Fast M\"obius and Zeta Transforms

M\"obius inversion of functions on partially ordered sets (posets) $\mathcal{P}$ is a classical tool in combinatorics. For finite posets it consists of two, mutually inverse, linear transformations called zeta and M\"obius transform, respectively. In this paper we provide novel fast algorithms for both that require $O(nk)$ time and space, where $n = |\mathcal{P}|$ and $k$ is the width (length of longest antichain) of $\mathcal{P}$, compared to $O(n^2)$ for a direct computation. Our approach assumes that $\mathcal{P}$ is given as directed acyclic graph (DAG) $(\mathcal{E}, \mathcal{P})$. The algorithms are then constructed using a chain decomposition for a one time cost of $O(|\mathcal{E}| + |\mathcal{E}_\text{red}| k)$, where $\mathcal{E}_\text{red}$ is the number of edges in the DAG's transitive reduction. We show benchmarks with implementations of all algorithms including parallelized versions. The results show that our algorithms enable M\"obius inversion on posets with millions of nodes in seconds if the defining DAGs are sufficiently sparse.Comment: 16 pages, 7 figures, submitted for revie

Algebraic number-theoretic properties of graph and matroid polynomials

PhDThis thesis is an investigation into the algebraic number-theoretical properties of certain polynomial invariants of graphs and matroids. The bulk of the work concerns chromatic polynomials of graphs, and was motivated by two conjectures proposed during a 2008 Newton Institute workshop on combinatorics and statistical mechanics. The first of these predicts that, given any algebraic integer, there is some natural number such that the sum of the two is the zero of a chromatic polynomial (chromatic root); the second that every positive integer multiple of a chromatic root is also a chromatic root. We compute general formulae for the chromatic polynomials of two large families of graphs, and use these to provide partial proofs of each of these conjectures. We also investigate certain correspondences between the abstract structure of graphs and the splitting fields of their chromatic polynomials. The final chapter concerns the much more general multivariate Tutte polynomials—or Potts model partition functions—of matroids. We give three separate proofs that the Galois group of every such polynomial is a direct product of symmetric groups, and conjecture that an analogous result holds for the classical bivariate Tutte polynomial

Positive Geometries for Scattering Amplitudes in Momentum Space

Positive geometries provide a purely geometric point of departure for studying scattering amplitudes in quantum field theory. A positive geometry is a specific semi-algebraic set equipped with a unique rational top form - the canonical form. There are known examples where the canonical form of some positive geometry, defined in some kinematic space, encodes a scattering amplitude in some theory. Remarkably, the boundaries of the positive geometry are in bijection with the physical singularities of the scattering amplitude. The Amplituhedron, discovered by Arkani-Hamed and Trnka, is a prototypical positive geometry. It lives in momentum twistor space and describes tree-level (and the integrands of planar loop-level) scattering amplitudes in maximally supersymmetric Yang-Mills theory. In this dissertation, we study three positive geometries defined in on-shell momentum space: the Arkani-Hamed-Bai-He-Yan (ABHY) associahedron, the Momentum Amplituhedron, and the orthogonal Momentum Amplituhedron. Each describes tree-level scattering amplitudes for different theories in different spacetime dimensions. The three positive geometries share a series of interrelations in terms of their boundary posets and canonical forms. We review these relationships in detail, highlighting the author's contributions. We study their boundary posets, classifying all boundaries and hence all physical singularities at the tree level. We develop new combinatorial results to derive rank-generating functions which enumerate boundaries according to their dimension. These generating functions allow us to prove that the Euler characteristics of the three positive geometries are one. In addition, we discuss methods for manipulating canonical forms using ideas from computational algebraic geometry.Comment: PhD Dissertatio

Positive Geometries for Scattering Amplitudes in Momentum Space

Positive geometries provide a purely geometric point of departure for studying scattering amplitudes in quantum field theory. A positive geometry is a specific semi-algebraic set equipped with a unique rational top form—the canonical form. There are known examples where the canonical form of some positive geometry, defined in some kinematic space, encodes a scattering amplitude in some theory. Remarkably, the boundaries of the positive geometry are in bijection with the physical singularities of the scattering amplitude. The Amplituhedron, discovered by Arkani-Hamed and Trnka, is a prototypical positive geometry. It lives in momentum twistor space and describes tree-level (and the integrands of planar loop-level) scattering amplitudes in maximally supersymmetric Yang-Mills theory. In this dissertation, we study three positive geometries defined in on-shell momentum space: the Arkani-Hamed–Bai–He–Yan (ABHY) associahedron, the Momentum Amplituhedron, and the orthogonal Momentum Amplituhedron. Each describes tree-level scattering amplitudes for different theories in different spacetime dimensions. The three positive geometries share a series of interrelations in terms of their boundary posets and canonical forms. We review these relationships in detail, highlighting the author’s contributions. We study their boundary posets, classifying all boundaries and hence all physical singularities at the tree level. We develop new combinatorial results to derive rank-generating functions which enumerate boundaries according to their dimension. These generating functions allow us to prove that the Euler characteristics of the three positive geometries are one. In addition, we discuss methods for manipulating canonical forms using ideas from computational algebraic geometry

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

The equifibered approach to ∞\infty-properads

We define a notion of $\infty$-properads that generalises $\infty$-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on $\infty$-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our $\infty$-properads as presheaves on a category of graphs by comparing them to the Segal $\infty$-properads of Hackney-Robertson-Yau. Combining these two perspectives yields a flexible tool for doing higher algebra with operations that have multiple inputs and outputs. The key ingredient to this paper is the notion of an equifibered map between $\mathbb{E}_\infty$-monoids, which is a well-behaved generalisation of free maps. We also use this to prove facts about free $\mathbb{E}_\infty$-monoids, for example that free $\mathbb{E}_\infty$-monoids are closed under pullbacks along arbitrary maps.Comment: 70 pages, 4 figure

Advances in Discrete Differential Geometry

Differential Geometr

Advances in Discrete Differential Geometry

Differential Geometr