Projects in analysis, Karl-Mikael Perfekt

I am happy to offer projects within complex analysis, operator theory, and harmonic analysis. Your specific project will be tailored to fit your interests and scientific background. If you are interested, please contact me. Your project will be co-supervised by Carlos Mudarra.

1a) Complex analysis: Picard's theorem (suitable for Bachelor or Master)

One of the most famous classical results in complex analysis is Picard's theorem: that any non-constant entire function can omit at most one point from its range. Schottky's theorem is a local quantitative version of Picard's theorem. There are many approaches to the proof of Picard's theorem, and therefore a range of subjects that could be considered in the project: 1) Elliptic functions. Normal families of holomorphic functions. 2) The hyperbolic metric and complex analysis. 3) Harmonic and subharmonic functions and classical potential theory.

1b) Harmonic analysis (suitable for Master)

Harmonic analysis is the study of real-variable function and their Fourier transforms, integral operators, oscillation, harmonic functions, etc. Topics of study may include: 1) The Poisson kernel, the Hilbert transform, and Calderón–Zygmund operators 2) Maximal functions, Ap weights and the Muckenhoupt theorem (the weighted maximal function is bounded in weighted Lp iff the weight is in Ap) 3) Functions of bounded mean oscillation (BMO), the John-Nirenberg theorem, and the connection with Ap.

1c) Holomorphic function and operator theory in the unit disc (suitable for Master)

A project may focus on any of the following topics. Value distribution theory and boundary values of harmonic and analytic functions in the unit disc. Hardy spaces. Toeplitz and Hankel operators. Composition operators. Variations of the above topics where Fourier and power series have been replaced by Dirichlet series; a modern topic where many number-theoretic objects naturally appear.

2023-11-20, Karl-Mikael Perfekt