Research Topics

My research interests lie at the crossroads of geometric analysis, convex geometry, complex geometry and mathematical physics. Currently, I am thinking about the following topics: Non-existence results for harmonic maps, stochastic completeness, pullvexity, geometric maximum principles and the moduli space of twisted monopoles

Preprints

• The moduli space of twisted Bogomolny monopoles, (in preparation)
• A theorem of Omori in the context of Riemannian manifolds, (in preparation)
• Remarks on the generalised Calabi-Yau problem in higher codimension, Arxiv By introducing a more flexible notion of convexity, we obtain a new Omori-Yau maximum principle for harmonic maps. In the spirit of the Calabi-Yau conjectures, this principle is more suitable for studying the unboundedness of certain totally geodesic projections of minimal submanifolds of higher codimension. We further explore this maximum principle by applying it to conformal maps, harmonic maps into Cartan-Hadamard manifolds, as well as cone, wedge and halfspace theorems.
• Perturbed cone theorems for proper harmonic maps, Arxiv Inspired by the halfspace theorem for minimal surfaces in R^3 of Hoffman-Meeks, the halfspace theorem of Rodriguez-Rosenberg, and the cone theorem of Omori, we derive new non-existence results for proper harmonic maps into perturbed cones in Rn, horospheres in Hn and also into perturbed Riemannian cones. The technical tool in use is an extension of the foliated maximum principle appearing in Assimos-Jost to the non-compact setting.

Youtube seminar

This seminar is based on the fact that on Youtube (and on other streaming services as well) there are excelent talks by outstanding mathematicans and physicists. The setup of the seminar is the following: Watch these videos together and discuss about its content. As a sidenote, the original idea for this seminar arose in a discussion with Erfan Rezaei.

Summer term 2024

• HyperKähler metrics and symmetries - Andrew Swann (31 May 2024) YoutubeAn interesting introduction to HyperKähler geometry with many examples.
• Integrable systems and representation theory: between geometry, algebra and analysis - Nicolai Reshetikhin (7 May 2024) Not YoutubeAfter a short introduction to Hamiltonian systems, some recent(ish) work of R. is presented
• QM and QFT from algebraic and geometric viewpoints 1/4 - Albert Schwarz (5 May 2024)YoutubeA (new?) geometric framework for quamtum mechanics is introduced. The space of states is promoted as the main object of study
• Symplectic geometry of Teichmueller spaces for surfaces with boundary 1/3 - Eckhard Meinrenken YoutubeAfter some motivation the main topic is a coordinate free definition of the Virasoro algebra
• The 3d A-model: generalized Seiberg-Witten equations, vortices and monopoles 1 - Justin Hilburn Youtube Covers a lot of material about mirror symmetry from the physical point of view.
• Generalized Seiberg-Witten equations 3 - Thomas Walpuski (19 Apr. 2024) Not Youtube Rescaling of the generalised SW leads to Z_2 harmonic spinors. These objects often prevent compactness results.
• Generalized Seiberg-Witten equations 2 - Thomas Walpuski (11 Apr. 2024) YoutubeThis lectures is devoted to the definition of the generalised SW equation.
• Generalized Seiberg-Witten equations 1 - Thomas Walpuski (9 Apr. 2024) Youtube The lecture was about the mathematical setup for gauge theory. The equations of Yang-Mills (ASD), Seiberg-Witten and Vafa-Witten were introduced as well. Lecture Notes

Pizza dough (like in Napoli)

• 100% Wheat flour of type 00
• 65% Cold water
• 3% Salt
• 0.4% Fresh yeast
• Some semolina

Mix the salt and the cold water in a bowl. Take ~20% of the flour and mix it in. Only now should the yeast be mixed in the water-salt-flour mix since otherwise the salt kills of the yeast and you created a non-rising dought. Mix everything until it is smooth and you mix in more and more of the flour.

This is an optional step but putting the dough into the fridge for up to 3 days is an essential step for achieving the well-known Napolitan texture.

Let the dough rise for two hours at room temperature. Thereafter, seperate off 250g - 300g dough balls and let them rise for 5-7 hours in an airtight enviroment

Put on a flat surface semolina and put the dough ball on top. Create a boundary by pressing down the middle part of the dough ball with your fingertips. Lastly, use your palms for extending the dough to a usual pizza shape.