Intro -- Introduction -- Aspects of multilinear harmonic analysis related to transversality -- 1. Introduction -- 2. The linear restriction theory for the Fourier transform -- 3. The bilinear restriction theory: the emergence of transversality -- 4. Multilinear transversality and the multilinear restriction theory -- 5. Transversal multilinear harmonic analysis: a bigger picture -- References -- Multi-frequency Calderón-Zygmund analysis and connexion to Bochner-Riesz multipliers -- 1. Notations and preliminaries -- 2. Unweighted estimates for multi-frequency Calderón-Zygmund operators -- 3. Weighted estimates for multi-frequency Calderón-Zygmund operators -- 4. Connexion to Bochner-Riesz multipliers -- References -- Sharp Bounds for -Haar Multipliers on ² -- 1. Introduction -- 2. Preliminaries -- 3. Main tools -- 4. Haar Multipliers -- 5. Appendix -- References -- Spectra of Frame Operators with Prescribed Frame Norms -- 1. Introduction -- 2. Background results about Schur-Horn type theorems -- 3. Spectral set -- References -- Regularity criteria for Hall-magnetohydrodynamics and the space-time Monopole equation in Lorenz gauge -- 1. Introduction -- 2. Proof of Theorem 1.1 -- 3. Proof of Theorem 1.3 -- 4. Proof of Theorem 1.7 -- Acknowledgments -- References -- The ₂ theorem: Remarks and complements -- 1. Introduction -- 2. Detailed outline of the lower-left route -- 3. Proof of Lerner's formula -- 4. Sharpness of the weak (1,1) estimate -- 5. A direct verification of the testing conditions -- 6. The two-weight 1 theorem for positive operators -- References -- An invitation to harmonic analysis associated with semigroups of operators -- 1. A sample case -- 2. What is a framework to setup the theory? -- 3. BMO spaces, interpolation and duality -- 4. Hörmander-Mihlin multipliers and Riesz transforms. 5. What are Calderón-Zygmund singular integrals? -- 6. Metric on noncommutative measure spaces -- References -- The higher order regularity Dirichlet problem for elliptic systems in the upper-half space -- 1. Introduction -- 2. Preliminaries -- 3. Poisson kernels -- 4. The Dirichlet problem with data in higher order Sobolev spaces -- 5. Examples of boundary problems of mathematical physics -- References -- Perturbations of elliptic operators in chord arc domains -- 1. Introduction -- 2. Preliminaries -- 3. Main Result -- References -- Cauchy non-integral formulas -- 1. Introduction -- 2. Carleson estimates of operators -- 3. Proof of the Cauchy formula -- 4. Applications to the Neumann and Dirichlet problem -- References.