Analysis of Dirac Systems and Computational Algebra

1 Background Material.- 1.1 Algebraic tools.- 1.2 Analytical tools.- 1.3 Elements of hyperfunction theory.- 1.4 Appendix: category theory.- 2 Computational Algebraic Analysis.- 2.1 A primer of algebraic analysis.- 2.2 The Ehrenpreis-Palamodov Fundamental Principle.- 2.3 The Fundamental Principle for hyperfunctions.- 2.4 Using computational algebra software.- 3 The Cauchy-Fueter System and its Variations.- 3.1 Regular functions of one quaternionic variable.- 3.2 Quaternionic hyperfunctions in one variable.- 3.3 Several quaternionic variables: analytic approach.- 3.4 Several quaternionic variables: an algebraic approach.- 3.5 The Moisil-Theodorescu system.- 4 Special First Order Systems in Clifford Analysis.- 4.1 Introduction to Clifford algebras.- 4.2 Introduction to Clifford analysis.- 4.3 The Dirac complex for two, three and four operators.- 4.4 Special systems in Clifford analysis.- 5 Some First Order Linear Operators in Physics.- 5.1 Physics and algebra of Maxwell and Proca fields.- 5.2 Variations on Maxwell system in the space of biquaternions.- 5.3 Properties of DZ-regular functions.- 5.4 The Dirac equation and the linearization problem.- 5.5 Octonionic Dirac equation.- 6 Open Problems and Avenues for Further Research.- 6.1 The Cauchy-Fueter system.- 6.2 The Dirac system.- 6.3 Miscellanea.