-the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction-the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic," in the sense that their eigenfunction statistics deviate from random matrix theory.

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems.

Bruhat interval polytopes

Let $u$ and $v$ be permutations on $n$ letters, with $u$ ≤ $v$ in Bruhat order. A Bruhat interval polytope $Q_{u,v}$ is the convex hull of all permutation vectors $z=(z(1),z(2),...,z(n))$ with $u$ ≤ $z$ ≤ $v$. Note that when $u=e$ and $v=w_0$ are the shortest and longest elements of the symmetric group, $Q_{e,w_0}$ is the classical permutohedron. Bruhat interval polytopes were studied recently in the 2013 paper “The full Kostant-Toda hierarchy on the positive flag variety” by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety. In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and $R$-polynomials, we also give a generalization of the standard recurrence for $R$-polynomials.

We define and study a variant of the center of mass of a polygon and, more generally, of a simplicial polytope which we call the Circumcenter of Mass (CCM). The CCM is an affine combination of the circumcenters of the simplices in a triangulation of a polytope, weighted by their volumes. For an inscribed polytope, CCM coincides with the circumcenter. Our motivation comes from the study of completely integrable discrete dynamical systems, where the CCM is an invariant of the discrete bicycle (Darboux) transformation and of recuttings of polygons. We show that the CCM satisfies an analog of Archimedes' Lemma, a familiar property of the center of mass. We define and study a generalized Euler line associated to any simplicial polytope, extending the previously studied Euler line associated to the quadrilateral. We show that the generalized Euler line for polygons consists of all centers satisfying natural continuity and homogeneity assumptions and Archimedes' Lemma. Finally, we show that CCM can also be defined in the spherical and hyperbolic settings.

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Circumcenter of Mass and Generalized Euler Line

We study the dynamics of the discrete bicycle (Darboux, Backlund) transformation of polygons in n-dimensional Euclidean space This transformation is a discretization of the continuous bicycle transformation, recently studied by Foote, Levi, and Tabachnikov We prove that the respective monodromy is a Moebius transformation Working toward establishing complete integrability of the discrete bicycle transformation, we describe the monodromy integrals and prove the Bianchi permutability property We show that the discrete bicycle transformation commutes with the recutting of polygons, a discrete dynamical system, previously studied by V Adler We show that a certain center associated with a polygon and discovered by Adler, is preserved under the discrete bicycle transformation As a case study, we give a complete description of the dynamics of the discrete bicycle transformation on plane quadrilaterals

On the discrete bicycle transformation

A toric domain is a subset of $(\mathbb{C}^n,\omega_{\text{std}})$ which is invariant under the standard rotation action of $\mathbb{T}^n$ on $\mathbb{C}^n$. For a toric domain $U$ from a certain large class for which this action is not free, we find a corresponding toric domain $V$ where the standard action is free, and for which $c(U)=c(V)$ for any symplectic capacity $c$. Michael Hutchings gives a combinatorial formula for calculating his embedded contact homology symplectic capacities for certain toric four-manifolds on which the $\mathbb{T}^2$-action is free. Our theorem allows one to extend this formula to a class of toric domains where the action is not free. We apply our theorem to compute ECH capacities for certain intersections of ellipsoids, and find that these capacities give sharp obstructions to symplectically embedding these ellipsoid intersections into balls.

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Emmanuel Tsukerman

Papers

Bruhat interval polytopes

Bruhat interval polytopes

Circumcenter of Mass and Generalized Euler Line

On the discrete bicycle transformation

On symplectic capacities of toric domains