About

I’m Noam Borgnia, a master’s student at the Technical University of Munich. I studied mathematics at Princeton University, graduating in 2024. My interests sit between probability, mathematical physics, and theoretical computer science.

I’m drawn to the point where a complicated system admits a simpler description. In kinetic theory, that means understanding how local interaction rules give rise to mean-field equations, and when that passage can be justified. I’m especially interested in high-density regimes, and in identifying which correlations a mean-field description can safely neglect.

Related questions show up in the scaling exponents of Ising and percolation, and in hydrodynamic limits of lattice dynamics. I’m interested in the connections between these models and descriptions. I particularly like arguments in which a well-chosen coupling turns a difficult question into one we already know how to approach.

My master’s thesis, supervised by Noam Berger, studies cluster–cluster aggregation. I study how clusters grow as they move and merge, the geometry they leave behind, and what changes when the dynamics conserve momentum. It is a concrete setting in which to ask how much a description based only on cluster masses can capture, and when spatial geometry becomes essential.

My bachelor’s thesis, supervised by Jacob Shapiro, asked how the complex zeros of the Ising partition function relate to the decay of spin correlations. That question grew into ongoing work with Jacob and Jui-Hui Chung on the Lee–Yang gap and mass gap. I like this project for the bridge it builds between two ways of seeing a phase transition: the geometry of zeros in a complex parameter and the distance over which spins remain correlated.

I’m also interested in the interplay between stochastic processes and quantum information, especially phase transitions in entanglement entropy, as well as quantum walks in disordered environments.

On the computer science side, I’m interested in average-case complexity: what makes typical instances of a problem hard, and what low-degree methods can tell us about computational barriers. I think there is a case for complexity theory as the most valuable mathematical heuristic in the age of AI. It gives us a language for asking what structure makes a problem tractable, what resources a solution needs, and which obstacles better algorithms might overcome.

Alongside my research, I mentor students on long-term passion projects shaped by their own interests. I enjoy helping a student turn an initial curiosity into a question they can investigate, a model they can build, or an argument they can make their own. This site collects that work alongside my research, simulations, and expository writing.