Institute for Advanced Study
Princeton, NJ.
Email: earlnick@gmail.com
At the intersection of theoretical physics and pure mathematics lies a fascinating discovery: the mathematics of geometric spaces and combinatorial patterns directly shapes the behavior of fundamental particles. My research explores this connection through the lens of scattering amplitudes – mathematical functions that describe how particles interact – and their deep relationships with algebraic geometry and combinatorics.
In the past decade, this interdisciplinary field has witnessed remarkable breakthroughs. The Cachazo-He-Yuan (CHY) formula revealed that quantum field theory amplitudes emerge naturally from the geometry of certain moduli spaces – mathematical objects that parametrize configurations of points in space. This connection has opened new avenues in both physics and mathematics, particularly in understanding del Pezzo surfaces and related geometric structures.
My recent work focuses on an ambitious program initiated by Cachazo, Early, Guevara, and Mizera that generalizes traditional particle interactions. This framework, which began with geometric insights about points in higher dimensional projective spaces, has revealed unexpected connections to combinatorial structures called matroids, their subdivisions and orientations. These findings suggest that the mathematical language of particle physics might be even richer than previously imagined, potentially leading to new physical theories and mathematical insights.
Publications
Positive del Pezzo Geometry. (Joint with Alheydis Geiger, Marta Panizzut, Bernd Sturmfels and Claudia Yun). To Appear in Annales de l'Institut Henri Poincaré D: Combinatorics, Physics and their Interactions.
Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes (Joint with Jianrong Li), Published in Journal of Physics A: Mathematical and Theoretical.
Generalized permutohedra in the kinematic space. (Published in JHEP).
Generalized Color Orderings: CEGM Integrands and Decoupling Identities, (Joint with Freddy Cachazo and Yong Zhang). Published in Nuclear Physics B.
Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity (Joint with Freddy Cachazo and Yong Zhang). Published in SIGMA.
Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes (joint with Freddy Cachazo); Published in JHEP.
Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes (joint with Freddy Cachazo). Published in Annales de l'Institut Henri Poincaré D: Combinatorics, Physics and their Interactions.
Smoothly Splitting Amplitudes and Semi-Locality (joint with Freddy Cachazo and Bruno Umbert); published in JHEP.
From weakly separated collections to matroid subdivisions (Published in Combinatorial Theory )
Minimal Kinematics: An all k and n peek into Trop+G(k,n) (joint with Freddy Cachazo); (published in SIGMA)
Scattering Equations: From Projective Spaces to Tropical Grassmannians (joint with F. Cachazo, A. Guevara and S. Mizera); (published in JHEP)
Delta algebra and Scattering Amplitudes (joint with F. Cachazo, A. Guevara and S. Mizera); (published in JHEP)
On configuration spaces and Whitehouse's lifts of the Eulerian representations (joint with V. Reiner); (published in The Journal of Pure and Applied Algebra)
Canonical Gelfand-Zeitlin modules over orthogonal Gelfand-Zeitlin algebras (joint with V. Mazorchuk and E. Vishnyakova); (published in International Mathematics Research Notices).
Supplementary materials:
Notebook to accompany Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes.
Contains explicit expressions for 6 and 7-point CEGM 3-amplitudes.
Preprints
The Chirotropical Grassmannian (Joint with Dario Antolini)
Minimal Kinematics on M_(0,n) (Joint with Anaelle Pfister and Bernd Sturmfels)
Positive del Pezzo Geometry. (Joint with Alheydis Geiger, Marta Panizzut, Bernd Sturmfels and Claudia Yun)
Factorization for Generalized Biadjoint Scalar Amplitudes via Matroid Subdivisions (30 pages)
Weighted blade arrangements and the positive tropical Grassmannian
Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams
Honeycomb Tessellations and Graded Permutohedral Blades(53 pages). Submitted.
Conjectures for Ehrhart h∗-vectors of Hypersimplices and Dilated Simplices
Generalized Permutohedra, Scattering Amplitudes, and a Cubic Three-Fold
Combinatorics and Representation Theory for Generalized Permutohedra I: Simplicial Plates
Recent and Upcoming Conferences and Workshops
Combinatorics of Fundamental Physics Workshop. (IAS Princeton)
Amplitudes 2024 (IAS Princeton).
Amplituhedra, Cluster Algebras, and Positive Geometry (CMSA)
Tropical Geometry and Infrared Divergences: Heated Discussions on Cool Topics (IAS Princeton).
Minimal Kinematics on M_{0,n}, and beyond. Harvard Center of Mathematical Sciences and Applications (CMSA), Amplituhedra, Cluster Algebras, and Positive Geometry.
Scattering amplitudes, moduli space tilings and their tropicalization. Durham, UK. Tropical Mathematics & its Applications. March, 2024.
Figure 1: it's the degeneration of a permutohedron to a "period solid" or fundamental root parallelepiped, part of the development of a quantum analog of plates, in my Ph.D. thesis. The middle rows of the matrices give the destinations of the vertices in the degeneration.
Figure 2: uses machinery developed in From weakly separated collections to matroid subdivisions. It is the exchange graph for a "condensation" around certain multi-split matroid subdivisions of the hypersimplex D(5,12). However, the algorithm used here to generate the figure requires only elementary properties of weakly separated collections.
Selected posters and invited talks:
Scattering Amplitudes and Tilings of Moduli Spaces. University of Hertfordsire, Physics Seminar, February 2024.
Scattering Amplitudes and Tilings of Moduli Spaces, Perimeter Institute for Theoretical Physics, September 2023
From matroid subdivisions of hypersimplices to generalized Feynman diagrams. Higgs Centre for Theoretical Physics, Cluster Algebras and the Geometry of Scattering Amplitudes, March, 2020.
Harvard University, Mathematical Physics Seminar, February 2018
University of Minnesota, Combinatorics Seminar, December, 2017
University of Minnesota, Combinatorics Seminar, April, 2017
University of Sao Paulo, Seminar on Lie and Jordan Algebras and their Representations
University of Montreal, (Scattering Amplitudes and the Positive Grassmannian)
University of Michigan, Combinatorics Seminar
Symmetry and Geometric Structure for the Worpitzky identity. ( Grassmannian Geometry of Scattering Amplitudes)
Abstract: The classical Worpitzky identity for the symmetric group $S_n$ decomposes a cubical lattice into $n!$ simplices of different sizes, each with a multiplicity encoded by the number of permutations of $n$ with a fixed number of descents. Traditionally, this is understood to be a purely combinatorial identity. We unpack its full symmetry, observing that the Worpitzky identity amounts to the decomposition of a polyhedral representation of the symmetric group into vector spaces spanned by localized polyhedral cones. It is well-known that the Eulerian numbers become volumes of hypersimplices; on the other hand, it is not known that the hypersimplices are actually vector spaces of dimensions the Eulerian numbers. We show how the Worpitzky identity encodes localization data for our polyhedral cones, and finally we introduce the Character Polyhedron, whose volume equals the number of fixed points of a given permutation.
