Semester page for MAT4810 - Spring 2017

Krantz's book (Function theory of SCV) has a whole chapter with detailed treatment of the question of weighted L2 spaces, d-bar operator, and regularity. It is the same approach as in Hormander's book ,but with much more detail. You may also want to consult Bo BERNDTSSON's lecture notes, which are more leisurely:

May 2, 2017 2:53 PM

Narasimhans book explains that usc functions are decreasing limits of continuous functions. The passage of Hörmander's book attached is a shorter proof of characterization of pseudoconvexity, as well as a different proof that Levi pseudoconvexity is a manifestation of this (this approach is left as an exercise in Range).

I am adding  to the documents the book of Krantz,  "Function theory of several complex variables", 2nd ed for several reasons.  I think he does a great job at explaining the real/complex tangent spaces, differentiable characterization of convexity, and Real/complex hessian. (Lecutres 10,11)

Second, I like his proof of Narasimhan's lemma best (Lectures 12,13)

Finally if you disliked my clumsy argument that all domains in C are holomorphically convex, you may find a cleaner proof in Proposition 3.1.12.However the results that we take from here are also presented in different order and spirit in Range, Chapter II.2....

Feb. 15, 2017 10:17 AM

I have added a copy of Rosay's article from 1982 wherein he gives a greatly simplified proof of what I have wrongly called Osgood's theorem (apparently it's due to a certain Clements).

Feb. 1, 2017 2:17 PM

Concerning the "reminder" on differential forms, which we used in Lecture 5: this is stuff you should probably already know. The book of Range gives a short account. I have added some reference material with more detail (and proofs), which I think are quite good. They come from "Geometry of differential forms" by Morita and "Calculus on Manifolds" by Spivak.

Note that the notation used by Spivak, and myself on Lecture 5, namely Lamda(V) to denote the alternating forms on V, is not really standard: it conflicts with the algebraic meaning of the exterior algebra, which is a quotient of the tensor algebra. What is really meant is Lamda(V*), sometimes written Omega(V*), and generalizes to the somewhat standard Omega(T*M): the space of forms, i.e. alternating tensors on the tangent space at each point. In fancy words, a section of the exterior product of the cotangent bundle.

Jan. 31, 2017 11:17 AM

The achievement requirements are described in the main page for the course (MAT4810).

More precise contents of the lectures are listed in the schedule.

The main reference is: Range, "Holomorphic functions and integral representations in several complex variables"; Chapters 1,2,3,5,6

Jan. 17, 2017 10:53 AM

The course will be taught in English. 

The main reference is: Range, Holomorphic functions and integral representations in several complex variables.

(Althought we will not study integral representations...)

Dec. 7, 2016 1:24 PM