An Affine Sieve

该建议涉及将经典的和较新形式的组合筛应用于仿射n-空间的一组保持整点的态射的轨道的设置。这种“仿射筛法”统一和推广了寻找一个多项式在其上取值为素数或素因数较少的点的问题。这种“仿射筛子”在经典和新的丢番图问题上都有许多应用。在这种情况下，用于开发有效筛子的方法涉及自同构形式、扩展图以及意外算术和加性组合数学。孪生素数猜想断言，有无限多对素数相差两个。它是数学中历史最悠久的悬而未决的问题之一。虽然这类问题首先是由好奇心驱动的，但为他们的研究而发明的技术已被证明是更广泛的基础。该提案涉及对孪生素数猜想的深远推广，并开发新的技术来证明这些一般性猜想的一部分。近年来，数论、组合学和理论计算机科学之间的相互作用一直非常活跃。在这一背景下，数论思想多次应用于其他两个领域。在本项目中，反向应用也是至关重要的。

The proposal is concerned with applying classical and more recent forms of the combinatorial sieve, in the setting of an orbit of a group of morphisms of affine n-space,which preserves integer points. The setting unifies and generalizes the problem of finding points at which a polynomial takes on values which are prime or has few prime factors.This "affine sieve" has numerous applications to both classical and novel diophantine problems. The methods used to develop an effective sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic and additive combinatorics. The Twin Prime Conjecture asserts that there are infinitely many pairs of prime numbers which differ by two. It is one of the longest standing unsolved problems in mathematics. While such problems are driven first by curiosity, the techniques that have been invented for their study have proven to be fundamental more broadly. The proposal is concerned with far-reaching generalizations of the twin prime conjecture and with developing new techniques to prove parts of these general conjectures. The interplay between number theory, combinatorics and theoretical computer science has been a very active one in recent years. Many times in this context the applications have been of number theoretic ideas to the other two fields. In the present project the reverse application is also critical.