Julian Sahasrabudhe

Julian Sahasrabudhe (born May 8, 1988) is a Canadian mathematician who is an assistant professor of mathematics at the University of Cambridge, in their Department of Pure Mathematics and Mathematical Statistics. His research interests are in extremal and probabilistic combinatorics, Ramsey theory, random polynomials and matrices, and combinatorial number theory.

Life and education Sahasrabudhe grew up on Bowen Island, British Columbia, Canada. He studied music at Capilano College and later moved to study at Simon Fraser University where he completed his undergraduate degree in mathematics. After graduating in 2012, Julian received his Ph.D. in 2017 under the supervision of Béla Bollobás at the University of Memphis. He currently holds a position as an assistant professor in the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge. in 2011. He proved that for every finite colouring of the natural numbers there exists such that the triple is monochromatic, demonstrating the partition regularity of complex exponential patterns. This work marks a crucial development in understanding the structure of numbers under partitioning.

In 2023, Sahasrabudhe submitted a paper titled An exponential improvement for diagonal Ramsey along with Marcelo Campos, and Robert Morris. In this paper, they proved that the Ramsey number

This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.

Sahasrabudhe has also worked with Marcelo Campos, Matthew Jenssen, and Marcus Michelen on random matrix theory with the paper The singularity probability of a random symmetric matrix is exponentially small. The paper addresses a long-standing conjecture concerning symmetric matrix with entries in . They proved that the probability of such a matrix being singular is exponentially small. The research quantifies this probability as where is drawn uniformly at random from the set of all symmetric matrices and is an absolute constant.

In 2020, Sahasrabudhe published a paper named Flat Littlewood Polynomial exists, which he co-authored with Paul Ballister, Bela Bollobás, Robert Morris, and Marius Tiba. This work confirms the Littlewood conjecture by demonstrating the existence of Littlewood polynomials with coefficients of that are flat, meaning their magnitudes remain bounded within a specific range on the complex unit circle. This achievement not only validates a hypothesis made by Littlewood in 1966 but also contributes significantly to the field of mathematics, particularly in combinatorics and polynomial analysis.

In 2022, the authors worked on Erdős covering systems with the paper On the Erdős Covering Problem: The Density of the Uncovered Set. They confirmed and provided a stronger proof of a conjecture proposed by Micheal Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, and Gang Yu, which states that for distinct moduli within the interval , the density of uncovered integers is bounded below by a constant. Furthermore, the authors establish a condition on the moduli that provides an optimal lower bound for the density of the uncovered set. for his contribution to applying combinatorial methods to problems in harmonic analysis, combinatorial number theory, Ramsey theory, and probability theory. for his contribution to harmonic analysis, probability theory, and combinatorics. More specifically, Sahasrabudhe improved the bound on the singularity probability of random symmetric matrices and obtained a new upper bound for diagonal Ramsey numbers.

Personal life Sahasrabudhe's parents are television producers Deepak Sahasrabudhe and Susan Miller. His grandfather, Madhu Sahasrabudhe, was the acting director of the Food Research Institute of Agriculture and Agri-Food Canada.

Publications

Selected research articles Exponential Patterns in Arithmetic Ramsey Theory (2018)

Flat Littlewood polynomials exist (2020) The singularity probability of a random symmetric matrix is exponentially small (2021)

On the Erdős Covering Problem: the density of the uncovered set (2022)

An exponential improvement for diagonal Ramsey (2023)