The well-known moment map maps the Grassmannian Gr k+1,n and the positive Grassmannian Gr + k+1,n onto the hypersimplex ∆ k+1,n , which is a polytope of codimension 1 inside R n . Over the last decades there has been a great deal of work on matroid subdivisions (and positroid subdivisions) of the hypersimplex [Kap93, Laf03, Spe08, AFR10, ARW16, RVY19, Ear19]; these are closely connected to the tropical Grassmannian [SS04, Spe08, HJS14] and positive tropical Grassmannian [SW05]. Meanwhile any n×(k +2) matrix Z with maximal minors positive induces a map 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 [AHT14]. As the positive Grassmannian has a decomposition into positroid cells [Pos], one may ask when the images of a collection of cells of Gr + k+1,n give a dissection of the hypersimplex ∆ 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 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 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 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. ContentsLW was partially supported by NSF grants DMS-1854316 and DMS-1854512.

The hypersimplex ∆ k+1,n is the image of the positive Grassmannian Gr ≥0 k+1,n under the moment map. It is a polytope of dimension n − 1 which lies in R n . Meanwhile, the amplituhedron A n,k,2 (Z) is the projection of the positive Grassmannian Gr ≥0 k,n into the Grassmannian Gr k,k+2 under the amplituhedron map Z. It is not a polytope, and has full dimension 2k inside Gr k,k+2 . Nevertheless, as was first discovered in [LPW20], these two objects appear to be closely related via T-duality. In this paper we use the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -images of positroid cells of Gr ≥0 k+1,n under the moment map -translate into sign conditions characterizing the T-dual Grasstopes -images of positroid cells of Gr ≥0 k,n under the amplituhedron map. Moreover, we subdivide the amplituhedron into chambers enumerated by the Eulerian numbers, just as the hypersimplex can be subdivided into simplices enumerated by the Eulerian numbers. We use this property to prove one direction of the conjecture of [LPW20]: whenever a collection of positroid polytopes gives a triangulation of the hypersimplex, the T-dual Grasstopes give a triangulation of the amplituhedron. Along the way, we prove several more conjectures: Arkani-Hamed-Thomas-Trnka's conjecture that A n,k,2 (Z) can be characterized using sign conditions, and Lukowski-Parisi-Spradlin-Volovich's conjectures about characterizing generalized triangles (the 2k-dimensional positroid cells which map injectively into the amplituhedron A n,k,2 (Z)), and m = 2 cluster adjacency. Finally, we discuss new cluster structures in the amplituhedron.

We advance the exploration of cluster-algebraic patterns in the building blocks of scattering amplitudes in N = 4 super Yang-Mills theory. In particular we conjecture that, given a maximal cut of a loop amplitude, Landau singularities and poles of each Yangian invariant appearing in any representation of the corresponding Leading Singularities can be found together in a cluster. We check these adjacencies for all one-loop amplitudes up to 9 points. Along the way, we also prove that all (rational) N 2 MHV Yangian invariants are cluster adjacent, confirming original conjectures.

En 1840 se publica Voyage en Icarie una obra que revela uno de los proyectos de socialismo utópico más importantes del siglo XIX. Su autor, Étienne Cabet, probablemente se había inspirado en la lectura de Utopía de Tomás Moro. Esta lectura produjo en el político francés tal impresión que decidió cambiar totalmente su estrategia política. A los pocos años de la primera edición del Viaje a Icaria, un grupo de republicanos catalanes emprende la publicación por entregas de esta obra en el periódico La Fraternidad, y empieza a difundir las ideas de Cabet a través de la publicación de los artículos de Le Populaire. Es el inicio del grupo cabetiano más importante fuera de Francia. El objetivo de este texto es analizar el legado que Moro tuvo en el proyecto utópico de Cabet para ver, efectivamente, que propuestas del político inglés seguían siendo actuales a los ojos de los pensadores utópicos del siglo XIX; sobre todo a la luz del interés que despertó la propuesta utópica cabetiana en el contexto político español.