Student Differential Geometry Seminar — Spring 2020: The Inhomogeneous ˉ∂-Equation

Organizers

Meeting times

Location

Description

Welcome to the webpage for the Spring 2020 iteration of the SDGS! This semester, we decided to go for the inhomogenous \(\bar{\partial}\)-equation and its (complex) geometric applications.

More precisely, the goal of this seminar is three-fold: (1) studying bits and pieces of several complex variables (SCV) with an analytic introduction to the \(\bar{\partial}\)-operator, (2) covering some basics in complex differential geometry with a geometric take on the \(\bar{\partial}\)-operator, and then (3) exploring applications of the (\(L^2\) theory) of the \(\bar{\partial}\)-operator to various geometric problems.

Any suggestions are welcome!

References

Here are some links to relevant materials in different forms:

Introductory articles:

Books:

  1. Tasty Bits of Several Complex Variables, A whirlwind tour of the subject, By: Jiří Lebl. Version 3.2, October 1st, 2019, 182 pages.
  2. Complex Analytic and Differential Geometry. By: Jean-Pierre Demailly. Version of Thursday June 21, 2012.
  3. Holomorphic Function Theory in Several Variables - An Introduction. By: Christine Laurent-Thiébaut.
  4. Analytic and algebraic geometry : common problems, different methods, volume 17 of IAS/PARK CITY Mathematics Series. American Mathematical Soc., 2010. By: Jeffery D. McNeal and Mircea Mustata.

Papers:

Lecture notes:

  1. Bo Berndtsson’s notes.
  2. Notes for the course “MATH 710: TOPICS IN MODERN ANALYSIS II – \(L^2\)-METHODS” (taught by Mattias Jonsson), taken by Matt Stevenson.
  3. MSRI Summer School notes (cf. below.)
  4. SEVERAL COMPLEX VARIABLES. By: ZBIGNIEW BLOCKI.
  5. Complex analysis, the \(\bar{\partial}\)-Neumann problem, and Schrödinger operators. By: Friedrich Haslinger.
  6. Dror’s notes for the “CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS” program held at ICTS, Banglore.

Videos:

MSRI Summer School — The ˉ∂-Problem in the Twenty-First Century:

Schedule

SpeakerDateTopic
Jae Ho Cho01/31/2020An introduction to the \(\bar{\partial}\)-method
Pranav Upadrashta02/07/2020Basic properties of holomorphic functions of several variables
Willie Rush Lim02/14/2020Plurisubharmonic functions
Emily Schaal02/21/2020Pseudoconvexity
Roberto Albesiano02/28/2020\(L^2\) estimates for the \(\bar{\partial}\)-operator
El Mehdi Ainasse03/06/2020Hörmander’s theorem and its twisted versions
Marlon de Oliveira Gomes03/13/2020The geometry of bundles and notions of positivity
Jordan Rainone04/03/2020\(L^2\) estimates for \(\bar{\partial}\) on Riemann surfaces and applications
Conghan Dong04/17/2020\(L^2\) estimates for the \(\bar{\partial}\)-equation on Kähler manifolds
Conghan Dong04/24/2020Applications of \(L^2\) techniques on complex manifolds
El Mehdi Ainasse04/24/2020Berndtsson’s Nakano-positivity theorem and optimal \(L^2\) extension theory (Slides)
Roberto Albesiano05/1/2020The Kodaira embedding theorem