Stanford University, Department of Mathematics
450 Serra Mall, Building 380
Stanford, CA 94305
Research interestsAlgebraic Topology, Differential Topology, Topological Data Analysis
- fastcluster: Fast hierarchical clustering routines for R and Python
- xypdf, PDF output for diagrams in LaTEX with the XY-pic package
- Daniel Müllner, Modern hierarchical, agglomerative clustering algorithms, arXiv: 1109.2378
- Daniel Müllner, Orientation reversal of manifolds, Algebr. Geom. Topol. 9 (2009), no. 4, 2361–2390
Dissertation: Orientation reversal of manifoldsAdvisor: Prof. Matthias Kreck
I study the phenomenon of chirality in the context of manifolds. A connected, orientable manifold is called amphicheiral if it admits an orientation-reversing self-map and chiral if it does not. Many familiar manifolds like spheres or orientable surfaces are amphicheiral: they can be embedded mirror-symmetrically into Rn, as the following figure illustrates.
|Reflect at the equator:|
On the other hand, examples of chiral manifolds have been known for many decades, e. g. the complex projective spaces CP2k and some lens spaces in dimensions congruent 3 mod 4. A fundamental question was in which dimensions chiral manifolds exist. While every manifold in dimensions 1 and 2 is amphicheiral, one of my results is that in all other dimensions there exist chiral manifolds.
For more information, read a summary or the dissertation itself:
- Daniel Müllner, Orientation reversal of manifolds, Bonner Mathematische Schriften, vol. 392, Bonn, 2009