Here are some notes to accompany my 2023 class at the Fields Institute on the tangent groupoid in noncommutative geometry (that, at least, is the official name of the class). Hopefully I will persevere with them until the end, but no guarantees!
The notes are not meant to stand by themselves; instead my aim is recall the main topics covered, the main definitions, the statements the main theorems, and so on. There are also some homework exercises. And I'll use the notes to correct mistakes that I made during the lectures (even while making fresh mistakes here) and to include remarks that I wish I had made during the lectures, but didn't.
I may occasionally update the individual lectures if there are enough corrections to justify doing so. In that case I'll indicate below when the updates were made. The most up to date versions of all the (published) lectures together can be found here.
Nigel Higson, September 2023
Lecture 1: Introduction: Fundamental Solutions and Parametrices. The Laplace operator. Fundamental solutions. Smoothing operators. Parametrices. Hypoellipticity. Properly supported operators. The Fourier transform. A glimpse at pseudodifferential operators. Lecture notes
Some Background Reading: It will be helpful, but not completely essential to have some familiarity with the theory of distributions. A nice introduction is Friedlander's text Introduction to the theory of distributions. The earlier editions are shorter, and in my opinion they are better for that.
Lecture 2: Scalable Operators. Rescaling operators. Scaling families. Examples. Scalable operators. Lecture notes (updated 18 September)
Lecture 3: First Properties of Scalable Operators. Pseudolocality. Scalable operators of sufficiently negative order. Symbol families of infinite negative order. Symbol families. The symbol family of a scaling family of smoothing operators. Lecture notes
Lecture 4: Asymptotic Expansions. Symbols of scalable operators. The operator associated to a symbol family. Scalable operators from asymptotic series.
Lecture notes
Lecture 5: Elliptic Operators. Elliptic symbols and elliptic operators. Parametrices. Scalable operators versus pseudodifferential operators. Lecture notes
Lecture 6: Symbols and the Fourier Transform. Fourier transform of a symbol family. Inverse Fourier transform. More on elliptic operators. Lecture notes
Lecture 7: Hilbert Space Issues. Adjointability of scalable operators. Compactness of negative order operators. Boundedness of order zero operators. The pseudodifferential C*-extension. Lecture notes
Lecture 8: A First Look at the Tangent Groupoid. Transformation groupoids. The tangent groupoid for Euclidean space. Differential operators on the source fibers. Category structure. Diffeomorphisms. Lecture notes
Lecture 9: The Scaling Action. The tangent groupoid of a smooth manifold. The scaling action. Scalable operators revisited.
Lecture notes
Lecture 10: The Deformation to the Normal Cone. The character spectrum of an algebra over the reals. Smooth functions on a smooth manifold. Submanifolds and normal bundles. The Rees algebra. The deformation to the normal bundle. The tangent groupoid, revisited. Lecture notes
Lecture 11: Guest Lecture on Euler-Like Vector Fields by Dan Hudson. Summary
Lecture 12: Functions on the Tangent Groupoid Densities. Haar integrals on a Lie group. Families of Densities. Haar systems on a Lie groupoid. The reduced groupoid C*-algebra. The case of the tangent groupoid. Scaling automorphisms. Lecture notes
Lecture 13: Continuous Field, Multipliers and Scalable Operators Continuous fields of C*-algebras. Multiplier algebras. Scaling families as multipliers. Order zero families as C*-multipliers. Lecture notes
Lecture 14: The index problem Fredholm operators. A quick look at K-theory. The index problem. Lecture notes
Lecture 15: The Atiyah-Singer Index Theorem An action groupoid. The special case of the tangent groupoid. Steps in the proof of the index theorem. Lecture notes
Lecture 16: The Dirac Operator Clifford matrices and the Dirac operator on Euclidean space. Dirac operators on submanifolds. Essentially self-adjoint operators with compact resolvent. Families and K-theory. Lecture notes
Lecture 17: The Index Theorem for the Dirac Operator The index formula, a family of operators parametrized by points in the deformation to the normal cone. The harmonic oscillator. A final homotopy. Lecture notes
Lecture 18: Deformations of Group C*-Algebras The full and the reduced group C*-algebra. A continuous field of group C*-algebras. The Connes-Kasparov isomorphism. A glimpse of the Baum-Connes conjecture. Lecture notes
Lecture 19: The Mackey Bijection K-Theory and discrete series. Morita equivalences. The Mackey Bijection. Lecture notes
Lecture 20: Scalable Operators and Representation Theory, I Reductive groups of real rank one. Tempiric representations. Equivariant scalable operators on the symmetric space. C*-categories from norm-completions. Scaling families and the Connes-Kasparov isomorphism. Lecture notes
Lecture 21: Scalable Operators and Representation Theory, II Multiplicities for scalable operators. K-theory for scalable operators and Vogan's theorem. A Fourier transform for scalable operators. Lecture notes