Hyperplane arrangements in the grassmannian

The Euler characteristic of a smooth very affine variety encodes the algebraic complexity of solving likelihood (or scattering) equations on this variety. We study this quantity for the Grassmannian with d hyperplane sections removed. We provide a combinatorial formula, and explain how to compute this Euler characteristic in practice, both symbolically and numerically. Our particular focus is on generic hyperplane sections and on Schubert divisors. We also consider special Schubert arrangements relevant for physics. We study both the complex and the real case

Tropicalizing binary geometries

The 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

The chirotropical grassmannian

Recent 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 $n = 6,7,8$, 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)

Algebraic approaches to cosmological integrals

Cosmological 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

Moduli spaces in positive geometry

These are lecture notes for five lectures given at MPI Leipzig in May 2024.  We study the moduli space M0,n of n distinct points on P1 as a positive geometry and a binary geometry.  We develop mathematical formalism to study Cachazo-He-Yuan\u27s scattering equations and the associated scalar and Yang-Mills amplitudes.  We discuss open superstring amplitudes and relations to tropical geometry

Points on rational normal curves and the ABCT variety

The 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

Cyclic polylopes through the lense of iterated integrals

The volume of a cyclic polytope can be obtained by forming an it-erated integral, known as the path signature, along a suitable piecewiselinear path running through its edges. Different choices of such a path arerelated by the action of a subgroup of the combinatorial automorphismsof the polytope. Motivated by this observation, we look for other polyno-mials in the vertices of a cyclic polytope that arise as path signatures andare invariant under the subgroup action. We prove that there are infinitelymany such invariants which are algebraically independent in the shufflealgebra

From Feynman diagrams to the amplituhedron: a gentle review

In 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

Bivariate exponential integrals and edge-bicolored graphs

We show that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex incidence structures. The asymptotic behavior is governed by the critical points of a polynomial. As an application, we discuss the Ising model on a random 4-regular graph and show how its phase transitions arise from our formula

Uniqueness of MHV gravity amplitudes

We investigate MHV tree-level gravity amplitudes as defined on the spinor-helicity variety. Unlike their gluon counterparts, the gravity amplitudes do not have logarithmic singularities and do not admit Amplituhedron-like construction. Importantly, they are not determined just by their singularities, but rather their numerators have interesting zeroes. We make a conjecture about the uniqueness of the numerator and explore this feature from a more mathematical perspective. This leads us to a new approach for examining adjoints. We outline steps of our proposed proof and provide computational evidence for its validity in specific cases