Nathan Hayford

Hi!

I am currently a postdoc in the department of mathematics at the Royal Institute of Technology (KTH) working under the supervision of Maurice Duits. Before that I was a PhD student at the University of South Florida (USF) of Razvan Teodorescu and Seung-Yeop Lee.

If you have any questions, feel free to send me an email at:
nhayford [at] kth.se

Research Interests

I work mostly on problems in complex analysis and mathematical physics.

A few of my interests are:

Random matrix theory
Conformal field theory and 2D quantum gravity
Statistical mechanics and exactly solvable lattice models
Riemann-Hilbert problems and orthogonal polynomials
Free boundary problems

List of Publications

N. Hayford, The Ising Model Coupled to 2D Gravity: Higher-order Painlevé Equations/The (3,4) String Equation . arXiv preprint, 2024.
M. Duits, N. Hayford, & S.-Y. Lee, The Ising Model Coupled to 2D Gravity: Genus Zero Partition Function. arXiv preprint, 2024.
N. Hayford & F. Wang. A Note on Electrified Droplets. Comput. Methods Funct. Theory 54, 2021.
A. Abanov, N. Hayford, D. Khavinson, & R. Teodorescu. Around a theorem of F. Dyson and A. Lenard: Energy Equilibria for Point Charge Distributions in Classical Electrostatics. Expositiones Mathematicae 39 (2), 2021.

Upcoming Work

My most recent project (joint work with Maurice Duits and Seung-Yeop Lee) is on the 2-matrix model, and its relation to the Ising model on random graphs. In what will probably be a series of 2-3 papers, we make rigorous the results of physicists, which may be found in the following works (among others):

V. A. Kazakov, Ising Model on a Dynamical Planar Random Lattice: Exact Solution. Phys. Lett. A. 119 (3), 1986.
V. A. Kazakov and D. V. Boulatov, The Ising Model on a Random Planar Lattice: The Structure of the Phase Transition and the Exact Critical Exponents. Phys. Lett. B. 186 (3), 1986.
E. Brézin and V. A. Kazakov, Exactly solvable field theories of closed strings. Phys. Lett. B. 236 (2), 1990.
M. R. Douglas, Strings in less than one dimension and generalized KdV hierarchies. Phys. Lett B. 238 (2-4), 1990.
D.J. Gross and A. A. Migdal, A non-perturbative treatment of two-dimensional quantum gravity. Nucl. Phys. B. 340 (2-3), 1990.

The first two references pertain to the calculation of the planar free energy of the quartic 2-matrix model; the last three references pertain to the critical partition function for the same matrix model. Using Riemann-Hilbert methods, we are able to compute the planar free energy, and provide a more explicit characterization of the phase portrait of this model. We are also able to show that the critical partition function is a tau function for a certain integrable equation.

Recent talks

(May 2024) "The Ising Model Coupled to 2D Gravity: Genus 0 Partition Function", ARNO 2024
(Dec. 2023) "Hamiltonian Structure of an equation of Painlevé Type", Analysis seminar, Stockholm University
(Oct. 2023) "Critical Phenomena in Random Matrices: Old and New", Probability and Mathematical Physics seminar, KTH
(Jun. 2023) "Painlevé Equations and Critical Phenomena in Random Matrix Theory", Conference at Bar-Ilan University

Me talking at Great Bay University in June 2024.

Teaching Experience

While at USF, I taught:

Engineering Calculus III (Fall 2020)
Vector Calculus (Spring 2023)

Here are my lecture notes for Stokes' Theorem.

Translational Work (Not much, but I gave it an attempt)
I recently translated a paper by A. A. Kapaev ("Asymptotic behavior of the solutions of the Painlevé equation of the first kind", Differ. Uravn. 24 (10), 1986, p. 1684-1695), and am sharing it here for anyone who wants it.