Quasiconformal Analysis and Conformal Invariants
A mini-lecture series
Shanshuang Yang
Emory University
Lecture 1: Introduction to Quasiconformal Analysis (10:00-11:30, Dec 31, 2012, Math building
102)
Abstract: In this lecture, we will give a brief introduction to the theory of quasi-
conformal mappings, both in the plane and in higher dimensions. We will discuss the
connections between the theory of QC maps and other branches of mathematics, the
major tools and methods used in the study, and the development of some important
problems.
Lecture 2: Conformal Invariants - Modulus and Capacity (14:30-16:00, Jan 3, 2013, Math
building 129)
Abstract: This lecture will be devoted to the discussion of two conformal invariants
- modulus and capacity, which are indispensable tools in the study of quasiconformal
mappings. We will present some fundamental properties of these invariants, which
will be used in later lectures.
Lecture 3: QED Domains and QED Constants (14:30-16:00, Jan 8, 2013, Math building 129)
Abstract: The concept of QED (for quasi-extremal distance) domains will be intro-
duced. This class of domains play an important role in the study of QC maps and
geometric function theory. In this lecture an important geometric and topological
characterization will be given for domains with QED constant M(D) = 2. The val-
ues of the QED constants will be computed for some special domains such as polygons
and cubes.
Lecture 4: QED Domains and Garnett-Yang Conjecture (14:30-16:00, Jan 10, 2013, Math
building 127)
Abstract: In this lecture we will continue the discussion of QED domains. In partic-
ular, we will present the connection of QED constant with other conformal invariants
of domains such as the QC reflection constant R(D). The Garnett-Yang conjecture
and a counterexample will also be presented.
Lecture 5: Dilatations of quasisymmetric homeomorphisms (10:00-11:30, Jan 14, 2013, Math
building 102)
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Abstract: In this lecture we introduce several conformal invariants associated with
a quasisymmetric homeomorphism of the unit circle (or real line). We will discuss
how these invariants are related to each other and how they are related to the QED
constant and reflection constant (introduced in the previous lectures). Topics will
include some earlier results of Shen, Wu and Yang, as well as new results of Cheng-
Yang on related problems.
Lecture 6: QED constants and boundary dilatation, and open discussion (14:30-16:00, Jan 15,
2013, Math building 129)
Abstract: I will divide this lecture into two parts. The first one I will discuss how
the boundary dilatation is related to QED constant in degenerate case. The second
one of this lecture I will devote to open discussion on any topics that are of interests
to the audience and myself.
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