# Tao Lecture Series

## Terence Tao

Analysis Group, Department of Mathematics, UCLA, USA

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## Lecture 1: Pseudorandomness of the Liouville function

May 3, 2021, 5:30 p.m. (Bonn local time, CEST)

The Liouville pseudorandomness principle (a close cousin of the Mobius pseudorandomness principle) asserts that the Liouville function $\lambda(n)$, which is the completely multiplicative function that equals $-1$ at every prime, should be "pseudorandom" in the sense that it behaves statistically like a random function taking values in $-1,+1$. Various formalizations of this principle include the Chowla conjecture, the Sarnak conjecture, the local uniformity conjecture, and the Riemann hypothesis. In this talk we survey some recent progress on some of these conjectures.

## Lecture 2: Singmaster's conjecture in the interior of Pascal's triangle

July 16, 2021, 5 p.m. (Bonn local time, CEST)

An old conjecture of Singmaster asserts that every integer greater than 1 occurs only a bounded number of times in Pascal's triangle. In this talk we survey some results on this conjecture, and present a recent result in joint work with Kaisa Matomaki, Maksym Radziwill, Xuancheng Shao, Joni Teravainen that establishes the conjecture in the interior region of the triangle. Our proof methods combine an "Archimedean" argument due to Kane (and reminiscent of the Bombieri-Pila determinant method) with a "non-Archimedean argument" based on Vinogradov's exponential sum estimates over primes.

(Note: The first five minutes of the lecture are missing in the video recording due to technical difficulties, sorry. Please us the slides to fill the gap.)

## Lecture 3: The circle method from the perspective of higher order Fourier analysis

July 23, 2021, 5 p.m. (Bonn local time, CEST)

Higher order Fourier analysis is a collection of results and methods that can be used to control multilinear averages (such as counts for the number of four-term progressions in a set) that are out of reach of conventional linear Fourier analysis methods (i.e., out of reach of the circle method). One notable feature of this theory is that the role of linear phase functions is replaced by the notion of a nilsequence. On the other hand, key identities from linear Fourier analysis, such as the Plancherel identity or the Fourier inversion formula, are notably absent in the higher order theory. In this survey talk we give an introduction to the higher order Fourier theory by revisiting the linear circle method from a higher order perspective, in particular downplaying as much as possible the role of Fourier identities.