arXiv subjects
Publications and source records attributed to Bailee Zacovic.
The CHY scattering equations on the moduli space $M_{0,n}$ play a prominent role at the interface of particle physics and algebraic statistics. We study the scattering correspondence when the Mandelstam invariants are restricted to a fixed graph on $n$ vertices.
Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations, organizing the several naturally arising variants into a single unified framework. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size $9$, and the connected quotient of the simple loopless regular complex through ground-set size $15$. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids.
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.
We study chip-firing on a signed graph $G_ϕ$, employing a general theory of chip-firing on invertible matrices introduced by Guzmán and Klivans. Here a negative edge designates an adversarial relationship, so that firing a vertex incident to such an edge leads to a loss of chips at both endpoints. The chip-firing rule for $G_ϕ$ is described by its reduced Laplacian matrix $L_{G_ϕ}$, which also defines the critical group ${\mathcal K}(G_ϕ)$. The valid chip configurations are given by the lattice points of a rational cone determined by $G_ϕ$ and the underlying graph $G$. This gives rise to notions of critical as well as $z$-superstable configurations, both of which are counted by the determinant of $L_{G_ϕ}$. We establish general results regarding these configurations, focusing on efficient methods of verifying the underlying properties. We then study the critical groups of signed graphs in the context of vertex switching and Smith normal forms. We use this to compute the critical groups of various classes of signed graphs including signed cycles, wheels, complete graphs, and fans, in the process generalizing results of Biggs and others.
We give combinatorial proofs of some enumeration formulas involving labelled threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs. In each case we count by number of vertices and number of components. For threshold graphs, we also count by number of dominating vertices, and for loop-threshold graphs we count by number of looped dominating vertices. We also obtain an analog of the Frobenius formula (connecting Eulerian numbers and Stirling numbers of the second kind) in the context of labelled threshold graphs.