Speaker: Lukas Schönlinner
University of Augsburg
This lecture series explores the deep connections between spin geometry and scalar curvature rigidity phenomena. We will develop the foundational tools of Dirac operators and their applications to understanding geometric constraints on Riemannian manifolds.
Part 1 of the series
Wednesday, October 8, 2025
05:00 PM IST
Abstract: We will define Clifford algebras, Clifford bundles and Dirac bundles. Every Dirac bundle carries an associated Dirac operator and we will prove the Weitzenböck formula in this general setup, which is one of the most important tools in spin geometry.
This talk has ended — see the recording.
Part 2 of the series
Thursday, October 16, 2025
03:00 PM IST
Abstract: In this talk, we will introduce the spin groups and discuss the properties necessary to define spinable manifolds. We will then associate a canonical connection, the so-called spin connection, with every spin manifold $M$. This connection induces a Dirac bundle structure on $M$. The associated Dirac operator satisfies the Schrödinger-Lichnerowicz formula, a special case of the Weitzenböck formula. Additionally, we will discuss the Atiyah-Singer index theorem, which links the index of the Dirac operator to topological properties of the manifold. These equations are the key features of spin geometry that we will exploit later.
This talk has ended — see the recording.
Part 3 of the series
Wednesday, October 22, 2025
05:00 PM IST
Abstract: The theorem of Llarull is a famous rigidity result in scalar curvature which states the following: Let $(M,g)$ be a closed connected $n$-dimensional smooth Riemannian spin manifold with $n\geq2$ even and $\mathrm{scal}_g \geq n(n-1)$. If $f\colon M\to S^n$ is a smooth 1-Lipschitz map of non-zero degree then $f$ is a Riemannian isometry.
We will discuss a generalization of this theorem, which states that it is sufficient to assume that $g$ is a metric of Sobolev regularity $W^{1,p}$ with $p> n$ and that $f$ is merely 1-Lipschitz instead of smooth.
This is joint work with Simone Cecchini, Bernhard Hanke, and Thomas Schick.
This talk has ended — see the recording.
Part 4 of the series
Wednesday, October 29, 2025
05:00 PM IST
Abstract: In this talk, we will discuss another generalization of Llarull's theorem in even dimensions where we allow cone-like singularities in the domain. To do so, we will discuss an abstract functional analytic setup that enables us to show that the Dirac operator on these singular spaces is Fredholm. Moreover, we will apply an index formula of Chou together with a deformation argument to show that the index is non-zero.
This is joint work with Simone Cecchini, Bernhard Hanke, and Thomas Schick.
This talk has ended — see the recording.
University of Augsburg
Lukas Schönlinner is a PhD student of Bernhard Hanke at the University of Augsburg. His research interests includes scalar curvature geometry and spin geometry.