This seminar is dedicated to spin geometry and it's applications. The main source of material will be Spin Geometry by Lawson & Michelsohn. The book can be obtained for free (online access) by UC Berkeley students via the library website. During the semester we will cover most of Chapters 1 & 2 from Lawson & Michelsohn plus additional topics decided upon by the group. Seminar outline is detailed below, though it is subject to change.
For the second half of the seminar we intend to cover the index theorem, which is detailed in chapter III of Lawson & Michelsohn. Other useful sources and supplemental material include,
Atiyah, M. (1967). K-Theory. New York: W.A Benjamin.
Landweber, GD. "K-Theory and Elliptic Operators." 2018. eprint: arXiv:math/0504555v1
Seminar will be at 12 in rm 891 every Wednesday (Except 2/19 we are in 939!!).
| Week No. | Date | Lecture Topic | Speaker | Sources |
|---|---|---|---|---|
| 1 | 1/25 | Planning Meeting | N/A | |
| 2 | 1/30 | Introduction to Clifford algebras, Clr,s, examples. | Meredith | I.1, I.3, I.4 |
| 3 | 2/6 | The adjoint map and the groups Spin and Pin. | Meredith | I.2 |
| 4 | 2/13 | Representations of Clifford algebras I. | Calvin | I.5 |
| 5 | 2/20 | Representations of Clifford algebras II. **(RM 939)** | Calvin | I.5 |
| 6 | 2/27 | Spin structure on vector bundles. | Anthony | II.1 |
| 7 | 3/6 | Principal G-bundles, Clifford & spinor bundles. | Meredith | II.3, Appendix A |
| 7 | 3/13 | Connections on spinor bundles. | Alex | II.4 |
| 8 | 3/20 | The Dirac operator. | Eugene | II.5 |
| 9 | 4/3 | K-Theory and the Atiyah-Bott-Shapiro construction. | Xiaohan | I.9 |
| 10 | 4/10 | Differential operators, elliptic operator, and defining the topological index. | Meredith | III.1, III.13 |
| 11 | 4/17 | Topological index, its axioms and an outline of the proof of the index theorem. | Meredith | III.12, III.13, [GDL] |
| 12 | 4/24 | Redefining the analytic index and showing it satisfies the axioms. | Charlie | III.13, [GDL] |
Determine an isomorphism Cl3 ≅ C(2), and determine the possible forms of elements in Pin3 and Spin3 in terms of the generators of Cl3 (use the canonical basis of R3).
Prove that Clr,s ⊗ Cl1,1 ≅ Clr+1,s+1. Use this to compute Cl2,2 and Cl3,3.
Determine the center of Spinn for all n, then determine the center of Spinr,s for all r,s.
Show that the mapping i → e2e3, j → e3e1, k → e1e2 determines an isomorphism of the quaternions onto the even part of Cl3 such that conjugation in H corresponds to conjugation in the clifford algebra. Show that this maps Sp(2) isomorphically onto Spin3. (from Fulton & Harris p. 312, note in the text they define Clifford algebras by v2 = 1)
Prove that a genus g surface admit 22g inequivalent spin structures.
Desribe the spin structures on the 2-shpere and torus.
Compute the Dirac operator explicitly for the Dirac bundle S = R3 × V where V = C ⊕ C is a Cl2 module.
Using the clutching construction, show that K(S2) ≅ Z ⊗ Z.