An Affine Sieve

该建议涉及应用经典和最近形式的组合筛，在设置的轨道的一组态射的仿射n-空间，保持整数点。该设置统一和推广的问题，找到点的多项式采取的值是素数或有几个素factors.This“仿射筛”有许多应用经典和新颖的丢番图问题。在这种情况下，用于开发有效筛子的方法涉及自守形式，扩展图和意想不到的算术和加法组合学。 孪生素数猜想断言有无穷多对素数相差2。这是数学中最长的未解决问题之一。虽然这些问题首先是由好奇心驱动的，但为研究这些问题而发明的技术已被证明是更广泛的基础。该提案涉及孪生素数猜想的广泛推广，并开发新的技术来证明这些一般猜想的一部分。近年来，数论、组合数学和理论计算机科学之间的相互作用非常活跃。许多时候在这方面的应用一直数论思想的其他两个领域。在本项目中，反向应用也很关键。

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.