Introduction to CR Geometry

We will meet on Microsoft Teams. The team link/code is available on the IISc intranet (see Department of Mathematics). Please add yourself to this team even if you just want to attend the first class. If you are not from IISc, drop me an email to be added to the team.
Course Description. The aim of this course is to provide an introduction to CR (Cauchy Riemann/Complex Real) geometry, which is broadly the study of the structure(s) inherited by real submanifolds in complex spaces. We will first give a parallel introduction to the fundamental objects of SCV and CR geometry. These include holomorphic functions in several variables, CR manifolds (embedded and abstract) and CR functions. Next, we will cover some examples, results, and techniques from the following range of topics:
  • embeddability of abstract CR structures;
  • holomorphic extendability of CR functions;
  • CR singularities.
Wherever possible (and time permitting), we will highlight the connections of this field to other areas of analysis and geometry. For instance, abstract CR structures will be discussed in the broader context of involutive structures on smooth manifolds.

Prerequisites.
  • MA 224 (Complex Analysis).
  • Basic familiarity with: differentiable manifolds, tangent and cotangent bundles, and systems of (first order) PDEs.
  • Although this is a Topics in Several Complex Variables course, MA 328 (Introduction to SCV) is not a prerequisite. All the necessary concepts from SCV will be rigorously introduced along the way.
Suggested References.
  • A. Boggess, CR Manifolds and the tangential Cauchy-Riemann complex, CRC Press, Boca Raton, FL (1991).
  • M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Real Submanifolds in Complex Space and their Mappings, Princeton Math. Series., Princeton Univ. Press (1999).
Instructor. Purvi Gupta. The best way to contact me is via email (purvigupta(at)iisc(dot)ac(dot)in).

Course timings. Tuesdays & Thursdays, 11:30 am - 1:00 pm.

Office hours. Drop-in office hours (on Teams): Thursdays, 2:30 - 3:30 pm.
To consult me outside of these, drop me an email to set up an appointment.

Lecture Format. The lectures will be live on Microsoft Teams.
Resources. All the material for this course (videos, notes, assignments, etc.) will be posted on the MA 381 team page on Microsoft Teams.
Evaluation Scheme.
Course Calendar.

# Date Time Video Availability Topics Quiz/Assignment Tracker
1. Oct. 1 11:30 - 1:00 Oct. 1, by 5:00 PM. Introduction+Overview Quiz 0 (Sample Quiz)
2. Oct. 6 11:30 - 1:00 TBA (Forgot to record!) Holomorphicity in several variables Quiz 1
3. Oct. 8 11:30 - 1:00 Oct. 8, 5:00 PM Holomorphicity (Contd.) Quiz 2
4. Oct. 13 11:30 - 1:00 Oct. 13, 5:00 PM Holomorphicity (Contd.) Quiz 3
5. Oct. 15 11:30 - 1:00 Oct. 15, 5:00 PM Holomorphicity (Contd.) Assignment 1 posted
6. Oct. 20 11:30 - 1:00 Oct. 20, 5:00 PM Basic Notions: Linear Algebra Quiz 4
7. Oct. 22 11:30 - 1:00 Oct. 22, 5:00 PM Basic Notions: Linear Algebra, Differentiable Manifolds Quiz 5
8. Oct. 27 11:30 - 1:00 Oct. 27, 5:00 PM Embedded CR Manifolds Quiz 6
9. Oct. 29 11:30 - 1:00 Oct. 29, 5:00 PM Embedded CR manifolds contd. Assignment 1 due by 5:00 pm
10. Nov. 03 11:30 - 1:00 Nov. 03, 5:00 PM Embedded CR manifolds contd. Quiz 7
11. Nov. 05 11:30 - 1:00 Nov. 05, 5:00 PM Embedded CR manifolds contd. Quiz 8
12. Nov. 10 11:30 - 1:00 Nov. 10, 5:00 PM Frobenius Theorem(s) No quiz
13. Nov. 12 11:30 - 1:00 Nov. 12, 5:00 PM Involutive structures Assignment 2 to be posted
14. Nov. 17 11:30 - 1:00 Nov. 17, 5:00 PM Embeddability of abstract CR manifolds Quiz 9
15. Nov. 19 11:30 - 1:00 Nov. 19, 5:00 PM Embeddability of CR manifolds contd. No Quiz
No meeting in the week Nov. 23-27 Assign. 2 due Nov. 26
16. Dec. 01 11:30 - 1:00 Dec. 01, 5:00 PM The Baouendi-Treves Approximation Theorem No Quiz
17. Dec. 03 11:30 - 1:00 Dec. 01, 5:00 PM The Levi form of an embedded CR manifold Quiz 10
Dec. 08 Class cancelled due to technical issues
18. Dec. 10 11:30 - 1:00 Dec. 10, 5:00 PM Two holomorphic extension theorems Quiz 11
19. Dec. 15 11:30 - 1:00 Dec. 15, 5:00 PM The analytic disc method: hypersurfaces
20. Dec. 17 11:30 - 1:00 Dec. 17, 5:00 PM The analytic disc method: Bishop's equation Quiz 12