Research

December 2025

Cutting Open Matrix Models

Pre Thesis Presentation

📍 Princeton University, NJ

In this pre-thesis talk, we develop a conceptual bridge between two sources of rich combinatorics: Hermitian matrix models and the recently introduced surface functions and cut equation. We begin by reviewing how matrix models generate ribbon graphs, how the free energy organizes a topological expansion, and how classical tools such as Schwinger–Dyson (Virasoro) constraints and Tutte-style recursions encode the counting and gluing of vacuum fatgraphs. We then introduce surface functions as generating functions for topologically inequivalent polyangulations of a surface, built from variables attached to curve classes up to mapping-class-group action, and we emphasize the cut equation as a remarkably direct rule: differentiating with respect to a curve variable corresponds to cutting the surface along that curve, recursively reducing general vacuum surfaces to products of disk data. Taking a suitable "matrix model limit" of surface functions, we show how these objects resemble matrix-model contributions and use this perspective to sharpen the central question of the talk: how the cut-equation recursion relates to the Virasoro constraints that traditionally control matrix models. We further organize the constraints using correlators and resolvents, highlighting loop-equation recursions that mirror Berends–Giele-type structures and providing explicit diagram-counting checks. Finally, we present stringy integrals as alpha-prime deformations of surface functions that capture basic open-string combinatorics and connect naturally to moduli-space geometry and Weil–Petersson volumes, while also explaining why such deformations cannot, in general, be absorbed into a single one-matrix model potential across all topologies.

April 2025

Motivating the Associahedron

Research Group Meeting

📍 Princeton University, NJ

In this talk, we explain how planar scattering amplitudes can be organized and understood using combinatorial and geometric ideas rather than relying only on individual Feynman diagrams. We start by showing how the relevant kinematic channels can be encoded using a momentum-polygon picture and a related "kinematic mesh," which helps separate genuine physical singularities from spurious ones that arise from particular representations. We then introduce the associahedron as a concrete geometric object whose faces and boundaries systematically capture the allowed factorization channels of the amplitude, illustrating the construction with simple low-point examples. The main message is that factorization becomes a statement about approaching specific boundaries of this polytope, so the amplitude naturally splits into products of lower-point amplitudes in a way that is controlled by geometry. We conclude by noting that this perspective extends beyond the simplest cubic theory to broader classes of colored theories and connects to familiar combinatorial structures, matrix-model intuition, and modern amplitude relations.

October 2022

Liouville Theory as a 2D Bulk Quantum Gravity and Matrix Models

American Physical Society Far West Annual Conference

📍 University of Hawaii, Honolulu

The aim of this program is to study the case of c=1 Liouville Theory having a dual description in terms of Matrix Quantum Mechanics (MQM) of N-ZZ D0 Branes. Here, instead of the conventional approach, where one interprets Liouville Theory as a worldsheet Conformal Field Theory (CFT, String Theory) embedded in a 2-dimensional target space, we take Liouville Theory as the Quantum Gravity Theory in bulk spacetime. This approach is corroborated by the fact that a holographic connection can be seen as in the case of a single Hermitian matrix model describing (2,p) minimal models coupled to gravity, where the physics of JT-gravity can be reached as a limit of these models. We study the aforementioned theory since it is a richer UV-Complete theory of 2D-gravity with matter. The Matrix Models here do not play the role of their boundary duals, but give a direct link to the third quantized Hilbert Space description, i.e The target space of c=1 string plays the role of the superspace in which these two dimensional geometries are embedded. From the Matrix Model point of view, we introduce appropriate loop operators to create macroscopic boundaries on the bulk geometry. We do this so that the boundary is of fixed size 𝑙 and is related to the temperature 𝛽 of the holographic dual theory. Here we are currently looking at two-point macroscopic loop operator correlators corresponding to Euclidean wormhole geometry and three-point correlators with a (local) Vertex operator on the same Geometry, which corresponds to the insertion of an operator on the boundary. We initially look at these objects at genus zero and then use MQM to study them at higher genera.