Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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Kinematic varieties for massless particles
Full text linkWe study algebraic varieties that encode the kinematic data for n massless particles in d-dimensional spacetime subject to momentum conservation. Their coordinates are spinor brackets, which we derive from the Clifford algebra associated to the Lorentz group. This was proposed for d=5 in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices
Algebraic approaches to cosmological integrals
Full text linkCosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition
Logarithmic discriminants of hyperplane arrangements
Full text linkA recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate critical point in the corresponding hyperplane arrangement complement. We study properties of this discriminant, exploiting its connection with the Hurwitz form of a reciprocal linear space
Points on rational normal curves and the ABCT variety
Full text linkThe ABCT variety is defined as the closure of the image of G(2, n) under the Veronese map. We realize the ABCT variety V(3,n) as the determinantal variety of a vector bundle morphism. We use this to give a recursive formula for the fundamental class of V(3,n). As an application, we show that special Schubert coefficients of this class are given by Eulerian numbers, matching a formula by Cachazo-He-Yuan. On the way tothis, we prove that the variety of configuration of points on a common divisor on a smooth variety is reduced and irreducible, generalizing a result of Caminata-Moon-Schaffler
The positive orthogonal grassmannian
Full text linkThe Plücker positive region OGr+(k,2k) of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian OGr+(k,n) for general values of k, n. We determine the boundary structure of the quadric OGr+(1,n) in Pn-1+ and show that it is a positive geometry. We show that OGr+(k,2k+1) is isomorphic to OGr+(k+1, 2k+2) and connect its combinatorial structure to matchings on [2k+2]. Finally, we show that in the case n > 2k+1, the positroid cells of Gr+(k,n) do not induce a CW cell decomposition of OGr+(k,n)
The chirotropical grassmannian
Full text linkRecent developments in particle physics have revealed deep connections between scattering amplitudes and tropical geometry. From the heart of this relationship emerged the chirotropical Grassmannian TropχG(k,n) and the chirotropical Dressian Drχ(k,n), polyhedral fans built from uniform realizable chirotopes that encode the combinatorial structure of Generalized Feynman Diagrams. We prove that TropχG(3,n) = Drχ(3,n) for , and develop algorithms to compute these objects from their rays modulo lineality. Using these algorithms, we compute all chirotropical Grassmannians TropχG(3,n) for n = 6,7,8 across all isomorphism classes of chirotopes. We prove that each chirotopal configuration space Xχ(3,6) is diffeomorphic to a polytope and propose an associated canonical logarithmic differential form. Finally, we show that the equality between chirotropical Grassmannian and Dressian fails for (k,n) = (4,8)
How to stab a polytope
Full text linkWe study the set of linear subspaces of a fixed dimension intersecting a given polytope. To describe this set as a semialgebraic subset of a Grassmannian, we introduce a Schubert arrangement of the polytope, defined by the Chow forms of the polytope’s faces of complementary dimension. We show that the set of subspaces intersecting a specified family of faces is defined by fixing the sign of the Chow forms of their boundaries. We give inequalities defining the set of stabbing subspaces in terms of sign conditions on the Chow form
From Feynman diagrams to the amplituhedron: a gentle review
Full text linkIn this article we review, for a mathematical audience, the computation of (tree-level) scattering amplitudes in Yang-Mills theory in detail. In particular we demonstrate explicitly how the same formulas for six-particle NMHV helicity amplitudes are obtained from summing Feynman diagrams and from computing the canonical form of the n=6, k=1, m=4 amplituhedron
Tropicalizing binary geometries
Full text linkThe type A cluster configuration space, commonly known as M0,n, is the very affine part of the binary geometry associated with the associahedron. The tropicalization of M0,n can be realized as the space of phylogenetic trees and its signed tropicalizations as the dual-associahedron subfans. We give a concise overview of this construction and propose an extension to type C. The type C cluster configuration space MCl arises from the binary geometry associated with the cyclohedron. We define a space of axially symmetric phylogenetic trees containing many dual-associahedron and dual-cyclohedron subfans. We conjecturally realize the tropicalization of MCl as the defined space and its signed tropicalizations as the aforementioned subfans
Elastic beam qquations with variable coefficients: multiple solutions under mixed nonlinearities
No full textThis paper investigates the existence of multiple solutions for a fourth-order differential equation modelling an elastic beam, where the coefficients are variable, and the nonlinearities exhibit both concave and convex characteristics. Our approach is based on variational methods and critical point theorems, particularly those formulated by Ricceri, which provide a powerful framework for proving the existence of solutions in reflexive Banach spaces. By leveraging these mathematical tools, we establish that the considered problem admits at least three distinct weak solutions under specific conditions. To validate our theoretical findings, we present an illustrative example demonstrating how our results can be applied in practice