Sieve Methods with Applications

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The sieve theory offers tools for selecting subsequences of particular interest from a larger sequence which is typically more accessible by other means. For example in the recent developments prime numbers were captured in polynomial values of degree four. The proposal goes further to solve problems not only concerning prime numbers but also for solving some diophantine equations and estimating the rational points on some cubic surfaces. In more theoretical aspects of sieve theory the goal of the Proposal is to investigate the intrinsic limitations of the methods (parity barrier of sieve) and to find ways to break these limits. Sieve ideas when enhanced with arguments of harmonic analysis (spectral methods) become powerful and versatile tools which can even exceed the capability of the Grand Riemann Hypothesis. The Proposal makes a few suggestions in this direction.Sieve methods turned out to be very attractive for researchers working in cryptography. Although this project does not address such applications directly, it seems likely that advances in the theory of sieves will open new possibilities. The implementation of Fourier analysis to sieve methods creates a lot of demand in modern harmonic analysis and will certainly have valuable impact on shaping the latter. These developments in the interface of combinatorial ideas and analysis will be quite inspiring for graduate students.