Daniel Müllner

Address Daniel Müllner Stanford University, Department of Mathematics 450 Serra Mall, Building 380 Stanford, CA 94305 USA e-mail:  Research interests Algebraic Topology, Differential Topology, Topological Data Analysis

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Teaching

• Applied Algebraic Topology, Spring 2012

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Software

• fastcluster: Fast hierarchical clustering routines for R and Python
• xypdf, PDF output for diagrams in LaTEX with the XY-pic package

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Preprints

• Daniel Müllner, Modern hierarchical, agglomerative clustering algorithms, arXiv: 1109.2378

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Publications

• Daniel Müllner, Orientation reversal of manifolds, Algebr. Geom. Topol. 9 (2009), no. 4, 2361–2390

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Dissertation: Orientation reversal of manifolds

Advisor: 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:

	Similarly
		and

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