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Research interests

My current research focuses on random polynomial theory, a field at the intersection of probability, stochastic processes, analysis, and combinatorics. Random polynomials arise naturally in models with randomness or noise, including differential and difference equations with random coefficients, partial differential equations with random initial data, and the spectral theory of random matrices. They have applications in numerical analysis, statistical regression, quantum chaos, filtering theory, communication systems, mathematical physics, and economics. A central theme of my work is understanding the behavior of their random roots, especially the distribution and number of real roots, which form intricate random point processes. I study the variance, limiting distributions, and concentration of the number of real roots across a variety of polynomial ensembles. Earlier in my career, I also worked on real functions, integral transforms, operational calculus, functional analysis, ODEs, PDEs, and operator theory. 

News and updates

• June 23, 2025: Yen Q. Do and Nhan D. V. Nguyen, Real roots of random polynomials: Asymptotics of the variance, Electron. J. Prob. 30 (2025), article no. 112, 1-37. [Link]

• June 17, 2025: I was awarded the 2025 AMS-Simons Travel Grant, which provides up to $2,500 per year for two years to support research-related travel. [Link]

• May 19, 2025: I am an invited speaker at the upcoming workshop Random Polynomials and Their Applications, to be held August 4-8, 2025, at the Institute for Computational and Experimental Research in Mathematics (ICERM), Brown University. [Link]

• May 12, 2025: Summer REU, Real roots of random polynomials, May 12 - June 27, 2025. Undergraduates: Farid Ahmadov, Melanie Fouque, Ayush Khadka, and Connor Shanley. [Link]