### Topics in Analysis - Quasiconformal mappings in the plane

Course Description and Grading Breakdown

Course Meeting Time and Location
Monday, Wednesday and Friday
10:00 - 10:55 am
104 Math Building (Building 15)

Course Instructor Contact Information and Office Hours
Kirill Lazebnik
210-7 Math Building (Building 15)

Course Schedule

 Date Topic 04/02/18 Overview. Preliminary "C^1" definition of quasiconformal homeomorphisms (equivalent to the standard definition of a q.c. mapping in the case that said mapping is C^1), weakening "C^1" to "absolutely continuous on lines" or equivalently "lying in the appropriate sobolev space" to obtain the analytic definition of a quasiconformal mapping. Statement of Measurable Riemann mapping theorem. Equivalent "metric/circular" definition of quasiconformality. 04/04/18 Definition of a quadrilateral. Quadrilaterals can always be mapped conformally to some Euclidean rectangle (vertices -> vertices) via elliptic integral (schwarz-christoffel). Two non-similar rectangles are never conformally equivalent. Geometric definition of quasiconformal as orientation-preserving homeomorphisms which do not change the moduli of quadrilaterals beyond bounded factor.
 04/06/18 Some basic properties of q.c. maps (inverses/compositions are q.c.). Definition of a "regular" q.c. mapping as a C^1 homeomorphism with a certain uniform bound on the dilatation quotient. Grotzsch's inequality (length-area method), and in particular "regular" q.c. maps are quasiconformal in the geometric sense.
 04/09/18 Characterization of the modulus of a quadrilateral using extremal length of a certain path family. Deducing Rengel's inequality.
 04/11/18 Superadditivity/monotinicity of the Modulus. One can only have equality in Rengel's inequality when the quadrilateral in question is a Euclidean rectangle (length-area method again). Deducing that 1-q.c. maps are conformal.
 04/16/18 A homeomorphism which is a uniform limit of K-q.c. maps is also K-q.c. A proof (https://mathoverflow.net/questions/261535/mapping-the-doubly-connected-domain-to-an-annulus) that all doubly connected planar domains are conformally equivalent to a Euclidean annulus.
 04/18/18 K-q.c. mappings do not distort the modulus of a ring domain (topological annulus) by more than a factor of K. Conversely, if a sense preserving homeomorphism does not distort the modulus of ring domains by more than K, then this homeomorphism is K-q.c. (i.e. does not distort the moduli of quadrilaterals by more than K).
 04/20/18 Extendability of q.c. mappings to the boundary. Removability of an analytic arc for quasiconformal maps. Consequences: the reflection principle and local nature of quasiconformality.
 04/23/18 Reconciling, for C^1 q.c. maps, the two notions of local dilatation (one in terms of the distortion of small quadrilaterals near a point, the other in terms of the ratio of two quantities involving the derivative of the map).
 04/25/18 The moduli of Grotzsch/Teichmuller extremal domains.
 04/27/18 Deriving functional equations relating the moduli of Grotzsch's extremal domain for different parameters.
 04/30/18 finding upper/lower bounds for the modulus of Grotzsch's extremal domain by considering conformal images of ellipses (under the Joukowsky map), an asymptotic expression for the modulus of Grotzsch's extremal domain. Distortion of distance for normalized q.c. mappings of the unit disc.
 05/02/18 Distortion of distance for normalized q.c. mappings of the unit disc (under normalization 0->0), distortion of hyperbolic vs. euclidean distance of two points in unit disc.
 05/04/18 Notions of normality, equicontinuity, and holder continuity for a family of K-q.c. mappings. A family of K-q.c. mappings each of which omit two points of at least some fixed distance d>0 is equicontinuous.
 05/07/18 Equivalence of holder continuity, normality, and equicontinuity for a family of K-q.c. mappings.
 05/09/18 A necessary condition for a boundary homeomorphism of two jordan domains to be a K-q.c. mapping: $1/K \leq M(Q') \leq K$ for quadrilaterals $Q$ of modulus one. Translating this in the upper half plane setting to the definition of quasisymmetry.
 05/11/18 The Buerling-Ahlfors extension of a quasisymmetric map on the real line to a q.c. map of the upper half plane.
 05/14/18 Quasiconformal mappings are absolutely continuous on lines (i.e. lie in $W^{1,2}_{loc}(\mathbb{R}^2)$). "Devil's staircase" example of a homeomorphism which is conformal outside of an area-measure zero set but is not quasiconformal (is not absolutely continuous on lines!).
 05/16/18 Gehring-Lehto theorem that homeomorphisms which have partial derivatives a.e. are differentiable a.e..
 05/18/18 Some remarks about analytic properties of q.c. mappings. Analytic definition implies geometric definition, i.e. an orientation-preserving homeomorphism which is ACL and satisfies a dilatation condition does not distort the moduli of quadrilaterals by more than a bounded factor.
 05/21/18 Complex dilatation (Beltrami coefficient) of a quasiconformal mapping - geometric interpretation. Calculation for the Beltrami coefficient of a composition of two quasiconformal mappings.
 05/23/18 Uniform convergence on compact subsets of a sequence of K-q.c. maps to a K-q.c. map does not imply convergence of the associated dilatations. If the dilatations of the sequence DO converge, they must converge to the dilatation of the limiting K-q.c. map. (Good approximation lemma)
 05/30/18 Proof of measurable Riemann mapping theorem modulo a conformal welding/sewing theorem.
 06/01/18 Proof of conformal welding/sewing theorem, concluding the proof of MRT.

Textbooks:

The main resource will be the text of Lehto/Virtanen. There are many other good textbooks on the subject.