I will describe recent progress on our understanding of the factorization of the integers, specifically consecutive integers. The main theme is the tension between the additive and multiplicative structure of the integers. This is a central topic in number theory, connected among others to problems of equidistribution of arithmetic objects (subconvexity) or more classical problems such as the twin prime conjecture (parity obstruction).

The first significant result towards Chowla's conjecture goes back to my work with Matomaki from 2015. In the last six years this particular sub-area gave rise to several new ideas in analytic number theory, specifically ideas related to entropy, expander graphs and additive combinatorics. Among the recent achievements are results on local Fourier uniformity and expansion in thin graphs connected with prime divisors of integers.

In turn progress on this basic question gave back various results beyond number theory in areas as distinct as combinatorics (Erdos discrepancy problem), mathematical physics (spacing betweens eigenfunctions of the Laplacian on generic rectangular tori), cryptography (smooth numbers in short intervals), ergodic theory (Sarnak's conjecture), etc.

I will discuss various papers joint with Matomaki, Helfgott, Ziegler, Tao and Teravainen and also progress by others, e.g Tao, Walsh, Frantzikinakis and Host.

https://umich.zoom.us/j/92876413255?pwd=UUs5MitnN1FhVVRCb3RIOTlWWThkdz09

Meeting ID: 928 7641 3255
Passcode: UMColloq Speaker(s): Maksym Radziwill (Caltech)