Lecture notes: Analysis auf Mannigfaltigkeiten

typed and coauthored by Florian Munkelt

The lecture notes can be found here on the lecture's homepage.

This is still work in progress and the total number of pages is currently 137.

abstract :
"We begin with an overview of topological basics and measure theory, and in particular Radon and Lebesgue measures. Afterwards, we present the theory of smooth manifolds and explain thoroughly arbitrary tensors. We continue with de Rham cohomology, Poincare's Lemma, orientability and Stokes' Theorem.
The lecture's most important part is the real Hodge theory. The main subjects are the Hodge star operator, the de Rham Laplace operator and its spectral decomposition with a proof copied from Warner's book. Then, we identify the set of harmonic p-forms with the pth cohomology group and conclude the Poincare duality.
We finish with a chapter on Riemannian curvature and Bochner's method."

These lecture notes are based on lecture notes by Eberhard Freitag, Frank W. Warner's book Foundations of differentiable manifolds and Lie groups , Martin Speight's lecture notes and Gallot, Sylvestre Hulin, Dominique und Lafontaine, Jacques: Riemannian geometry.

Vector Valued Siegel Modular Forms for \Gamma_2[2,4] and Sym^2

The article can be found here arXiv:1309.1766.

This is a shortened version of my PhD thesis and has just 23 pages.

abstract :
"We develop two structure theorems for vector valued Siegel modular forms for Igusa’s subgroup \Gamma_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular forms with rational tensors with easily handleable poles on P^3C. It follows that the observed modules of modular forms are generated by the Rankin-Cohen brackets of the four theta series of the second kind."

Structure Theorems for Certain Vector Valued Siegel Modular Forms of Degree Two

PhD thesis supervised by Eberhard Freitag and Rainer Weissauer

The article can be found here http://archiv.ub.uni-heidelberg.de/volltextserver/15327/.

University of Heidelberg, Summer 2013, 88 pages

abstract :
"We develop two structure theorems for vector valued Siegel modular forms for Igusa’s subgroup \Gamma_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular forms with rational tensors with easily handleable poles on P^3C. It follows that the observed modules of modular forms are generated by the Rankin-Cohen brackets of the four theta series of the second kind."