The purpose of this talk is to present an (apparently) new way to look at the intersection complex of a singular variety over a finite field, or, more generally, at the intermediate extension functor on pure perverse sheaves, and an application of this to the cohomology of noncompact Shimura varieties.

pdf/proceedings-of-the-international-congress-of-mathematicians-32epad3tqi.pdf

Proceedings of the International Congress of Mathematicians

Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field.

Bounded Gaps Between Primes: The Epic Breakthroughs of the Early Twenty-First Century

Let $P^+(n)$ denote the largest prime factor of the integer $n$ and $P_y^+(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, firstly we show that the triple consecutive integers with the two patterns $P^+(n-1)>P^+(n) P^+(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any $J\in \mathbb{Z}, J\geqslant3$, the $J-$tuple consecutive integers with the two patterns $P^+(n+j_0)= \min\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ and $P^+(n+j_0)= \max\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ also have a positive proportion respectively. Secondly for $y=x^{\theta}$ with $0<\theta\leqslant 1$ we show that there exists a positive proportion of integers $n$ such that $P_y^+(n)<P_y^+(n+1)$. Specially, we can prove that the proportion of integers $n$ such that $P^+(n)<P^+(n+1)$ is larger than 0.1356, which improves the previous result "0.1063" of the author.

Sur les plus grands facteurs premiers d'entiers cons\'ecutifs

This article proposes a synthesized classification of some Goldbach-like conjectures, including those which are “stronger” than the Binary Goldbach’s Conjecture (BGC) and launches a new generalization of BGC briefly called “the Vertical Binary Goldbach’s Conjecture” (VBGC), which is essentially a metaconjecture, as VBGC states an infinite number of conjectures stronger than BGC, which all apply on “iterative” primes with recursive prime indexes (i-primeths).

The “Vertical” Generalization of the Binary Goldbach’s Conjecture as Applied on “Iterative” Primes with (Recursive) Prime Indexes (i-primeths)

In this paper we prove that every sufficiently large odd integer can be written as a sum of a prime and 2 times a product of at most two distinct odd primes. Together with Chen’s theorem and Ross’s observation, this shows every sufficiently large integer can be written as a sum of a prime and a square-free number with at most three prime divisors.

On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors

In this paper, we give a modified proof of Chen's theorem "every sufficiently large even inte- ger is a sum of a prime and a product of at most 2 primes".

On the representation of every large even integer as a sum of a prime and an almost prime

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On the representation of every large even integer as a sum of a prime and an almost prime