Such a rotation g can be described by three parameters. For instance, if you know the axis of rotation (specified by a direction or a point on S or two angles) and the angle of rotation (one parameter), g is determined uniquely. Alternatively, a matrix g ∈ R3×3 has 9 entries, but the requirement that gg = 1 gives 6 conditions on these entries, and again 9− 6 = 3. (The condition that det g = 1 s...

Harmonic analysis on the line is the theory of Fourier transforms, more complicated than Fourier series, due to the line’s non-compactness. On R the exponential functions, while still eigenfunctions for d dx and still giving group homomorphisms, are no longer in L(R). Entangled with this point is the fact that Fourier inversion expresses functions as integrals of exponential functions, not as s...

Chapter 4 is concerned with meromorphic continuation and examples of various Riemann surfaces, and Chapter 5 deals with Fuchsian groups. The chapter finishes with a proof of Hurwitz's theorem that a compact Riemann surface of genus g has at most 84(g—1) automorphisms, together with a discussion of possible equality. There are four appendices of which the first consists of a review in six pages ...

These notes were taken in the second week of the 2011 Study Analysis and Geometry summer program at Princeton.

In this set of notes we shall investigate applications of commutative harmonic analysis, which for us will mean the Fourier transform on commutative groups. The focus will be on examples, and these will come from a range of settings including functional analysis, number theory, and probability. Most of what we do will focus on quantitative estimates related to finite groups, and in light of thi...

In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold Γ\Hn. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of L(Γ\H) into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and charac...

‎Einstein‎, ‎M"{o}bius‎, ‎and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time $bkR^{n,1}.$ Using the gyrolanguage we study their gyroharmonic analysis‎. ‎Although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them‎. ‎Our...

1. A doomed-but-plausible trial calculation 2. Matrix exponentiation 3. Rotation-invariant derivatives on R 4. Instrinsic derivatives 5. Aside: GLn(R)-invariance of the Euler operator 6. ∑ i<j X 2 ij in coordinates on R 7. X` commutes with ∑ i<j X 2 ij 8. Stabilization of Hd by Xγ 9. Integration by parts for Xγ 10. Sup norms of derivatives of harmonic polynomials 11. Termwise differentiation of...