Topics in multiplicative number theory

In this thesis we use a blend of analytic and combinatorial techniques to address a variety of problems about primes and related multiplicative structures: We introduce a modified linear sieve, and obtain new upper bound for twin primes, refining a 2004 result of Wu. This represents the largest improvement on the problem since 1986. We prove the equidistribution of primes p in arithmetic progressions to moduli of size p^{0.5313}, and consequently obtain the infinitude of shifted primes p-1 without prime factors above p^{0.2844}, refining a 1998 result of Baker and Harman. We introduce a new hybrid conjecture of the famous conjectures of Hardy--Littlewood and Chowla on correlations of the Mobius and von Mangoldt functions. We prove several `on average' results on the problem (in part jointly with J. Teravainen), extending work of Matomaki, Radziwill, and Tao. We study primitive sets. The Erdos primitive set conjecture, posed in 1986, asserts that the set of primes is maximal among all primitive sets, in a precise sense. We explore variations of this problem in multiple directions, where many open questions remain. We disprove a natural generalization of the conjecture, due to Banks and Martin in 2013, by showing the k-almost primes are minimal when k=6, in the same sense of Erdos. We also show a translated analogoue of the conjecture is false already for translates h~1.04, highlighting the subtlety of the original problem. Finally, this thesis culminates with a proof of the Erdos primitive set conjecture

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Quanta of Maths

The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics

The Material Theory of Induction

The fundamental burden of a theory of inductive inference is to determine which are the good inductive inferences or relations of inductive support and why it is that they are so. The traditional approach is modeled on that taken in accounts of deductive inference. It seeks universally applicable schemas or rules or a single formal device, such as the probability calculus. After millennia of halting efforts, none of these approaches has been unequivocally successful and debates between approaches persist. The Material Theory of Induction identifies the source of these enduring problems in the assumption taken at the outset: that inductive inference can be accommodated by a single formal account with universal applicability. Instead, it argues that that there is no single, universally applicable formal account. Rather, each domain has an inductive logic native to it.The content of that logic and where it can be applied are determined by the facts prevailing in that domain. Paying close attention to how inductive inference is conducted in science and copiously illustrated with real-world examples, The Material Theory of Induction will initiate a new tradition in the analysis of inductive inference