Efficient algorithms and applications of symbolic computation

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The overarching theme of this proposed program is the design, analysis and implementation of efficient algorithms for a number of important non-commutative algebraic structures in symbolic computation (also called computer algebra), such as Ore (skew) polynomials, polynomials over some non-commutative rings, generalized differential-difference algebras and matrices over these non-commutative algebras. Their study is motivated by many applications in coding theory, control theory, cryptography and engineering, etc.******The challenge in this area is that these non-commutative algebras generally have a much more complex structure when compared with commutative case.  In particular, many of the algorithmic breakthroughs in symbolic computation over the past three decades do not obviously apply in non-commutative domains. This is due both to the non-existence of particular mathematical properties in these rings or their inherently more difficult formulations. My research under this program will be to consider some of the most important computational problems in these non-commutative algebras at both a theoretical and practical level, which include fast algorithms for factoring Ore polynomials (both simple variable and multivariate cases), computing normal forms and inverses of matrices over Ore polynomial rings, and fast algorithms for computing Groebner bases in generalized differential-difference algebra and hope to achieve a unified and efficient approach to many well-known non-commutative domains. We also wish to give some applications, e.g.,  computing Gelfand-Kirillov dimensions,  and classifying Post-Lie algebras. The algorithmic advances of this proposal will be implemented in computer algebra software such as Maple, SAGE and Singular.  ****The proposed research plans all involve extensive training of highly qualified personnel (HQP) for their future positions in academia and industry.********