Fields Academy Shared Graduate Course: Several Complex Variables: from Hartogs to Stein
Description
Instructor: Prof. Rasul Shafikov
Email: shafikov@uwo.ca
Course Dates: September 7th - December 7th, 2023
Mid-Semester Break: October 30th - November 3rd, 2023
Lecture Times: Tuesdays & Thursdays | 10:30 AM - 12:00 PM (ET)
Office Hours: Via Zoom on Wednesdays, 10:00 AM - 12:00 PM (ET), starting September 13th, 2023, or by appointment
Registration Fee: PSU Students - Free | Other Students - CAD$500
Capacity Limit: N/A
Format: Online with synchronous delivery via Zoom
Course Description
Please view the course syllabus HERE.
The plan is to cover the fundamentals of several complex variables (SCV), including geometric and analytic aspects of the subject. Throughout its development SCV produced many important and deep techniques that are now widely used in other areas of mathematics, such as algebraic geometry, symplectic geometry and topology, PDEs, and dynamical systems, to name a few.
Prerequisites: Multivariable Calculus; Linear Algebra; Complex Analysis in one variable
Evaluation: Students will be required to submit 5 assignments during the term.
Textbooks: Several texts will be used to cover different topics, including
- Lars Hörmander, An Introduction to Complex Analysis in Several Variables, North Holland.
- Michael Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer GTM 108.
- Klaus Fritzsche and Hans Grauert, From Holomorphic Functions to Complex Manifolds, Springer GTM 213.
- Boris Shabat, Introductions to Complex Analysis, Part II Functions of Several Variables, AMS.
Course Plan: (approximate, will be updated as needed)
- Week 1. Review of complex analysis in dimension 1.
- Week 2. Definitions and basic properties of holomorphic functions of several variables; Cauchy-Riemann equations.
- Week 3. Cauchy integral formula; Hartogs theorem on separate holomorphicity.
- Week 4. Holomorphic mappings; Biholomorphic equivalence of domains.
- Week 5. Zeros of holomorphic functions; Weierstrass preparation theorem.
- Week 6. Martinelli-Bochner integral formula; Hartogs' Kugelsatz.
- Week 7. Holomorphic, polynomial and rational convexity; Oka-Weil theorem.
- Week 8. Pseudoconvexity I: psh functions.
- Week 9. Pseudoconvexity II: exhaustion functions; Levi problem.
- Week 10. Pseudoconvexity III: equivalent conditions.
- Week 11. Envelopes of holomorphy, Stein manifolds.
- Week 12. Stein theory.
