B9075 - TOPICS IN HOLOMORPHIC FUNCTION THEORY

Learning outcomes

Starting from the basic notions of holomorphic function theory, the course addresses some more advanced topics in such a theory, in one or several complex variables.

Course contents

Prerequisites:

the complex analysis course in the laurea triennale, or equivalent course from other universities.

Course contents:

FIRST PART (Professor Arcozzi):

1. disc automorphisms and hyperbolic geometry;

2. normal families and Montel’s theorem;

3. the Riemann mapping theorem and some of its topological consequences;

4. harmonic/subharmonic functions, the reflection principle, Schwarz-Christoffel’s construction of Riemann maps onto polygons.

5. In the remaining time, one or two of the following topics might be covered:

-Charathéodory’s theorem on the extension of Riemann’s map to the boundary of a Jordan domain, and its topological consequences;

-Jensen’s formula and factorizations;

- Phragmén-Lindelof principle, with some applications;

-the Beltrami equation and the local conformal equivalence of all two-dimensional Riemannian metrics to the Euclidean metric.

 

SECOND PART (Professor Parmeggiani):

1.Holomorphic functions in several variables, Cauchy-Riemann equations.

2. The Cauchy formula in several variables (the general case of C^1 functions).

3. Power series in several variables, power series expansion of holomorphic functions in C^n.

4. Properties of holomorphic functions in C^n (invertibility, implicit function theorem, holomorphic submanifolds of C^n).

5. Domains of holomorphy.

6. Hartogs’ Theorem (using PDEs); Doulbeault-Grothendieck’s Lemma.

7. Pseudoconvexity and Levi-form.

 

THIRD PART (Professor Lanzani):

1. The polydisc is not biholomorphic to the ball (that is, in higher dimension there is no Riemann mapping theorem)

2. Holomorphic Hardy Spaces and representations via the Bochner-Martinelli formutla

3. Cauchy-Leray formula and L^p-regularity problem for the Cauchy-Leray integral associated to strictly C-linearly convex domains of class C^2 (and, time permitting, class C^{1,1})

4. Cauchy-Szego projection: definition and basic properties

5. Kerzman-Stein operator: definition and main properties

6. L^p regularity of the Cauchy-Szego projection for strictly pseudoconvex domains of class C^2.

 

Readings/Bibliography

FIRST PART (Professor Arcozzi):

L.V. Ahlfors Complex Analysis, 3^rd edition, McGrow Hill, 1978

M. Andersson, Topics in Complex Analysis, Springer Universitext, 2013

D. Sarason, Complex Function Theory, AmMathSoc, 2007

T. Tao, https://terrytao.wordpress.com/, note del corso 246A/B

SECOND PART (Professor Parmeggiani):

L. Hörmander: An introduction to complex analysis in several variables. 3rd edition. North Holland Mathematical Library, Vol. 7. 1990.

J. Lebl: Tasty bits of several complex variables. May 20, 2025 (version 4.2).

V. Scheidermann: Introduction to the complex analysis in several variables. 2nd edition. Birkhäuser 2023.

THIRD PART (Professor Lanzani):

Range M.R., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer 1986

Stein E. M., Boundary Behavior of Holomorphic Functions in Several Complex Variables, Princeton University Press 1972

Kerzman N. & Stein E.M., The Cauchy kernel, the Szego kernel, and the Riemann mapping function. Math. Ann. 236 (1978), no. 1, 85–93.

Kerzman N. & Stein E.M., The Szego Kernel in terms of the Cauchy-Fantappie’ kernels, Duke Math. J., 45 (1978) 197 - 224

Lanzani L. & Stein E. M., The Cauchy Integral in C^n for domains with minimal smoothness, Advances Math. {\bf 264} (2014) 776 -- 830.

Lanzani L. & Stein E. M., Hardy Spaces of Holomorphic functions for domains in Cn with minimal smoothness, Harmonic Analysis, Partial Differential Equations, Complex Analysis, and Operator Theory: Celebrating Cora Sadosky’s life, AWM-Springer vol. 1 (2016), 179 - 200.

Lanzani L. & Stein E. M., The Cauchy-Szeg ̋o projection for domains with minimal smoothness, Duke Math. J. 166 no. 1 (2017), 125-176.

Teaching methods

In-class lectures.

Assessment methods

Each student will be evaluated with three, short oral colloquia (about 15-20min) with each of the three teachers on a day and time to be agreed upon on an individual basis.

Each teacher will give a numerical score; the final grade will be the average of the three scores rounded by excess (a score of 30L in a module counts as 31 points towards the computation of the average ).

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

Office hours upon request.

Office hours

See the website of Loredana Lanzani

See the website of Nicola Arcozzi

See the website of Alberto Parmeggiani