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Research interests

My research lies at the intersection of probability theory and stochastic processes, analysis, combinatorics, and mathematical physics, with a focus on random polynomials, which are classical polynomials with random coefficients. These objects naturally arise in models involving randomness or noise, such as differential and difference equations with random coefficients, partial differential equations with random initial data, and the spectral theory of random matrices. Applications span numerical analysis, statistical regression, quantum chaos, filtering theory, communication systems, and mathematical economics. A central challenge is to understand the behavior of their random roots, particularly the distribution and enumeration of real roots, which form complex random point processes. My work investigates the variance, limiting distributions, and concentration of the number of real roots across various 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]